Aida ; Cuni-Sanchez

Martin J P ; Sullivan

Philip J ; Platts

Simon L ; Lewis

Rob ; Marchant

Imani

Gérard

Gérard Imani

Wannes Hubau

Iveren Abiem

Hari Adhikari

Tomas Albrecht

Jan Altman

Christian Amani

Abreham B Aneseyee

Valerio Avitabile

Lindsay Banin

Rodrigue Batumike

Marijn Bauters

Hans Beeckman

Serge K Begne

Amy C Bennett

Robert Bitariho

Pascal Boeckx

Jan Bogaert

Achim Bräuning

Franklin Bulonvu

Neil D Burgess

Kim Calders

Colin Chapman

Hazel Chapman

James Comiskey

Thales De Haulleville

Mathieu Decuyper

Ben Devries

Jiri Dolezal

Vincent Droissart

Corneille Ewango

Senbeta Feyera

Aster Gebrekirstos

Roy Gereau

Martin Gilpin

Dismas Hakizimana

Jefferson Hall

Alan Hamilton

Olivier Hardy

Terese Hart

Janne Heiskanen

Andreas Hemp

Martin Herold

Ulrike Hiltner

David Horak

Marie-Noel Kamdem

Charles Kayijamahe

David Kenfack

Mwangi J Kinyanjui

Julia Klein

Janvier Lisingo

Jon Lovett

Mark Lung

Jean-Remy Makana

Yadvinder Malhi

Andrew Marshall

Emanuel H Martin

Edward T A Mitchard

Alexandra Morel

John T Mukendi

Tom Muller

Felix Nchu

Brigitte Nyirambangutse

Joseph Okello

-H Peh

Petri Pellikka

Oliver L Phillips

Andrew Plumptre

Lan Qie

Francesco Rovero

Moses N Sainge

Christine B Schmitt

Ondrej Sedlacek

Alain S K Ngute

Douglas Sheil

Demisse Sheleme

Tibebu Y Simegn

Murielle Simo-Droissart

Bonaventure Sonké

Teshome Soromessa

Terry Sunderland

Miroslav Svoboda

Hermann Taedoumg

James Taplin

David Taylor

Sean C Thomas

Jonathan Timberlake

Darlington Tuagben

Peter Umunay

Eustrate Uzabaho

Hans Verbeeck

Jason Vleminckx

Göran Wallin

Charlotte Wheeler

Simon Willcock

John T Woods

Etienne Zibera

University of Dundee

Tropical forests store 40-50% of terrestrial vegetation carbon 1 . Spatial variations in aboveground live tree biomass carbon (AGC) stocks remain poorly understood, in particular in tropical montane forests 2 . Owing to climatic and soil changes with increasing elevation 3 , AGC stocks are lower in tropical montane compared to lowland forests 2 . Here we assemble and analyse a dataset of structurally intact old-growth forests (AfriMont) spanning 44 montane sites in 12 African countries. We find that montane sites in the AfriMont plot network have a mean AGC-stock of 149.4 Mg C ha -1 (95% CI 137.1-164.2), comparable to lowland forests in the African Tropical Rainforest Observation Network 4 and about 70 per cent and 32 per cent higher than averages from plot networks in montane 2,5,6 and lowland 7 forests in the Neotropics, respectively. Notably, our results are two-thirds higher than the IPCC default values for these forests in Africa 8 . We find that the low stem density and high abundance of large trees of African lowland forests 4 is mirrored in the montane forests sampled. This carbon store is endangered: we estimate that 0.8 million ha of old-growth African montane forest have been lost since 2000. We provide country-specific montane forest AGC stock estimates modelled from our plot network to help guide forest conservation and reforestation interventions. Our findings highlight the need for conserving these biodiverse 9,10 and carbon-rich ecosystems.

Main text

Tropical forests cover less than 10% of the global land area yet store 40-50% of terrestrial vegetation carbon [START_REF] Erb | Unexpectedly large impact of forest management and grazing on global vegetation biomass[END_REF] and contribute more than one third of primary productivity [START_REF] Pan | A large and persistent carbon sink in the world's forests[END_REF] so are a key component of the global carbon cycle [START_REF] Booth | High sensitivity of future global warming to land carbon cycle processes[END_REF][START_REF] Hubau | Asynchronous carbon sink saturation in African and Amazonian tropical forests[END_REF] . There is substantial variation in carbon stocks across the biome, with lowland forests in Africa and Borneo storing more carbon per unit area than lowland forests in the Neotropics [START_REF] Lewis | Above-ground biomass and structure of 260 African tropical forests[END_REF][START_REF] Sullivan | Long-term thermal sensitivity of Earth's tropical forests[END_REF] . This variation arises partly from structural differences: the signature feature of African forests is their low stem density but relatively high abundance of large trees (>70 cm diameter) which store large quantities of carbon, while Bornean forests are characterised by high stem density and basal area [START_REF] Lewis | Above-ground biomass and structure of 260 African tropical forests[END_REF][START_REF] Feldpausch | Tree height integrated into pantropical forest biomass estimates[END_REF][START_REF] Bastin | Pan-tropical prediction of forest structure from the largest trees[END_REF] . Despite increased understanding of biogeographic differences in tropical lowland forests, patterns of spatial variation in carbon stocks remain poorly understood in the 880,000 km 2 of tropical montane forests located ≥ 1,000 m asl [START_REF] Spracklen | Tropical montane forests are a larger than expected global carbon store[END_REF] . Montane forests are expected a priori to have lower aboveground live tree biomass carbon (AGC) stocks than lowland forests because (1) temperature decreases with increasing elevation, reducing net primary productivity and slowing nutrient recycling, (2) long periods of cloud immersion in montane forests suppresses productivity, (3) soil waterlogging slows nutrient recycling and (4) high epiphyte load, local wind exposure in crests and nutrient-limited soils limit tree size and increase investment in roots over shoots [START_REF] Fahey | Tropical montane cloud forest: environmental drivers of vegetation structure and ecosystem function[END_REF] . While forest inventory plots provide some support for these assumptions 2 data from African mountain regions are exceptionally sparse. Indeed, in the most recent IPCC guidelines, there is no specific AGC default value for old-growth montane forests in Africa: the value given of 89.3 Mg C ha -1 is simply a mean of secondary and oldgrowth forests found ≥ 1,000 m asl [START_REF] Domke | Refinement to the 2006 IPCC Guidelines for National Greenhouse Gas Inventories[END_REF] . Mountain areas also pose special challenges for remote-sensing approaches for estimating carbon stocks, as radar data are affected by geometric distortions [START_REF]CCI BIOMASS Product User Guide Year 1 Version 1.0[END_REF] while steep slopes bias spaceborne LiDAR estimates towards overestimating canopy height [START_REF] Lefsky | Revised method for forest canopy height estimation from the Geoscience Laser Altimeter System waveforms[END_REF] . These issues are reflected in the limited correlation between estimates of AGC-stocks at mountain locations from different recent remote-sensing derived carbon maps (Supplementary Information Table S1). Better understanding of montane carbon stocks is important for many African countries, particularly in eastern Africa where montane forests represent most of the extant evergreen old-growth forest cover. Quantifying carbon stocks in these ecosystems is critical for estimating national carbon losses from deforestation and forest degradation [START_REF] Willcock | Quantifying and understanding carbon storage and sequestration within the Eastern Arc Mountains of Tanzania, a tropical biodiversity hotspot[END_REF] . Quantifying carbon stocks in old-growth montane forests also serves to constrain potential carbon uptake by restored natural forests, given the high commitment of most African nations to the Bonn Challenge effort to restore 150 million ha of degraded and deforested lands by 2020 (see Table 1), and 350 million by 2030. Here we measured, compiled and analysed a new dataset of 226 plot inventories spanning 44 sites in 12 African countries, covering most major mountain regions on the continent (the "AfriMont" dataset). Plots range from 800 to 3,900 m asl to include submontane forests (800-1,000 m asl) in smaller mountains closer to the ocean [START_REF] Bussmann | Vegetation zonation and nomenclature of African Mountains -An overview[END_REF][START_REF] Hamilton | Vegetation, climate and soil, altitudinal relationships on the East Usambara Mountains of Tanzania[END_REF] . For all plots, stem diameter and species were recorded for each tree ≥10 cm diameter at breast height (or above buttress) following standard methods [START_REF] Phillips | RAINFOR field manual for plot establishment and remeasurement[END_REF] . Tree height was sampled in 23 montane sites, allowing variation in height-diameter allometry to be incorporated into the calculation of aboveground biomass. A total of 72,336 stems with diameter ≥10 cm were measured. For each tree, we computed AGC (in Mg C ha -1 ) according to standard procedures (see methods). We find that the mean plot-level AGC-stock across sampled African tropical montane forests is 149.4 Mg C ha -1 (95% confidence interval (CI) 137.1-164.2), two-thirds more than the IPCC default value of 89.3 Mg C ha -1 . Our estimates are robust to subsampling our dataset (Extended Data Fig. 1) and excluding small plots (Extended Data Fig. 2) and are not affected by the sampling strategy used to establish plots in each study site (Extended Data Fig. 2). Comparing our dataset to previous syntheses of montane [START_REF] Spracklen | Tropical montane forests are a larger than expected global carbon store[END_REF][START_REF] Vilanova | Environmental drivers of forest structure and stem turnover across Venezuelan tropical forests[END_REF][START_REF] Alvarez-Davila | Forest biomass density across large climate gradients in northern South America is related to water availability but not with temperature[END_REF] and lowland [START_REF] Sullivan | Long-term thermal sensitivity of Earth's tropical forests[END_REF] forest plot networks reveals that tropical montane forests in Africa have significantly higher AGC-stocks per unit area than both montane (95% CI = 50.4 -71.9 Mg C ha -1 ) and lowland (95% CI = 124.0 -147.9 Mg C ha -1 ) forests in the Neotropics, and that they do not differ significantly from lowland forests in Africa (95% CI = -27.6 -9.6 Mg C ha -1 ) (Fig. 1, Table S2). The similar AGC-stocks in montane and lowland forests in Africa contrasts with the Neotropics and Southeast Asia, where carbon stocks are lower in montane forests than lowland forests (albeit not significantly different in Southeast Asia due to the small sample size, Fig. 1). These differences are robust to accounting for differences in elevation among montane datasets: removing African plots 800-1,000 m asl slightly reduces estimated montane forest AGC-stock to 145.0 Mg C ha -1 (95% CI 129.6 -163.2), but observed differences in AGC-stock among continents remain when plots are restricted to elevations well represented in all continents (Extended Data Fig. 3). The characteristic structural properties of lowland African forests (relatively low stem density and greater importance of large trees compared to elsewhere in the tropics [START_REF] Lewis | Above-ground biomass and structure of 260 African tropical forests[END_REF] ) are also evident in the African montane forests we sampled. In these montane forests mean stem density is 483.3 stems ha - 1 (± 177.7 s.d.) and mean basal area is 39 m 2 ha -1 (± 14.8 s.d.). We find a high density of large stems (>70 cm diameter, 19.1 stems ha -1 ± 15.4 s.d.) which contribute 35.3% (95% CI = 29.6 -41.8 %) to plot-level AGC-stock (Fig. 2). The contribution of large trees to plot-level AGC-stock is also similar in montane and lowland Africa (95% CI of difference in square-root transformed proportional contribution of large trees between lowland and montane forests = -0.100 -0.075, P = 0.80). There was no significant difference in the proportional contribution of any other size class to AGC-stocks between our montane dataset and 132 lowland plots from the AfriTRON network (P≥0.24, Table S3), although greater variation among plots is observed in montane forests (Fig. 2). To investigate if elevation affected AGC or forest structure, we modelled these variables as functions of elevation using random slopes mixed-effects models. This approach allows intercepts and relationships to vary among sites, which would be expected as mountains can have very different climate at the same elevation due to proximity to the ocean (generally the further, the drier) and because of the mass-elevation or telescopic effect [START_REF] Jarvis | The climate of cloud forests[END_REF] (larger mountains are better at warming the atmosphere above them). We found that AGC, stem density or density of large stems (>70 cm diameter) were not significantly related to elevation (Fig. 3, Table S4). Across sites these nonsignificant relationships were all negative, although there was some variation in strength and direction amongst sites (Fig. 3). Similarly, in the Neotropics and Southeast Asia montane forest plot datasets, AGC was not significantly correlated with elevation (Extended Data Fig. 4). To assess potential environmental drivers of AGC-stock variation across the AfriMont plot network, we related AGC to climate, soil and topography. We found that AGC-stocks increased with annual precipitation (albeit not statistically significantly), decreased with soil fertility and were higher in plots which were locally at higher elevation than their surroundings (Extended Data Fig. 5). Relationships with other environmental variables were non-significant (Extended Data Fig. 5). Although global datasets might not capture fine-scale variation in climate or soils in mountain regions [START_REF] Platts | AFRICLIM: high-resolution climate projections for ecological applications in Africa[END_REF] , leading to regression dilution [START_REF] Mcinerny | Fine-scale environmental variation in species distribution modelling: Regression dilution, latent variables and neighbourly advice[END_REF] , the general absence of strong climate effects combined with the lack of significant effect of elevation on AGC-stocks suggest that the high AGC-stock of African montane forests is a pervasive phenomenon across a wide environmental gradient. Although the AfriMont dataset covers most major mountain areas in tropical Africa (Fig. 4), some areas remain under-sampled relative to forest extents (Extended Data Fig. 6), resulting in some differences between the environmental conditions sampled by our plot network and the wider montane forest biome in Africa (Extended Data Fig. 7). Notably, the absence of plots from montane forests of eastern Democratic Republic of the Congo (Fig. 4, Extended Data Fig. 6) means that the AfriMont dataset samples forests are, on average, at higher elevations, and that are cooler and cloudier than the wider montane forest biome in Africa (Extended Data Fig. 7). Using relationships with environmental variables (Extended Data Fig. 5) to predict AGC-stocks in each 1-km grid cell containing montane forest gives a mean (weighted by remaining forest cover) AGC-stock of 176.9 Mg C ha -1 (± 32.0 s.d.) for the tropical montane forest biome in Africa. This indicates that the estimate we report based on our AfriMont plot network data (149.4 Mg C ha -1 ) is conservative. Several mechanisms could explain the high AGC-stock of montane forests in the AfriMont plot network. Firstly, large herbivores such as elephants (Loxodonta spp.) can have profound effects on forest structure by consuming biomass, destroying small stems, dispersing seeds and transporting nutrients [START_REF] Poulsen | Ecological consequences of forest elephant declines for Afrotropical forests[END_REF] . Studies for lowland forests suggest that elephants can increase carbon stocks [START_REF] Berzaghi | Carbon stocks in central African forests enhanced by elephant disturbance[END_REF][START_REF] Enquist | The megabiota are disproportionately important for biosphere functioning[END_REF] . We tested if AfriMont plots with known elephant presence as of 2019 had significantly higher AGCstocks, but found that they had significantly lower AGC-stocks, although significant differences were not observed in some countries (Extended Data Fig. 8). While the initial ecosystem response to elephant removal might be greater AGC-stocks due to reduced biomass consumption and small-stem destruction, the longer-term effects might differ. We were unable to fully disentangle such effects, as we lacked details on both i) time since elephant extirpation, and ii) elephant abundance and its determinants (see Table S5). A second potential explanation is a relatively low frequency of large-scale abiotic disturbances, allowing trees time to grow large and stands to self-thin, as is seen in lowland African forests [START_REF] Lewis | Above-ground biomass and structure of 260 African tropical forests[END_REF] . For example, tropical cyclones are largely absent in mainland Africa (except in Mozambique [START_REF] Lin | Tropical Cyclone Ecology: A Scale-Link Perspective[END_REF] ) and lava flows are limited even in the active volcano of Mt Cameroon [START_REF] Favalli | Lava flow hazard and risk at Mt. Cameroon volcano[END_REF] . Although fine-scale variability in landslide risk limits comparisons across large spatial scales, there are fewer areas with high landslide susceptibility in mountains in tropical Africa than in the Andes and most mountain ranges in Southeast Asia [START_REF] Stanley | A heuristic approach to global landslide susceptibility mapping[END_REF] . If forests have been ecologically stable over evolutionary timescales, tree species may be adapted to grow slowly but potentially reaching great sizes [START_REF] Lovett | Elevational and latitudinal changes in tree associations and diversity in the Eastern Arc mountains of Tanzania[END_REF] . On Mt Kilimanjaro Entandrophragma individuals reach enormous heights and ages [START_REF] Hemp | Africa's highest mountain harbours Africa's tallest trees[END_REF] . This low frequency of large-scale abiotic disturbances contrasts with the Andes and several mountains in Southeast Asia (e.g. Barisian mountains in western Sumatra), which are tectonically active, so the trees there are adapted to sudden disturbance followed by intense competition to get established and grow. Future monitoring of the AfriMont plot network will help determine the extent to which the high biomass of African tropical montane forests results from them being dynamic and productive, or adapted to stability. A third potential explanation could be the presence of conifers [START_REF] Culmsee | Forest aboveground biomass along an elevational transect in Sulawesi, Indonesia, and the role of Fagaceae in tropical montane rain forests[END_REF] . Mixed conifer/broad-leaved forests tend to have greater basal area than purely broad-leaved forests due to a more effective use of light and other resources [START_REF] Enright | The southern conifers -a synthesis[END_REF] . Podocarpaceae can be found in montane forests across the tropics [START_REF] Neale | The Conifers. In: The Conifers: Genomes, Variation and Evolution[END_REF] . Despite having fewer species in Africa than in other continents [START_REF] Mill | Towards a Biogeography of the Podocarpaceae[END_REF] , these could be more abundant at the sitelevel. However, there is no pantropical comparative study on Podocarpaceae abundance in tropical montane forests. In our dataset there was no significant correlation between plot-level AGC-stock and conifer (Podocarpaceae) abundance (Extended Data Fig. 9). Other explanations could be continental differences in mountain terrain (more gentle slopes or plateau regions in Africa) or types of montane forests investigated (less cloud forest existing/sampled in Africa). Within our dataset, slope did not have a significant effect on AGC-stocks (Extended Data Fig. 5). Contrary to the Neotropics [START_REF] Helmer | Neotropical cloud forests and paramo to contract and dry from declines in cloud immersion and frost[END_REF] , there is no high-resolution map of cloud forests available for Africa, so while we found no relationship between AGC-stock and cloud frequency (Extended Data Fig. 5), we were unable to investigate differences in AGC-stock between cloud forest vs non-cloud forest plots. To understand the policy implications of our findings for African countries, we calculated montane (≥800 m asl) forest cover change between 2000 and 2018, using forest cover from ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] clipped to 'primary humid forest' from ref. [START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . We show that tropical montane forests represent most -or allevergreen old-growth forests found in ten African countries (Fig. 4), and that the Democratic Republic of the Congo has two thirds of the remaining 16 million ha of montane forests in Africa.

Over 0.8 million ha (5%) have been lost in Africa since 2001, with the highest losses in the Democratic Republic of the Congo (536,000 ha), Uganda (65,000 ha) and Ethiopia (62,000 ha) (Fig. 4, Table 1). In terms of percentage, Mozambique and Côte d'Ivoire lost over 20% of their montane forests over this period (Fig. 4, Table 1). In some sites, however, a larger proportion of montane forests was lost before 2000, e.g. in Taita Hills in Kenya [START_REF] Pellikka | Airborne remote sensing of spatiotemporal change (1955-2004) in indigenous and exotic forest cover in the Taita Hills, Kenya[END_REF] . If absolute country-level deforestation rates continue, a further 0.5 million ha of tropical montane forests will be lost by 2030. African tropical montane forests are not only carbon-rich, but they also harbour some of the highest concentrations of biodiversity and endemism in the world 9,10 . They are important 'water towers' as, located at the headwaters of numerous river systems, including the Congo and the Nile, they regulate timing and magnitude of runoff 9 . They also regulate local temperatures [START_REF] Zeng | Deforestation-induced warming over tropical mountain regions regulated by elevation[END_REF] and provide numerous other services to people in the surrounding landscapes 9 . Clearly, more should be done to avoid the destruction of these important ecosystems. Logging, mining and clearing land for farming, but also political unrest and militia presence have affected -and continue to affect-these forests, e.g. in Itombwe Mts in the Democratic Republic of the Congo [START_REF] Spira | The socio-economics of artisanal mining and bushmeat hunting around protected areas: Kahuzi-Biega National Park and Itombwe nature reserve, eastern Democratic Republic of Congo[END_REF] . Protected areas are known to help reduce deforestation in the tropics 43 . Beyond protected areas, other forest conservation mechanisms could be implemented, including effective carbon finance. Previous IPCC AGC-stock estimates for montane forests in Africa (89.3 Mg C ha -1 ) may have contributed to low incentives for carbon finance mechanisms in these ecosystems. Our study shows the far greater carbon storage potential in these tropical montane forests, which will be even higher if soil carbon stocks are considered (e.g. > 200 Mg C ha -1 of organic carbon occurs in the top 0-30 cm soil on Mt Cameroon [START_REF] Tegha | Soil organic carbon stocks in Mt Cameroon National Park under different land uses[END_REF] and in the Usambara Mts, Tanzania [START_REF] Munishi | Carbon storage in afromontane rain forests of the eastern arc mountains of Tanzania: their net contribution to atmospheric carbon[END_REF] ). As well as conserving the remaining montane forests, efforts to restore them are critical. Forest restoration at one of our sites, Kibale National Park in Uganda, indicates the potential for rapid AGC accumulation 46 . Our study shows the high potential AGC-stock these montane forests can attain. The possible co-benefits of forest restoration, notably water regulation, control of soil erosion and landslides and biodiversity conservation should also be considered. Most African nations are committed to the Bonn Challenge; Ethiopia leading with 15 million ha committed (Table 1). We provide country-specific estimates of potential AGC-stocks based on forests sampled in the AfriMont dataset to help guide such interventions (Table 1, Extended Data Fig. 10). Caution is needed when scaling-up our estimates to the landscape scale, as not all forests are closed-canopy old-growth and structurally intact. Remote sensing or ancillary data (landcover maps, spatial environmental data) could be used to identify e.g. exotic plantations, degraded or bamboo forests, and thus help create detailed AGC maps at different spatial scales [START_REF] Willcock | Quantifying and understanding carbon storage and sequestration within the Eastern Arc Mountains of Tanzania, a tropical biodiversity hotspot[END_REF]47 . A closer collaboration between air-borne, spaceborne and ground approaches (such as the AfriMont and AfriTRON plot networks) is key for accurate quantification and monitoring of landscape-scale tropical forest AGC-stocks, particularly in mountain regions. Our newly compiled dataset and analysis provides a large-scale quantification of AGC-stock in African tropical montane forests, indicating it to be on average substantially higher than previously thought. While there is variation around this mean AGC-stock within and across sites, it is not systematically related to elevation. Apart from helping refine country-level estimates, IPCC guidelines and ground-calibration of remote-sensing estimates, continued on-the-ground monitoring of the AfriMont plot network will help determine ecosystem dynamics and carbon residence time in these extraordinarily carbon-rich forests, as well as their responses to climatic changes. ) are from ref. [START_REF] Spracklen | Tropical montane forests are a larger than expected global carbon store[END_REF][START_REF] Vilanova | Environmental drivers of forest structure and stem turnover across Venezuelan tropical forests[END_REF][START_REF] Alvarez-Davila | Forest biomass density across large climate gradients in northern South America is related to water availability but not with temperature[END_REF] , lowland forests in Africa (n = 290), the Neotropics (n = 416) and Southeast Asia (n = 60) arewfrom ref. [START_REF] Sullivan | Long-term thermal sensitivity of Earth's tropical forests[END_REF] . Coloured points show the AGC-stock in each plot, with point size proportional to square-root plot area. Black points show means for each continent-elevation category estimated using linear mixed-effects models with site as a random effect, and lines show 95% confidence intervals around means. Letters indicate signficiant differences between continent elevation category combinations (linear mixed-effects models with site as a random effect, P < 0.05). S4), while the black lines show the relationship within each site. Point sizes are proportional to square-root plot area. A polynomial model allowing a non-linear relationship with elevation was also tested but not supported over the linear model in any case (P ≥ 0.7, Table S4). The absence of a significant relationship with elevation is robust to removing the two highest elevation sites, Rwenzori and Virunga (Table S4). DBH diameter at breast height. Montane forests represent most (or all) evergreen humid old-growth forest in ten African nations: Burundi, Ethiopia, Kenya, Rwanda, Tanzania, Uganda and Zimbabwe (included in AfriMont); and Zambia, Malawi and South Sudan (no plot data available). Forest cover extracted from ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] and clipped to 'primary humid forest' using ref. [START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . See Table 1 for country-level absolute estimates.

Table 1 | Remaining forest area and aboveground carbon estimates for montane and lowland tropical forests in Africa

Forest cover circa 2018 was extracted from ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] and clipped to 'primary humid forest' using ref. [START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . Montane forest lost covers the period 2000-2018. Mean aboveground carbon (AGC, in MgC ha -1 ) estimates for montane (or lowland) forests were estimated from AfriMont and AfriTRON plot network data. Mean AGC values are in boldface, 95% confidence intervals in parentheses. For details on sites and plots used see Table S5

Methods AfriMont or montane Africa dataset

We compiled forest inventory plot data from the African Tropical Rainforest Observatory Network (AfriTRON; www.afritron.org ), with data curated at www.ForestPlots.net 50,51 and the TEAM network 52 , as well as from numerous site-specific publications detailed in Table S5 and mapped in Fig. 4. Plots were selected for the analysis when conforming to the following criteria: ≥800 m asl, closed-canopy evergreen wet or moist tropical forest, geo-referenced, old-growth and structurally intact (not impacted by recent selective logging, fire or coffee cultivation), with no exotic species present (e.g. Eucalyptus or Pinus spp.), all trees ≥10 cm diameter measured and majority of stems identified to species. We included plots from Virunga Massif in Rwanda/Uganda even when not 100% closed-canopy due to high abundance of naturally-occurring bamboo. In all plots, tree diameter was measured at 1.3 m along the stem from the ground, or above buttresses if present. In 23 sites tree height was sampled in the field for some stems, using a clinometer or a laser. Families and species names follow the African Plant Database (http://africanplantdatabase.ch). The AfriMont dataset consists of 72,336 stems, of which 92.9% were identified to species, 98.4% to genus and 98.5% to family. This dataset represents a standardised safe long-term repository of valuable historical data (four sites initially considered could not be included because tree-level data had already been lost by data owners).

AfriTRON or lowland Africa dataset

The 132 lowland-forest plots are all from AfriTRON 4,13,53 . They were selected using the same criteria as above (but with elevation <800 m asl), restricted to countries for which we also had montane plots plus neighbouring countries where the mountains span international borders (e.g. Mt Nimba spans Guinea and Liberia). The dataset includes 51,305 stems, of which 89.6% were identified to species, 97.3% to genus and 97.7 % to family. The plot data were retrieved from forestplot.net on 06/01/2019. The plot locations and details are in Table S6.

Literature dataset

We compiled data on AGC-stocks in tropical lowland and montane forests to compare to the AfriMont data. Data for lowland forests came from ref. [START_REF] Sullivan | Long-term thermal sensitivity of Earth's tropical forests[END_REF] and consisted of all multi-and single-census plots that were <800 m asl. Data for montane forests were obtained from ref. [START_REF] Spracklen | Tropical montane forests are a larger than expected global carbon store[END_REF] , with additional data from Venezuela (ref. [START_REF] Vilanova | Environmental drivers of forest structure and stem turnover across Venezuelan tropical forests[END_REF] ) and Colombia (ref. [START_REF] Alvarez-Davila | Forest biomass density across large climate gradients in northern South America is related to water availability but not with temperature[END_REF] ). Montane plots were defined as ≥800 m asl; elevation was not provided for the Colombian dataset so plots were selected based on the forest type, and these plots were excluded from analyses requiring elevation. To avoid double counting plots, Venezuelan and Colombian plots were removed from the ref. [START_REF] Spracklen | Tropical montane forests are a larger than expected global carbon store[END_REF] dataset.

Aboveground carbon

For each tree in the montane dataset we used the published allometric equation by ref. 54 to estimate aboveground biomass. This allometric equation was created using data from directly harvested trees at 58 sites across the tropics, including eight sites with elevation ≥800m asl (range 900-3,000m asl including sites in Africa). We then converted this biomass to carbon, assuming that aboveground carbon (AGC, in Mg C ha -1 ) is 45.6% of aboveground biomass 55 . AGC for each plot was estimated as the sum of the AGC of each living stem, divided by planimetric plot area (in hectares). If field measurements of slope were unavailable, we converted surface to planimetric area extracting slope from the NASA's Shuttle Radar Topography Mission (SRTM) product. We excluded tree ferns, bamboo and palms, as these were not measured in all plots. Ref. 54 includes tree diameter, wood mass density and tree height. The best taxonomic match wood density of each stem was extracted from a global database 56,57 following ref. 53 . For some sites, all trees in a plot had been sampled for height. If this was not the case, but some field measurements of height were available (typically ten stems per diameter class), we constructed a site-specific height-diameter model, using a Weibull equation following ref. [START_REF]Eco2[END_REF] . If no field measurements of height were available, we constructed a cluster-specific height-diameter model, using a Weibull equation, as explained in Table S7 in Supplementary Information. The same approach was used to calculate aboveground biomass for lowland forests. For these, height was estimated using a Weibull equation following ref. [START_REF]Eco2[END_REF] . Small plots and data subsampling For 22 sites where plots were small (<0.2 ha), we aggregated plots to groups of about 0.2 ha based on their geographic proximity, elevation, environmental affinity and the co-authors' knowledge of the site, to help reduce the variation among plots at site level. This is because the presence of an extremely large tree in a small plot can result in overestimates of AGC 59 . We investigated if using the aggregated-plot approach affected AGC-stock estimates at the site level, and this was not the case (Extended Data Fig. 2). We also investigated if including small plots affected the continental mean AGC-stock estimates, as small plots have greater edge surface, and there is a tendency of some field teams to include large trees inside plots when laying out the boundaries 60 . Including small plots did not significantly affect our continental mean AGC-stock estimates (Extended Data Fig. 2). We also explored the sensitivity of our continental mean AGC-stock estimates to data subsampling. Data were resampled at different sample sizes either at plot level (sampling with replacement) or at site level (sampling without replacement). The number of plots (n=226) and the number of sites (n=44) we sampled indicate that our estimates of AGC-stock at the continental level are robust (Extended Data Fig. 1). They are also not affected by the fact that we included plots 800-1,000 m asl (Extended Data Fig. 3).

Size classes

For all plots, we computed the proportion of AGC which was distributed in each size-diameter class, using the classes of ref. [START_REF] Bastin | Pan-tropical prediction of forest structure from the largest trees[END_REF] . We also computed stem density, basal area, density of large trees (>70 cm diameter, named SD70 in stems ha -1 ) and Podocarpaceae abundance (in percentage of plot-level basal area).

Environmental variables and their effects

Climate variables (temperature annual mean and seasonality, and precipitation mean and seasonality, i.e. Bio1, 4, 12 and 15) were extracted from WorldClimV2 61 at 30 arc-sec (~1-km) resolution. Mean temperature values were adjusted for the difference in elevation between the plot and the wider 1-km grid cell using the lapse rate of -0.005°C m -1 . We obtained data on cloud cover from ref. [START_REF] Albrecht | LIS 0.1 Degree Very High Resolution Gridded Lightning Climatology Data Collection[END_REF] and lightning frequency (0.1 degree, ~11 km) from the Lightning Imaging Sensor (LIS) very high resolution climatology [START_REF] Hengl | SoilGrids250m: Global gridded soil information based on machine learning[END_REF] . Values for soil variables (cation exchange capacity, CEC, representing soil fertility, and percentage clay representing soil texture) were extracted from SoilGrids 64 (~1-km resolution) and a depth-weighted mean taken for values from 0 to 30 cm depth to give a single value of each soil variable per plot. Elevation was obtained from SRTM (at 3 arc-second resolution, ~90 m). Topographic metrics were calculated from elevation data using the terrain function in the raster R package version 3.3-6. These were slope and topographic position index (TPI). TPI is the difference between the elevation of the plot and the mean value of the eight surrounding grid cells -positive values indicate locally high locations and negative values indicate locally low locations. Where small plots were aggregated for analysis, environmental variables were extracted for the ungrouped plot locations, and then an area-weighted mean taken to obtain a plot-level value.

Elephant and conifer effects on AGC-stocks

For the current elephant presence in the AfriMont plots, we created a binary variable (presence/absence) based on co-authors knowledge of elephant ranges and elevation distribution at each site as of 2019. Co-authors estimated that elephants were present in 2019 in 54 plots in 12 sites in five countries (see Table S5). For all plots which had at least one individual in the Podocarpaceae family (47 plots, 16 sites, 7 countries), we computed the contribution of Podocarpaceae to plot basal area and AGC-stock in terms of percentages.

Estimating forest cover and loss

We obtained estimates of forest cover and loss in the years 2000 through to 2018, using the 'loss year' dataset of the Global Forest Change database, version 1.6 (ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] ). To exclude plantation forests, 'dry' forests (e.g. miombo woodland) and degraded forests, we applied the 'primary humid forest' mask developed by ref. [START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . We distinguished montane from lowland forests using an elevational cut-off of 800-m elevation, using the SRTM v3 product at 1 arc-sec resolution (snapping to the ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] grid of the same resolution). Where there were gaps in the 1 arc-sec SRTM product, we filled these using a 1 arc-sec bilinear interpolation of the (gapless) 3 arc-sec SRTM product. Areal estimates of forest cover and loss were calculated at 30-m resolution using the Africa Sinusoidal projection. To estimate future forest loss by year 2030, we extrapolated absolute country-level deforestation rates for the period 2000-2018 (in ha per year).

Investigating AfriMont representativeness

To quantify AfriMont sampling effort within the montane forest biome in Africa, we used the map of tropical montane forest extent (see above) and calculated the amount of remaining forest in each 1degree grid-cell. By dividing the area sampled in the AfriMont dataset by the proportion of this biome in a grid-cell, we calculated the expected sampling intensity if sampling was proportional to remaining forest extent. To assess how representative our plot network was of the environmental conditions of the wider tropical montane forest biome in Africa, we extracted the environmental data (climate and soil variables presented above) at ~1-km resolution from grid-cells that contained montane forest. We then visually compared the distribution of each variable in our dataset to its distribution across the biome (Extended Data Fig. 7).

AfriMont vs global AGC maps

We extracted alternative AGC estimates for the AfriMont plots (unaggregated, n=666) from four different sources: Harris et al. (ref. [START_REF] Santoro | ESA Biomass Climate Change Initiative (Biomass_cci): Global datasets of forest above-ground biomass for the year 2017, v1[END_REF] ) (30-m resolution, dated 2000), the European Space Agency Climate Change Initiative Biomass map [START_REF] Saatchi | Benchmark map of forest carbon stocks in tropical regions across three continents[END_REF] S5). Where the plots were found within a single map pixel, we extracted that value. Where plots were larger than the pixel size, we averaged the values from the surrounding pixels weighted according to the proportion of the pixel that was in the plot.

Statistical analysis

Data were analysed using linear mixed-effects models, with site as a random effect. Site was included as a random intercept in all models, and as a random slope where relationships were assessed against elevation. Allowing the slope of the elevation effect to vary amongst sites in this way captures the a priori expectation for slopes to differ among sites, for example due to mass elevation effects. The effect of plot size on variation was accounted for by weighting observations by a power transformation of plot size; this was estimated during model fitting using the varPower function in the nlme R package (ref. [START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF] ), and then models refitted using the lme4 R package (ref. [START_REF] Lüdecke | Parameters: extracting, computing and exploring the parameters of statistical models using R[END_REF] ) using these estimated weights. Confidence intervals and P-values for mixed effects models parameters were estimated by bootstrapping models (1,000 iterations) using the bootstrap_parameters function in the parameters R package (ref. [START_REF] Hothorn | Simultaneous Inference in General Parametric Models[END_REF] ). AGC-stocks, stem density and SD70 were natural-log transformed (a small constant was added to SD70 before log transforming to avoid log-transforming zeros) to meet assumptions of normality and avoid heteroscedacity. Likewise, the proportional contribution of each size class was square-root transformed. Differences in AGCstocks between all combinations of lowland and montane forests amongst continents were assessed using Tukey post-hoc tests implemented in the multcomp R package (ref. [START_REF] Barton | MuMIn: Multi-Model Inference[END_REF] ). Relationships between AGC-stocks and environmental variables were investigated by fitting all subsets of the full model with all environmental covariates and averaging the best supported (difference in Akaike information criterion from the best supported model ΔAIC<4) models (using dredge and movel.avg functions in the MuMIn R package (ref. 73 ). We used these relationships with climate and soil to predict AGC-stocks in each 1-km grid cell containing montane forests (holding topographic variables at their dataset wide mean), and then took the forest-area weighted mean of these to obtain a single mean for the tropical montane forest biome in Africa. Differences in AGC-stocks between plots with and without elephants were tested using t-test with AGC-stocks natural-log transformed. We investigated if Podocarpaceae abundance (in terms of basal area) and plot AGC-stocks were significantly correlated using Spearman's rank correlation coefficient. To investigate if sampling design affected AfriMont AGC-stock estimates we used ANOVA to test whether site-level mean AGCstocks differed according to the sampling strategy used to establish plots at that site. To explore the relationship between AfriMont AGC-stock estimates and global maps, and among these global maps, we used Spearman's rank correlation test.

Extended Data Fig. 2 | Effect of plot area, aggregation procedure and plot design on estimates of aboveground carbon stocks across the AfriMont plot network. (a) Relationship between aboveground carbon stocks and plot area of plots prior to aggregation. The red line shows the fit of a locally weighted regression model (span = 0.75) relating these variables, with dashed lines showing the standard errors. (b) Variation in aboveground carbon stocks using either all plots prior to aggregation (unaggregated), plots prior to aggregation but excluding those < 0.2 ha (unaggregated, > 0.2 ha) or the aggregated plots used in the main analyses (aggregated). (c) Effects of plot design on aboveground carbon stocks (each site represents one dot). Sampling strategies include random or stratified random, plots positioned along transects, plots established within elevation bands, subjective measures such as choosing an area of forest considered representative of the wider area, and other strategies (one plot sampled per site or unclear strategy). Carbon stocks (log-transformed) did not differ significantly between sites with different sampling strategies (ANOVA: F4,39 = 0.432, P = 0.785). For specific site information see Table S5. Extended Data Fig. 6 | Expected sampling effort if effort was distributed in proportion to the area of tropical montane forest biome in Africa. Data are summarised at 1-degree resolution. Upward triangles show grid-cells where AfriMont sampling effort is more than double expected effort, downward triangles show grid-cells where AfriMont sampling effort is less than half expected effort. Circles denote AfriMont sampling effort being between half and double expected effort. The extent of the tropical montane forest biome was defined as closed-canopy forests ≥ 800 m asl in December 2018, extracted from ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] and clipped to 'primary humid forest' using ref. [START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . This grided map differs from Fig. 4 as numerous grids have very little tropical montane forest.

Extended

Extended Data Fig. 7 | Differences in the environmental conditions sampled by the AfriMont plot network and the tropical montane forest biome in Africa. The extent of the biome was defined as closed-canopy forests ≥ 800m asl in December 2018, extracted from ref. [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] and clipped to 'primary humid forest' using ref. [START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . Environmental variables for the biome were extracted at ~1-km resolution.

Extended Data

Figures main document Fig. 1 |

 1 Figures main document

Fig. 2 |

 2 Fig. 2 | Proportion of plot-level aboveground carbon stock and stems accounted for by each size class in montane and in lowland forests in Africa. Statistically significant differences in contribution of each size class between montane and lowland forest plot networks are shown by asterisks (linear mixed-effects model, P < 0.05). NS = non-significant difference. Montane (n = 226), lowland (n = 132). The thick line shows the median, and boxes cover the interquartile range (IQR). Values > 1.5 times IQR away from the IQR are shown by points.

Fig. 3 |

 3 Fig. 3 | Variables as a function of elevation. a-c Relationship between elevation and plot-level AGC stock (a), stem density (b) and stem density of large stems (>70 cm diameter; c) for the AfriMont dataset. Note log-scale of y-axis. Each response variable was log-transformed and modelled as a function of elevation with a linear mixed-effect models with random slopes. The dashed line shows the relationship across sites (non-significant in all cases, P ≥ 0.3, TableS4), while the black lines show the relationship within each site. Point sizes are proportional to square-root plot area. A polynomial model allowing a non-linear relationship with elevation was also tested but not supported over the linear model in any case (P ≥ 0.7, TableS4). The absence of a significant relationship with elevation is robust to removing the two highest elevation sites, Rwenzori and Virunga (TableS4). DBH diameter at breast height.

Fig. 4 .

 4 Fig. 4. | Old-growth evergreen humid forests in lowland and montane tropical Africa. Forest extends circa 2018. Note that montane includes submontane forests (800-1,000 m asl, light purple).Montane forests represent most (or all) evergreen humid old-growth forest in ten African nations: Burundi, Ethiopia, Kenya, Rwanda, Tanzania, Uganda and Zimbabwe (included in AfriMont); and Zambia, Malawi and South Sudan (no plot data available). Forest cover extracted from ref.[START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] and clipped to 'primary humid forest' using ref.[START_REF] Turubanova | Ongoing primary forest loss in Brazil, Democratic Republic of the Congo, and Indonesia[END_REF] . See Table1for country-level absolute estimates.

  (100-m resolution, 2017), Saatchi, et al. (ref. 67 ) (1-km resolution, 2007/8) and Avitabile et al. (ref. 68 ) (1-km resolution, circa 2000-2010). Most of the AfriMont plots were sampled between 2000 and 2019 (Table

Data Fig. 3 |

 3 Robustness of differences in tropical montane forest aboveground carbon (AGC) stocks among continents based on plot networks to differences in elevation. (a) Elevations of montane forests plots sampled in each continent. Violin plots show the distribution of data, with boxplots showing the median and interquartile range of elevation in each continent. (b) Effect of removing submontane plots (800-1,000 m asl) and high elevation plots (> 2,200 m asl, approximately the upper quartile of elevations for the African montane plot dataset) on AGC-stocks in montane forests sampled by plot networks in each continent. Mean AGC-stocks and 95% confidence intervals are shown as estimated by models using i) all data, ii) excluding plots 800-1,000 m, and iii) restricting plots to 1,000-2,200 m. Means for all plots differ from the analysis in Fig. 1 as literature plots without elevation data (plots in Colombia) were excluded from this analysis. Point symbols are proportional to square-root plot area. N = 324 plots. Extended Data Fig. 4 | Relationship between aboveground carbon (AGC) stocks and elevation for tropical montane forests in each continent based on plot networks. Dashed lines show relationships from a linear mixed-effects model of log-transformed AGC-stocks as a function of elevation, continent and their interaction. Site was included as a random effect, and AGC-stockelevation relationships allowed to vary among sites. Lines show fitted slopes across sites. Neither the overall relationship between elevation and AGC-stocks (slope = -0.039 [95% CI = -0.127 -0.057], P = 0.420) nor interactions between elevation and continent (Southeast Asia, change in slope = -0.074 [-0.294 -0.149], P = 0.503; Neotropics, change in slope = 0.006 [-0.132 -0.149], P = 0.913) are statistically significant. N = 324 plots. Extended Data Fig. 5 | Environmental drivers of aboveground carbon stocks across the AfriMont plot network. Coefficients are from a linear mixed-effects model with site as a random intercept. Results are following all-subsets regression and model averaging, in which variables that do not appear in well supported models are given coefficients of zero, leading to shrinkage in model coefficients. Statistically significant relationships (P < 0.05) are indicated with asterisks. TPI refers to topographic position index (positive values indicate higher than surroundings, negative values indicate lower than surroundings). T_mean: annual mean temperature, T_seasonality: temperature seasonality, Precip_total: annual precipitation, Precip_seasonality: precipitation seasonality.

Fig. 8 |

 8 Differences in aboveground carbon (AGC) stocks in AfriMont plots located in montane forests with and without elephants. a) Differences across all plots. AGC-stocks are statistically significantly lower in forests with elephants (t-test, t = 3.5, df=83.5, P = 0.001). The thick line shows the median, and boxes cover the interquartile range (IQR). Values > 1.5 times IQR away from the IQR are shown by points. b) Differences in countries where elephants are present in at least one of the montane sites studied. Black squares show means in each country in forests with or without elephants -solid lines denote statistically significant differences (t-tests, P < 0.05). Elephant presence in 2019 was estimated by co-authors (see TableS5

  ). Extended Data Fig. 9 | Relationship between aboveground carbon (AGC) stocks and Podocarpaceae. (a) Relationship between AGC-stocks and Podocarpaceae basal area across plots in the AfriMont network, expressed as a percentage of total plot basal area. These variables are not significantly correlated (rs = 0.083, n = 226, P = 0.212). (b) Distribution of plots with at least 20% basal area of Podocarpaceae (black points) in relation to elevation and AGC-stocks. AGC-stocks are not significantly related to elevation or Podocarpaceae basal area (Linear mixed effects model, P = 0.152 and 0.132 respectively). Extended Data Fig. 10 | Within country variation in aboveground carbon stocks based on the AfriMont plot network. Error bars show means and 95% confidence intervals estimated by linear mixed-effects models. Modelled means not shown for countries with fewer than five plots. Point size is proportional to plot area.

  . Bonn Challenge pledges for 2030 not yet available. a Ref. 48 report 192 MgC ha -1 for lowland. b Few plots sampled, or very small plots sampled, AGC estimates may not be robust, see Extended data Fig. 10. Montane forest loss in Mozambique, Uganda and Zimbabwe represents 27%, 13% and 10% of the existing montane forest in 2001, respectively. Montane forest loss in Côte d'Ivoire (no plot data available) was estimated to be 21% for the same period. e Ref. 49 report 132.2 MgC ha -1 for lowland. Wheeler, C.E., et al. Carbon sequestration and biodiversity following 18 years of active tropical forest restoration. Forest Ecol. Manage. 373, 44-55 (2016). 47. Avitabile, V., Baccini, A., Friedl, M.A. & Schmullius, C. Capabilities and limitations of Landsat and land cover data for aboveground woody biomass estimation of Uganda. Remote Sens.

	Country	Montane (ha) 46. Environ. 117, 366-380 (2012). Montane lost (ha) Montane AGC (MgC ha -1 , 95% CI) 48. Aneseyee, B A., Soromessa, T. & Belliethathan, S. Carbon Sock of Gambella National Park: Montane sites (plots) Lowland (ha) Lowland AGC (MgC ha -1 , 95% CI) Lowland plots Implication for Climate Change Mitigation. Int. J. Adv. Life Sci. 35, 41-56 (2015).	Bonn Challenge by 2020 (ha)
	Burundi Cameroon	25,000 49. Lisboa, S.N., et al. Biomass allometric equation and expansion factor for a mountain moist 300 94 (47-176) 1 (7) 0 0 840,000 30,200 153 (121-195) 7 (37) 17.7 million 166 (151-185) 72 evergreen forest in Mozambique. Carbon Balance Manage. 13, 23 (2018).	2 million 12 million
	DRC	10.2 million	536,500	129 (84-202)	2 (37)	90 million	158 (135-183)	48	8 million
	Ethiopia	1.7 million	62,100	165 (124-215)	8 (25)	145,000	a	0	15 million
	Guinea	29,000	1,700	314 (147-616) b	1 (2)	193,000	157 (122 -206) c	24	2 million
	Kenya	568,000	44,100	104 (79-136)	8 (38)	37,000		0	5.1 million
	Mozambique	18,000	6,600 d	226 (146-384) b	3 (4)	93,000	e	0	1 million
	Nigeria	42,000	1,400	120 (47-309) b	1 (1)	1.8 million	161 (105-262)	2	4 million
	Rwanda	53,000	300	106 (65-168)	2 (11)	0		0	2 million
	Tanzania	587,000	13,900	175 (129-234)	6 (29)	130,000	128 (101-163)	16	5.2 million
	Uganda	427,000	64,600 d	158 (111-209)	6 (23)	18,000		0	2.5 million
	Zimbabwe	7,000	800 d	203 (108-363)	1 (12)	<1,000		0	2 million

c Data from neighbouring Liberia. d

Field campaigns for AfriMont were funded by Marie Skłodowska-Curie Actions Intra-European Fellowships (number 328075) and Global Fellowships (number 74356), National Geographic Explorer (NGS-53344R-18), Czech Science Foundation (no. 21-17125S), Rufford Small Grant Foundation (16712-B, 19476-D), Ministry of Foreign Affairs of Finland (BIODEV project), the Academy of Finland (number 318645), Swedish International Development Cooperation Agency, the Leverhulme Trust, the Strategic Research Area Biodiversity and Ecosystem Services in a Changing Climate, the German Research Foundation (DFG), Gatsby Plants, Natural Science and Engineering Research Council of Canada and International Development Research Centre of Canada. This paper is also a product of the AfriTRON network, for which we are indebted to hundreds of institutions, field assistants and local communities for establishing and maintaining the plots, including: the Forestry Development Authority of the Government of Liberia, the University of Liberia, University of Ibadan (Nigeria), the University of Abeokuta (Nigeria), the University of Yaounde I (Cameroon), the National Herbarium of Yaounde (Cameroon), the University of Buea (Cameroon), Bioversity International (Cameroon), Salonga National Park (Democratic Republic of Congo), The Centre de Formation et de Recherche en Conservation Forestière (CEFRECOF, Epulu, Democratic Republic of Congo), the Institut National pour l'Étude et la Recherche Agronomiques (INERA, Democratic Republic of Congo), the École Régionale Postuniversitaire d'Aménagement et de Gestion intégrés des Forêts et Territoires tropicaux (ERAIFT Kinshasa, Democratic Republic of Congo), WWF-Democratic Republic of Congo, WCS-Democratic Republic of Congo, the Université de Kisangani (Democratic Republic of Congo), Université Officielle de Bukavu (Democratic Republic of Congo), Université de Mbujimayi (Democratic Republic of Congo), le Ministère de l'Environnement et Développement Durable (Democratic Republic of Congo), the FORETS project in Yangambi (CIFOR, CGIAR and the European Union; Democratic Republic of Congo), the Lukuru Wildlife Research Foundation (Democratic Republic of Congo), Mbarara University of Science and Technology (MUST, Uganda), WCS-Uganda, the Uganda Forest Department, the Commission of Central African Forests (COMIFAC), the Udzungwa Ecological Monitoring Centre (Tanzania) and the Sokoine University of Agriculture (Tanzania). The AfriTRION network has been supported by the European Research Council (291585, 'T-FORCES' -Tropical Forests in the Changing Earth System, Advanced Grant to O.L.P. and S.L.L.), the Gordon and Betty Moore Foundation, the David and Lucile Packard Foundation, the European Union's Seventh Framework Programme (283080, 'GEOCARBON').

We are particularly grateful to A. Daniels, F. Mbayu, T.R. Feldpausch, E. Kearsley, J. Lloyd, R. Lowe, J. Mukinzi, L. Ojo, A.T. Peterson, J. Talbot and L. Zemagho for giving us access to their plot data. We also thank C. Chatelain (Geneva Botanic Gardens) for access to the African Plants Database and to H. Tang for helping explore the use of GEDI data. Data from AfriTRON and most of AfriMont are stored and curated by ForestPlots.net, a long-term cyberinfrastructure initiative hosted at the University of Leeds that unites permanent plot records and their contributing scientists from the world's tropical forests. The development of ForestPlots.net and curation of African data have been funded by many sources, including the ERC (principally from AdG 291585 'T-FORCES'), the UK Natural Environment Research Council (including NE/B503384/1, NE/F005806/1, NE/P008755/1, NE/N012542/1 and NE/I028122/1), the Gordon and Betty Moore Foundation ('RAINFOR', 'MonANPeru'), the EU Horizon programme (especially 'GEOCARBON', 'Amazalert') and the Royal Society (University Research Fellowship to S.L.L.).

Author Contributions

A.C-S. conceived the study and assembled the AfriMont dataset. A.C-S. and M.J.P.S. analysed the plot data (with contributions from S.L.L.) and wrote the manuscript. P.J.P. analysed forest extents and contributed to writing. S.L.L. conceived and managed the AfriTRON forest plot recensus

programme. E.T.A.M. and V.A. helped compare plot data with remote sensing carbon maps. All coauthors read and approved the manuscript.

Competing interests

The authors declare no competing interests. Additional information Supplementary information is available for this paper at XX (to be added) Correspondence and requests for materials should be addressed to A. C-S. Reprints and permissions information are available at XX (to be added)

Data availability


Philippe Bertrand
email: philippe.bertrand@univ-amu.fr

Jean-Luc Prigent
email: jean-luc.prigent@cyu.fr

Performance Participation Strategies: OBPP versus CPPP

Keywords: JEL classication : G11, G12, G13 portfolio insurance, performance participation, OBPP, CPPP

The goal of this paper is to provide and examine an important extension of the usual portfolio insurance, namely to study the notion of portfolio performance participation. In this framework, the portfolio is based on two risky assets: the rst one corresponds to a reserve asset, while the second one is considered as an active asset which has usually both a higher mean and a higher variance. We aim at insuring a given percentage of the reserve asset return, whatever the market uctuations. The two main performance participation methods are the Option-Based Performance Participation (OBPP) and the Constant Proportion Performance Participation (CPPP). We compare these two portfolio strategies by means of various criteria such as their payo¤s at maturity, their four rst moments and their cumulative distributions functions. We also compare their dynamic hedging properties by computing in particular their deltas and vegas.

Introduction

The purpose of portfolio insurance is to allow the investor to limit the loss risk while beneting in part from a possible rise in the reference nancial market. At maturity, the investor recovers at least a given percentage of his initial investment, especially in bearish markets. (see e.g. Aftalion and Portait, 1988; Charlety-Lepers and Portait, 1997; Poncet and Portait, 1997). Two standard portfolio insurance methods are the Option Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI). The OBPI has been introduced by Leland and Rubinstein (1976). This portfolio strategy is based on an investment in a risky asset � (usually a nancial index such as the S&P 500) covered by a put option written on it. Thus, at maturity, the portfolio value is always higher than the strike of the put. The CPPI has been introduced by Perold (1986) (see also Black and Jones, 1987; Perold and Sharpe, 1988; Black and Perold, 1992). This strategy allocates assets dynamically over time as follows: the investor begins by setting a oor equal to the lowest acceptable portfolio value. The di¤erence between the portfolio value and the oor is called the cushion. Then, he allocates to the risky asset an amount ("the exposure") equal to the cushion multiplied by a predetermined multiple. The remaining funds are invested in the reserve asset, usually T-bills. The key parameter is the multiple. Indeed, the higher the multiple, the higher the portfolio return when the nancial market is bullish. However, the higher the multiple, the higher the gap risk (i.e. the portfolio becomes smaller than the oor).

The goal of this paper is to extend portfolio insurance by substituting the xed capital guarantee [START_REF] Aftalion | La technique de portfolio insurance[END_REF] for a participation in the performance of a risky asset viewed as a reserve asset. The performance participation constraint is to ensure that the portfolio return will be higher than a given percentage of the reserve asset return, whatever the future nancial market uctuations. Note that the portfolio value guaranteed at maturity is no longer deterministic and can be correlated to the active asset itself. Such structured product can be introduced instead of usual portfolio insurance to overcome problems due for example to too low interest rates. For example, the reserve asset is no longer a money market account but rather a long-term government bond while the active asset is an equity index such as the S&P 500. Black and Perold (1992) have introduced such model (that they still call CPPI), showing mainly that, in the absence of transaction costs, this portfolio strategy is equivalent to investing in perpetual American call options, and that it is optimal for a piecewise-HARA utility function with a minimum consumption constraint. However, the interest of such a methodology is also to be able to invest in signicantly risky assets (emerging markets, hedge funds...) while ensuring a minimum return relative to a more standard nancial index.

In this paper, rst we set up the two main performance participation methods, namely the Option-Based Performance Participation (OBPP) and the Constant Proportion Performance Participation (CPPP). We introduce a quite general model allowing to take account simultaneously of both the performance participation condition and the initial budget constraint. [START_REF] Bajeux-Besnamou | The numeraire portfolio: A new perspective on nancial theory[END_REF] We show that they can be both expressed in terms of respectively the Option-Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI) by choosing the reserve asset as numeraire. [START_REF] Bajeux-Besnamou | Dynamic asset allocation in a mean-variance framework[END_REF] We show also that the ratio of the two risky assets play a key role when analyzing the two portfolio strategies and especially its volatility. [START_REF] Ben Ameur | Portfolio insurance: gap risk under conditional multiples[END_REF] For the study of the OBPP portfolio, we use results of Margrabe (1978) about the valuation of exchange options. Second, as in Bertrand and Prigent (2005, 2011) for the OBPI and CPPI strategies, we examine and compare OBPI and CPPI strategies by using various criteria. In a rst part, we analyze and compare their properties at the end of the investment period by examining their payo¤s at maturity, their four rst moments and their cumulative distributions functions (cdf). [START_REF] Bernard | Dynamic preferences for popular investment strategies in pension funds[END_REF] As a by-product, we provide the explicit CPPP multiple value such that the two portfolios have the same expected return. For comparison criteria based on cdfs, we show that none of the two strategies stochastically dominates the other one at the rst order. We also examine the cdf of the portfolio values ratio. Finally, in a second part, we compare their dynamic properties. First, we prove that the OBPP method can be viewed as a generalized CPPP where the multiple can evolve stochastically over time. Second, we examine their hedging properties by computing in particular their deltas and vegas. For this latter Greek, as for the OBPI and CPPI strategies, the OBPP and CPPP di¤er very signicantly since their signs are opposite: the OBPP value is increasing with respect to the volatility of the ratio of the two risky assets whereas, for the CPPP, it is the converse.

The paper is organized as follows. Section 2 sets out the two strategies. Section 3 compares the two strategies at maturity by means of their payo¤s, their four rst moments and their cdfs. Section 4 is devoted to dynamic and hedging properties. Finally, section 5 concludes. [START_REF] Bertrand | Portfolio insurance strategies : OBPI versus CPPI[END_REF] In what follows, we begin by setting up both OBPP and CPPP portfolio values. Then, to assess these two strategies, we begin by comparing their payo¤s at maturity. Thereafter, to better take account of their respective probability distributions, we compare their rst four moments and their cumulative distribution functions (cdf). Finally, we examine their dynamic properties, especially the computation of their Greeks for hedging purpose.

The portfolio manager is assumed to invest in two nancial risky assets denoted by � 1 and � 2 . We assume that they are di¤usion processes which are solutions of:

�� 1�� = � 1�� £ � � 1 �� + � � 1 �� 1�� ¤ � (1)
�� 2�� = � 2�� h � � 2 �� + � � 2 ��� 1�� + � � 2 p 1 ¡ � 2 �� 2�� i �
where � � = (� 1�� � � 2�� ) � is a standard two-dimensional Brownian motion with respect to its own ltration (F � ) � . We assume that the volatilities � �1 of asset � 1 and � �2 of asset � 2 are strictly positive and that their instantaneous correlation � is not null. The period of time considered is [0� � ]. The strategies are self-nancing. The portfolio manager aims at providing a predetermined participation of a reserve asset (here the risky asset � 1 ) while investing in an active asset (here the risky asset � 2 ). Note that usually the active asset � 2 (typically an equity index) is riskier than the reserve asset � 1 (a government bond for example) while it provides a higher instantaneous expected return yielding to conditions � � 2 ¸�� 1 and � � 2 ¸�� 1 . The predetermined participation of the reserve asset � 1 is to guaranty that portfolio return � � �� 0 is higher than a given proportion � of asset return � 1�� �� 1�0 (with 0 � � � 1).

The Option-Based Performance Participation (OBPP)

The Option-Based Performance Participation strategy aims at providing a predetermined participation of a reserve asset (here the risky asset � 1 ) by using an adequate exchange option between the active asset � 2 and given � shares of asset � 1 , namely the option with payo¤ ���(� 2�� � �� 1�� ) which is the best of the two assets �� 1 and � 2 . The parameter � denotes the performance participation on asset � 1 in this exchange option whose payo¤ can also be expressed as � 2�� + (�� 1�� ¡ � 2�� ) + . This latter formula shows that this exchange option corresponds to the purchase of asset � 2 covered by a put written on it with "strike" �� 1�� . Equivalently, we get also the payo¤

�� 1�� + (� 2�� ¡ �� 1�� ) + .
Note that usually the active asset � 2 is riskier than the reserve asset � 1 while it provides a higher instantaneous expected return yielding to conditions � � 2 ¸�� 1 and � � 2 ¸�� 1 . The OBPP portfolio is based on this exchange option. To adjust the value of the portfolio, we adequately7 duplicate this basic structured product by purchasing � shares of it . Thus, the portfolio value � ��� � is given at the terminal date by:

� ��� � � = � £ �� 1�� + (� 2�� ¡ �� 1�� ) + ¤ � (2)
This relation shows that the insured amount at maturity corresponds to (��) � 1�� .

Remark 1 The performance participation is dened as follows: the portfolio return must be always higher than a predetermined fraction � of the reserve asset return. We must have:

¡ � ��� � � �� 0 ¢ ¸� (� 1�� �� 1�0 ) , with 0 � � � 1� (3)
Since at maturity we want to recover exactly � (� 1�� �� 1�0 ) if the call is not exercisable (i.e. � 2�� � �� 1�� ), the coe¢cient �� is set equal to �� 0 �� 1�0 .

Denote by � ��� (�� �� � 2 � �� 1 ) the value at time � of the exchange option between � 2 and �� 1 . Then the value � ��� � � of the OBPP portfolio at any time 0 is given by:

� ��� � � = ��� 1�� + �� ��� (�� �� � 2 � �� 1 ) � (4)

Valuation of the OBPP strategy

In what follows, we show how to valuate the OBPP strategy.

Remark 2

The value of the exchange option can be expressed in terms of asset � 1 considered as numeraire. Indeed, � ��� (�� �� � 2 � �� 1 ) �� 1�� is equal to the value of the call option with underlying �=� 2 �� 1 , strike �, zero interest rate and volatility e � which is the volatility of the process � equal to the ratio � 2 �� 1 (i.e. the value of asset � 2 expressed in terms of asset � 1 as numeraire) given by:

e � = q � 2 � 1 + � 2 � 2 ¡ 2�� � 1 � � 2 � (5)
We have:

� ��� (�� �� � 2 � �� 1 ) �� 1 = ���� (�� �� � 2 �� 1 � �� � = 0� e �) � (6)
Therefore, at any time � 2 [0� � ], the OBPP strategy is equivalent to the purchase of �� 1�� shares of the OBPI strategy based on the ratio � = � 2 �� 1 as underlying risky asset with initial investment value � 0 �� 1�0 . 8 Indeed, using asset � 1 as numeraire, we get the following relation:

� ��� � � � 1�� = � £ � + (� 2�� �� 1�� ¡ �) + ¤ = �� ��� � (�� �� � = � 2 �� 1 � � 0 �� 1�0 � �� � = 0� e �) � (7)
Previous result implies that, for any time � 2 [0� � ], we get:

� ��� � � � 1�� = �� ��� � (�� �� � = � 2 �� 1 � � 0 �� 1�0 � �� � = 0� e �) � (8)
8 Recall that the OBPI strategy satises:

� ��� � � = � � � + (�� ¡ �) + � �
where �� is set equal to a xed proportion of the initial invested amount �0, namely �� = ��0 (see e.g. Bertrand and Prigent, 2005). We denote its current value by � ��� � (�� �� �� � 0 � �� �� �) where � is the risk-free interest rate and � is the volatility of the risky asset �.

In the Black and Scholes framework, recall that the value of ����

(�� �� � 2 �� 1 � �� � = 0� e �) is equal to: ���� (�� �� � 2 �� 1 � �� � = 0� e �) = (� 2�� �� 1�� ) � (� 1�� ) ¡ �� (� 2�� ) �
where � denotes the cumulative distribution function (cdf) of the standard univariate Gaussian distribution and 8 > < > :

� 1�� = ��� � � 2�� �� 1�� � + 1 2 � � 2 (� ¡�) � � p (� ¡�) � 2�� = � 1�� ¡ e � p (� ¡ �) �
We deduce (see Margrabe, 1978):

� ��� (�� �� � 2 � �� 1 ) = � 2�� � (� 1�� ) ¡ �� 1�� � (� 2�� ) � (9)

Determination of the number of shares � and of the performance participation �

To take account of both participation and budget constraint � ��� � 0 = � 0 , we have to solve the following system of equations:

( �� = �� 0 �� 1�0 � 0 = � [�� 1�0 + � ��� (0� �� � 2 � �� 1 )] �
which implies that � must satisfy: Then, we set

1 ¡ � � = � ��� (0� �� � 2 � �� 1 ) �� 1�0 � = ���� (�� �� � 2 �� 1 � �� � = 0� e �) � � (10)
� ¤ = �� 0 �� 1�0 � ¤ .

The Constant Proportion Performance Participation (CPPP)

The CPPP method consists of managing a dynamic portfolio so that its value [START_REF] Black | Simplifying portfolio insurance[END_REF] Denote by � � the exposure, which is the total amount invested in the risky asset � 2 . The CPPP method consists of letting � � = �� � where � is a constant parameter called the multiple. The interesting case is when � � 1, that is, when the payo¤ function is convex with respect to the asset � 2 payo¤. The value of portfolio � �� � � � at any time � in the period [0� � ] is given by [START_REF] Black | Constant proportion portfolio insurance and the synthetic put option : a comparison[END_REF] : [START_REF] Black | Simplifying portfolio insurance[END_REF] To insure that the cushion is always positive, we can set

� �� � � � is above the oor � � = �� 1�� at any time �, where � is xed. The di¤erence � �� � � � ¡ � � at any time � in [0� � ] is called the cushion, denoted by � � .
� �� � � � = �� 1�� + � � �� 1¡� 1�� � � 2�� �
�� = ��� � � �� � � � ¡ ��1��� 0 � .
It implies that, if the portfolio value becomes smaller than the oor, then the whole portfolio value is invested in asset �1. [START_REF] Black | Constant proportion portfolio insurance and the synthetic put option : a comparison[END_REF] Details about this formula are provided in the Appendix.

where

� � = à � 0 ¡ �� 1�0 � 1¡� 1�0 � � 2�0 ! exp [��] and � = ¡ 1 2 �(� ¡ 1)e � 2 �
Thus, the CPPP method is parametrized by � and �.

Remark 3 Note that we have:

à � 0 ¡ �� 1�0 � 1¡� 1�0 � � 2�0 ! = µ � 0 �� 1�0 ¡ � (� 2�0 �� 1�0 ) � ¶ �
Additionally, e � is the volatility of the process � 2 �� 1 . Therefore, as for the OBPP case, by using asset � 1 as numeraire, the CPPP portfolio value can be expressed in terms of reserve asset � 1 as follows: 11

� �� � � � � 1�� = � + � � � (� 2�� �� 1�� ) � = � �� � � � (�� �� � = � 2 �� 1 � � 0 �� 1�0 � �� � = 0) �
3 Comparison between OBPP and CPPP at maturity

Comparison of the payo¤ functions

The OBPP has just one parameter, the participation coe¢cient �. In order to compare the two methods, rst the initial amounts � ��� � 0 and � �� � � 0 are assumed to be equal to the same initial invested amount � 0 . Secondly, the two strategies are assumed to provide the same participation coe¢cient �, which implies to choose the performance participation � of the CPPP strategy equal to ��. Hence, � � = ��� 1�� and then � 0 = ��� 1�0 . Note that these two conditions do not impose any constraint on the multiple, �. Therefore, we can consider several CPPP strategies for various values of the multiple �. [START_REF] Kraus | Theory of performance participation strategies[END_REF] Note that, due to the absence of arbitrage and since � ��� � 0 = � �� � � 0 = � 0 , neither of the two payo¤s is higher than the other for all terminal values of the risky asset. The two payo¤ functions intersect one another. We choose the following numerical example with typical values for a nancial market with a long term government bond as reserve asset and an equity index as active asset: [START_REF] Charlety-Lepers | Assurance et couverture de portefeuille, volatilité des prix et stabilité des marchés nanciers[END_REF] Recall that the CPPI strategy associated to the multiple � satises:

� 0�1 = � 0�2 = 100� � � 1 = 5%� � � 1 = 6%� � � 2 = 9%� � � 2 = 20%� � = ¡0�15� � = 1 year, � = 0�9� (11)
� �� � � � = ��0 + � �� � � � � � � �
where � is a guaranteed proportion of the initial invested amount �0 (see e.g. Bertrand and Prigent, 2005). The coe¢cient �� is given by:

� �� � � � = � �0 ¡ ��0� ¡�� � � � 0 � � � � � � � with � �� � � = ¡ 1 2 � (� ¡ 1) � 2 �
where � is the risk-free interest rate and � is the volatility of the risky asset �. We denote its current value by � �� � � (�� �� �� �0� �� �� �)

Figure 1 illustrates the OBPP payo¤ as functions of the two assets. We note that the payo¤ is convex and increasing with respect to the active asset. Figure 2 displays the CPPP payo¤ as functions of the two assets. As for the OBPP payo¤, the CPPP payo¤ is convex and increasing with respect to the active asset. We can check in this example that the two curves intersect one another for the di¤erent values of � considered (� = 2� � = 4� � = 6 and � = 8). CPPP performs better for large uctuations of the ratio � while OBPP performs better around the initial value of ratio � and for its moderate increases.

Comparison of the expectation, variance, skewness and kurtosis

In what follows, we examine the rst four moments of both strategies. Indeed, since their payo¤s are not linear with respect to the two assets, the standard mean-variance approach is not su¢ciently suitable due to signicant skewness and kurtosis. We can consider various values of the multiple � to compare the rst four moments. However, it is interesting to examine the special case corresponding to the equality of the expected returns. We determine the multiple corresponding to this additional constraint in the following subsection.

Equality of return expectations

Let us denote the rates of portfolio returns by � ��� � � and � �� � � � . Recall that we assume that

� �2 � � �1 .
Proposition 4 For any given participation coe¢cient �, there exists a unique value

� ¤ (�) of the multiple such that �[� ��� � � ] = �[� �� � � � ].
In the Black and Scholes framework, it is given by:

� ¤ (�) = 1 + à 1 ¡ � � 2 ¡ � � 1 ¢ � ! ln µ �(0� �� � 2 �� 1 � �� � = � � 2 ¡ � � 1 � e �) �(0� �� � 2 �� 1 � �� � = 0� e �) ¶ � ( 12
)
where �(0� �� �� �� �� �) denotes the Black-Scholes value of the call with underlying �, strike � , interest rate � and volatility �, evaluated at time 0 and with investment horizon � .

Note that, from relation 10, we deduce that coe¢cient � is increasing with respect to performance participation percentage �. Then, from relation 12, we deduce that the multiple � ¤ (�) is an increasing function of the performance participation coe¢cient �. Therefore, � ¤ is also increasing with respect to �, as shown in Figure 4.

�(0� �� � 2 �� 1 � �� � = � �2 ¡ � �1 � e �) and �(0� �� � 2 �� 1 � �� � = � �2 ¡ � �1 � e �)
are decreasing with respect to �. From proposition 4, as for the comparison of standard OBPI and CPPI, we can deduce that there exists at least one value for �, for example the value � ¤ , such that the OBPP strategy dominates in a mean-variance sense the CPPI.

Numerical comparison of the rst four moments

Using parameter values of the numerical base case given in relation 11, we provide the numerical values of the rst four moments for the equality of expected returns case in Table 1. We consider three values of the performance participation percentage �, namely � = 0�5� 0�7 and 0�9.

Table 1

Comparison of the rst four moments The OBPP dominates the CPPP in a mean-variance sense. However, the CPPP has a signicantly higher positive relative skewness than the OBPP. This is due to its more prominent convexity Hence with respect to this criterion, CPPP should be preferred to OBPP. Looking at relative kurtosis, it is the converse since the CPPP relative kurtosis is much higher than that of OBPP.

� = 0�5 � ¤ = 4�6 � = 0�7 � ¤ = 3�07 � = 0�9 � ¤ = 5�57
Remark 5 For a given performance participation percentage �, as soon as the multiple � is higher than � ¤ , both the expected return and variance of the CPPP strategy are higher than those of the OBPP strategy. Therefore, no strategy dominates the other with respect to the mean-variance criterion. If � � � ¤ , for a small di¤erence between � and � ¤ , the variance of the OBPP strategy remains smaller than that of the CPPP. Consequently, the OBPP strategy strictly dominates the CPPP strategy. For a su¢ciently large di¤erence � ¤ ¡ �, no strategy dominates the other one.

Comparison of quantiles

Due to the presence of asymmetry and fat tails in the probability distributions, it is more convenient to examine the whole distribution by comparing their cdfs. Figure 5 shows that the OBPP and CPPP cdf curves intersect, implying that there is no stochastic dominance at the rst order. To go further on the comparison of the two payo¤s, we investigate the probability distribution of the ratio OBPP value on the CPPP value. Note that using remarks 2 and 3, we deduce:

� ��� � � � �� � � � = � ��� � (�� �� � = 0� � = � 2 �� 1 � � 0 �� 1�0 � �) � �� � � (�� �� � = 0� � = � 2 �� 1 � � 0 �� 1�0 � �) �
Using parameter values of the numerical base case given in relation 11, Figure 6 displays the cumulative distribution function of

� ��� � � � �� � � �
for di¤erent values of the multiple, namely � = 2� 4� 6 and 8.

Figure 6 shows that the probability that the OBPP value is smaller than the CPPP value is minimal for the smallest multiple value but is not monotonic with respect to the CPPP multiple. However the higher the multiple, the smaller the probability that the OBPP value is signicantly higher than the CPPP value (ratio values higher than 10%).

Comparison of the dynamic behaviors of OBPP and CPPP

Even if the OBPP is mainly a buy-and-hold strategy (i.e. at initial date, the investor purchases the nancial assets once and for all), its optional component has to be dynamically hedged from the portfolio manager point of view. Additionally, as proved in what follows, the OBPP strategy can be considered as a generalized CPPP by means of a variable multiple which measures its risk exposure. In the Black and Scholes model, there exists a perfect hedging strategy and both performance participation methods can be analyzed and compared dynamically.

OBPP as a generalized CPPP

For the CPPP method, the multiple is the fundamental parameter since it determines the amount allocated in the active asset at any time. Therefore, from the theoretical point of view, it is interesting to express the OBPP as a generalized CPPP with a corresponding generalized multiple (at least in the Black and Scholes framework).

Proposition 6

The OBPP strategy is equivalent to an extension of the CPPP one, provided that the multiple satises

� ��� � (�� � � ) = (� 2�� �� 1�� ) � (� 1�� ) ���� (�� �� � 2�� �� 1�� � �� � = 0� e �) � (13)
Proof. Recall that:

� ��� � � = � ¡ �� 1�� + (� 2�� ¡ �� 1�� ) + ¢ Thus: � ��� � � = � (�� 1�� + ���� (�� �� � 2�� �� 1�� � �� � = 0� e �)) �
Therefore, the OBPP is always above the oor � � = ��� 1�� and �� 1�� ���� (�� �� � 2�� �� 1�� � �� � = 0� e �) corresponds to the cushion

� ��� � � = ¡ � ��� � � ¡ � � ¢ at any time � during the portfolio management period [0� � ].
By denition, this cushion is equal to

� ��� � � � ��� � � .
Here, the cushion is simply the call and the exposure is the total amount invested in the risky asset, equal to � 2�� �� ��� � � �� 2�� . We note that:

�� ��� � � �� 2�� = �� 1�� ����� (�� �� � 2�� �� 1�� � �� � = 0� e �) �� 2�� � which is equal to �� 2�� � (� 1�� ).
Finally, the desired result for the multiple is obtained:

� ��� � � = (� 2�� �� 1�� ) � (� 1�� ) ���� (�� �� � 2�� �� 1�� � �� � = 0� e �) �
Therefore, the OBPP multiple is a function of the ratio � of the two asset values [START_REF] Leland | The evolution of portfolio insurance[END_REF] . As illustrated by gure 7, this generalized multiple is a decreasing function with respect to the ratio �. Recall that the generalized multiple � ��� � � is the ratio of the exposure on the cushion � ��� � � . Therefore, it can be very high if either the cushion is too small, or the exposure is too high. In such a case, there exists a signicant gap risk if the market suddenly drops.

The Deltas

The deltas of the OBPP are obviously based on the delta of the call, which yields to:

¢ ��� � 1�� = �� ��� � � �� 1�� = �� [1 ¡ � (� 2�� )] � ¢ ��� � 2�� = �� ��� � � �� 2�� = �� (� 1�� ) �
For the CPPP, they are given by:

¢ �� � � 1�� = �� �� � � � �� 1�� = �� + � � (1 ¡ �) � ¡� 1�� � � 2�� = �� + � � (1 ¡ �) � � � � ¢ �� � � 2�� = �� �� � � � �� 2�� = � � �� 1¡� 1�� � �¡1 2�� = � � �� �¡1 � �
For the deltas ¢ 1�� , we note that ¢ ��� � 1�� is always positive but smaller than �� which is equal to the performance participation percentage � times the ratio of initial values � 0 �� 1�0 . For high values of the ratio process �, and high values of the multiple �, ¢ �� � � 1�� can be negative due to the leverage e¤ect on the reserve asset resulting from the strong convexity of CPPP value for high multiples (see Figure 8). For both the OBPP and CPPP strategies, the deltas ¢ 2�� are always positive (see similar results for the OBPI and CPPI strategies). Due to the strong convexity of CPPP value, ¢ �� � � 2�� can be signicantly higher than �, which is the upper bound of ¢ ��� � 2�� (see Figure 9).

The Gammas

The Gammas of the OBPP are given by:

¡ ��� � 1�� = � 2 � ��� � � �� 2 1�� = �� � 0 (� 2�� ) � 1�� e � p (� ¡ �) � 0� ¡ ��� � 2�� = � 2 � ��� � � �� 2 2�� = � � 0 (� 1�� ) � 2�� e � p (� ¡ �) � 0� e ¡ ��� � 1�2�� = � 2 � ��� � � �� 1�� �� 2�� = ¡� � 0 (� 1�� ) � 1�� e � p (� ¡ �) � 0�
The Gammas of the CPPP are equal to:

¡ �� � � 1�� = � 2 � �� � � � �� 2 1�� = � � � (� ¡ 1) � ¡(�+1) 1�� � � 2�� = � � � (� ¡ 1) � ¡1 1�� � � � � 0� ¡ �� � � 2�� = � 2 � �� � � � �� 2 2�� = � � � (� ¡ 1) � 1¡� 1�� � �¡2 2�� = � � � (� ¡ 1) � ¡1 1�� � �¡2 � � 0� e ¡ �� � � 1�2�� = � 2 � �� � � � �� 1�� �� 2�� = ¡� � � (� ¡ 1) � ¡� 1�� � �¡1 2�� = ¡� � � (� ¡ 1) � ¡1 1�� � �¡1 � � 0�
For both performance participation properties, the two derivatives at the rst order

� 2 � � �� 2 1�� and � 2 � � �� 2 2��
are positive while the cross-derivatives are negative. Note that both Hessians are null (see Appendix).

For the OBPP strategy, the sensitivities ¡ ��� � 1�� and ¡ ��� � 2�� are rst increasing then decreasing while, for the CPPP strategy, ¡ �� � � 1�� and ¡ �� � � 2�� are always increasing and can take high values when the ratio process � increases signicantly (see Figures 10 and11). This is still due to the strong convexity of the CPPP value. In what follows, we consider the sensitivity of the portfolio values to the volatility e � of the ratio process

� = � 2 �� 1 .
The Vega of the OBPP is dened as: 14 For the CPPP, the Vega is given by: 15

V ��� � � = �� ��� � � �e � = �� 2�� p (� ¡ �)� 0 (� 1�� ) �
V �� � � � = �� �� � � � �e � = ¡� (� ¡ 1) e ��� �� � � � �
Therefore, the sensitivity of the OBPP value with respect to the volatility e � is positive whereas, for the CPPP, it is negative since � � 1.

This means that an increase in volatility works against the strategy CPPP. It implies also that, if the ratio of the two assets � 2 �� 1 increases moderately while its volatility increases, then the OBPP strategy will be preferred to the CPPP one (see Figure 3). For the strategy CPPP to be the best, the increase in the ratio will have to be relatively large if its volatility increases, for example a large growth of the active asset and/or a signicant drop in the price of the reserve asset. This is similar to the OBPI versus CPPI case: as highlighted by Black and Rouhani (1989), when the volatility increases, the CPPI values decreases. Additionally, the higher the multiple, the more signicant the decrease. [START_REF] Margrabe | The value of an option to exchange one asset for another[END_REF] Recall that:

� �� � � � = ���1�� + ���� 1¡� 1�� � � 2�� � where � � = � �0 ¡ ���1�0 � 1¡� 1�0 � � 2�0 � exp [��] and � = ¡ 1 2 �(� ¡ 1)� � 2 �
Therefore, we get:

�� �� � � � �� � = � �0 ¡ ���1�0 � 1¡� 1�0 � � 2�0 � exp [��] � �� �� � = ¡�(� ¡ 1)� ��� �� � � � �

Conclusion

In this paper, we have analyzed and compared the two main performance participation methods, namely the OBPP and CPPP. We have shown that usual comparison criteria such as comparison of their payo¤s, of their cdfs with rst order stochastic dominance and of their rst four moments, do not allow a clear conclusion in favour of one of the two strategies. This is not so surprising since, as proved in the paper, their relative values expressed in terms of the reserve asset as numeraire correspond respectively to OBPI and CPPI strategies the comparison of which leads to similar conclusions. For example, when the ratio of the two processes is around or moderately higher than its initial value, the OBPP value is higher than the CPPP one (at least in the GBM framework). Otherwise, it is the converse. Note also that the OBPP is a buy-and-hold strategy while the CPPP is more exible but may bear transaction costs and is more sensitive to sudden drops. Further extensions could consider capped OBPP and CPPP, more general dynamics by using Monte Carlo simulations and also other decision criteria for absolute and relative loss control (see e.g. Mantilla-Garcia, 2014). The optimality of such products could be also examined (see e.g. Bajeux-Besnamou and Portait (1998) for the meanvariance case) and also according to various assumptions about the asset dynamics (see e.g. Bernard and Kwak, 2016). Finally, much exibility would be allowed for example by introducing a conditional multiple as in Ben Ameur and Prigent (2014).

6 Appendix

Determination of the CPPP value

The value at time � of the CPPP portfolio is given by:

�� �� � � � = (� �� � � � ¡ � � ) �� 1�� � 1�� + � � �� 2�� � 2�� � Recall that � �� � � � = � � + � � , � � = �� � and � � = �� 1�� .
Thus, the cushion value � must satisfy :

�� � = �(� �� � � � ¡ � � )� = (� �� � � � ¡ � � ) �� 1�� � 1�� + � � ��� � � ¡ �� � � = (� � + � � ¡ �� � ) �� 1�� �1�� + (�� � ) ��� �� ¡ �� � � = (� � ¡ �� � ) �� 1�� �1�� + (�� � ) �� 2�� �2�� �
Therefore, we get:

�� � � � = (1 ¡ �) �� 1�� � 1�� + � �� 2�� � 2�� �
from which we deduce:

� � = � 0 exp " ¡ (1 ¡ �)� � 1 + �� �2 ¡ 1�2 £ (1 ¡ �) 2 � 2 �1 + � 2 � 2 �2 + 2��(1 ¡ �)� � 1 � � 2 ¤¢ � +(1 ¡ �)� �1 � 1�� + �� �2 �� 1�� + �� �2 p 1 ¡ � 2 � 2�� # �
By using the relations:

� 1�� = � 1�0 exp • � �1 � 1�� + µ � � 1 ¡ 1 2 � 2 �1 ¶ � ¸� � 2�� = � 2�0 exp • � � 2 �� 1�� + � � 2 p 1 ¡ � 2 � 2�� + µ � � 2 ¡ 1 2 � 2 � 2 ¶ � ¸� we can deduce that: exp [� � 1 � 1�� ] = � 1�� � 1�0 exp • ¡ µ � � 1 ¡ 1 2 � 2 �1 ¶ � ¸� exp h � � 2 p 1 ¡ � 2 � 2�� + � � 2 �� 1�� i = � 2�� � 2�0 exp • ¡ µ � � 2 ¡ 1 2 � 2 �2 ¶ � ¸�
By substituting these expressions for � 1�� and � 2�� into previous expression for � � , we get:

� � = � 0 exp £¡ (1 ¡ �)� � 1 + �� �2 ¡ 1�2 £ (1 ¡ �) 2 � 2 � 1 + � 2 � 2 � 2 + 2��(1 ¡ �)� � 1 � � 2 ¤¢ � ¤ £ exp [� � 1 � 1�� ] (1¡�) exp h � � 2 p 1 ¡ � 2 � 2�� + � � 2 �� 1�� i � � � � = � 0 exp £¡ (1 ¡ �)� � 1 + �� �2 ¡ 1�2 £ (1 ¡ �) 2 � 2 � 1 + � 2 � 2 � 2 + 2��(1 ¡ �)� � 1 � � 2 ¤¢ � ¤ exp • ¡(1 ¡ �) µ � � 1 ¡ 1 2 � 2 � 1 ¶ � ¸exp • ¡� µ � � 2 ¡ 1 2 � 2 � 2 ¶ � ¸µ � 1�� � 1�0 ¶ [1¡�] µ � 2�� � 2�0 ¶ � �
Therefore, we deduce:

� � = � 0 µ � 1�� � 1�0 ¶ [1¡�] µ � 2�� � 2�0 ¶ � £ exp "à ¡1�2 £ (1 ¡ �) 2 � 2 �1 + � 2 � 2 �2 + 2��(1 ¡ �)� � 1 � � 2 ¤ +(1 ¡ �) ¡ 1 2 � 2 � 1 ¢ + � ¡ 1 2 � 2 � 2 ¢ ! � # � = � � �� (1¡�) 1�� � � 2�� = � � �� 1�� µ � 2�� � 1�� ¶ � �
where

� � = � 0 µ 1 � 1�0 ¶ [1¡�] µ 1 � 2�0 ¶ � exp [��] � and � = ¡1�2 £ (1 ¡ �) 2 � 2 � 1 + � 2 � 2 � 2 + 2��(1 ¡ �)� � 1 � � 2 ¤ + (1 ¡ �) µ 1 2 � 2 � 1 ¶ + � µ 1 2 � 2 � 2 ¶ � = ¡1�2 ££ (1 ¡ �) 2 ¡ (1 ¡ �) ¤ � 2 � 1 + £ � 2 ¡ � ¤ � 2 � 2 + 2��(1 ¡ �)� �1 � �2 ¤ � = ¡1�2�(� ¡ 1)e � 2 �
Therefore, the portfolio value is given by:

� �� � � � = �� 1�� + � � �� (

Equality of expectations

Recall that:

� ��� � � = � [�� 1�� + � ��� (�� �� � 2 � �� 1 )] � (14)
with

� = � 0 �� 1�0 + � ��� (0� �� � 2 � �� 1 )
�

where the value � ��� (0� �� � 2 � �� 1 ) is equal to:

� ��� (0� �� � 2 � �� 1 ) = � 1 ���� (�� �� � 2 �� 1 � �� � = 0� e �) � (15)
Recall also that (� = ��)

� �� � � � = ��� 1�� + � � �� 1¡� 1�� � � 2�� �

22

where

� � = à � 0 ¡ ��� 1�0 � 1¡� 1�0 � � 2�0 ! exp [�� ] and � = ¡ 1 2 �(� ¡ 1)e � 2 �
We are looking for the value of the multiple � such that the expected portfolio values are equal:

�[��� 1�� + � (� 2�� ¡ �� 1�� ) + ] = �[��� 1�� + � � �� (1¡�) 1�� � � 2�� ]�
We get the equivalent condition:

�� £ (� 2�� ¡ �� 1�� ) + ¤ = � � � h � (1¡�) 1�� � � 2�� i � �� £ (� 2�� ¡ �� 1�� ) + ¤ = (� 0 ¡ ��� 1�0 ) exp [�� ] � 2
£ ¡�� + � � 1 � + � ¡ � � 2 ¡ � � 1 ¢ � ¤ ; (2)
(� 0 ¡ ��� 1�0 ) = �� 1 ���� (�� �� � 2 �� 1 � �� � = 0� e �) ; (3)
� £ (� 2�� ¡ �� 1�� ) + ¤ = exp £ � �2 � ¤ �(0� �� � 2 �� 1 � �� � = � �2 ¡ � �1 � e �)�
Therefore, we deduce:

�� 1 exp £ � �2 � ¤ �(0� �� � 2 �� 1 � �� � = � �2 ¡� �1 � e �) = �� 1 ���� (�� �� � 2 �� 1 � �� � = 0� e �) exp £ � �1 � + � ¡ � �2 ¡ � �
Recall that we assume � � 1 � � � 2 . Finally, we obtain the value of the multiple � as a function � ¤ (�) of the performance participation �:

� ¤ (�) = 1 + à 1 ¡ � �2 ¡ � �1 ¢ � ! ln µ �(0� �� � 2 �� 1 � �� � = � �2 ¡ � �1 � e �) �(0� �� � 2 �� 1 � �� � = 0� e �) ¶ �

Hessians of the two portfolio values

For the OBPP strategy, the Hessian � (which corresponds to the mathematical determinant of the matrix of derivatives at the second order) is given by:

� ��� � = � 2 � ��� � � �� 2 1�� £ � 2 � ��� � � �� 2 2�� ¡ µ � 2 � ��� � � �� 1�� �� 2�� ¶ 2 = � 2 � 0 (� 1�� ) e � 2 (� ¡ �) � 1�� µ � � 0 (� 2�� ) � 2�� ¡ � 0 (� 1�� ) � 1�� ¶ � Since we have � 0 (� 1�� ) � 0 (�2��) = �� 1��
� 2�� , we deduce that the Hessian of the OBPP is null. For the CPPP, the Hessian � is equal to:

� �� � � = � 2 � �� � � � �� 2 1�� £ � 2 � �� � � � �� 2 2�� ¡ µ � 2 � �� � � � �� 1�� �� 2�� ¶ 2 = � 2 � � 2 (� ¡ 1) 2 ³ � ¡1 1�� � � � � ¡1 1�� � �¡2 � ¡ � ¡2 1�� � 2�¡2 � = � 2 � � 2 (� ¡ 1) 2 � ¡2 1�� ¡ � � � � �¡2 � ¡ � 2�¡2 � ¢ = 0

  Since � (�) = 1¡� � is strictly decreasing and bijective from ]0� 1[ to ]0� +1[ and � (�) = ����(��� ��2��1����=0�� �) � is strictly decreasing and bijective from ]0� +1[ to ]0� +1[, there exists one and only one solution � ¤ .

Figure 1 :

 1 Figure 1: OBPP return as function of the two assets � � �1 and � � �2

Figure 2 :Figure 3 Figure 3 :

 233 Figure 2: CPPP return as function of the two assets � � �1 and � ��2

Figure 4 :

 4 Figure 4: Multiple of the CPPP strategy such that its mean is equal to that of the OBPP. Illustration as function of the performance participation percentage �

Figure 5 :Figure 6 :

 56 Figure 5: Cumulative distribution functions

Figure 7 :

 7 Figure 7: OBBP multiple as function of the ratio of the two assets (at t=T/2).

Figure 8 :Figure 9 :

 89 Figure 8: Comparison of Delta 1 as function of the ratio of the two assets �.

Figure 10 :Figure 11 :

 1011 Figure 10: Gamma 1 (times S1) as function of the ratio of the two assets

14Figure 12 :

 12 Figure 12: Vega in terms of the reserve asset as function of the ratio of the two assets

  Note that we have:� �� � � � = � 1�� � �� � � � (�� �� � = 0� � = � 2 �� 1 ).

In France, several nancial structured products have such implicit "capital guarantee". However, the "Autorité des Marchés Financiers" (AMF) does no longer consider them as portfolio insurance but rather prefers to label them "fonds à formule" as those which have not such property.

See Kraus et al (2010) for the study of a similar model but without taking explicitly the budget constraint into account.

See Black and Perold (1992) for a similar result. See also Bajeux-Besnamou and Portait (1997) for another illustration of the use of a numeraire.

See subsection 2.1.2.

Note that the multiple must not be too high as shown for example in[START_REF] Prigent | Assurance du portefeuille : analyse et extension de la méthode du coussin[END_REF] or in[START_REF] Bertrand | Portfolio insurance strategies : OBPI versus CPPI[END_REF].

Such more general multiples have been introduced and studied in[START_REF] Prigent | Assurance du portefeuille : analyse et extension de la méthode du coussin[END_REF].

This work was supported by French National Research Agency Grant ANR-17-EURE-0020.