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Estimating Economic Capital for Private Equity Portfolios
Mark Johnston, Macquarie Group
22 September, 2008
• What is private equity and how is it different to public equity and credit?
• How much capital does it require?
• Modelling capital requirements for a diversified equity portfolio
• Estimating parameters for unlisted positions
• Results
Today’s presentation
What is Private Equity?Private equity investment involves taking a stake in unlisted equity; for example: – Leveraged Buyouts (LBO)– Venture Capital– Infrastructure Projects – Unlisted Managed Funds
Significant increase in global private equity investment in recent years:– US$800b (2006) versus US$125b (2000) in equity and debt*
No well-established method for estimating the capital requirement for private equity portfolios
A variety of regulatory treatments are available under Basel 2: Deductions, Simple RWA, internal models, PD/LGD.
*Reserve Bank of Australia, Financial Stability Review, Private Equity in Australia, March 2007, p. 60.
Macquarie’s equity investment portfolioPortfolio Risk by Industry
Property26%
Commercial Services & Supplies
1%
Infrastructure16%
Transportation11%
Energy8%
Materials9%
Consumer Services8%
Health Care3%
Semiconductors2%Media
3%Telecommunication
2%
Miscellaneous11%
Portfolio Risk by Region
Africa & Middle East3%
Australia & New Zealand
18%
Europe & UK11%
MIX21%
North America28%
Asia17%
South America2%
As at 30 September 2007
What is an appropriate economic capital ratio for a private equity portfolio?
What is an appropriate economic capital ratio for a private equity portfolio?
• Some benchmarks (99.9% confidence level, 1 year time horizon):
– Basel 2 simple risk weight method: 32% for non-publicly traded equity holdings
– Basel 2 PD/LGD approach*: 34% for a portfolio of B-rated LBO equity holdings
– Banks (differing portfolios):31% - 68% – Solvency 2 standards for EU life
insurers: 54%– Statistical analysis of annual
movements in the S&P500: 40% -55%
*LGD: 90%, PD: satisfy same requirements as if holding was debt, Maturity: 5 years
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.40.0
0.5
1.0
1.5
2.0
2.5
3.0-0.51-0.37
Simple return of annual average
Prob
abili
tyde
nsity
S&P 500 - Distribution of Simple Annual Returns
Equity Risk vs. Credit RiskAny decline in asset value results in a decline in equity value
Debt value begins to decline materially only after a significant deterioration in asset value
Question appropriateness of applying the Basel 2 credit model directly to equity portfolios
State Credit Loss Equity Loss
Default Loss Given Default
Close to 100%
No Default 0% Some
Classical Portfolio Model
Applying to private equity we have a couple of problems:
– Can’t observe the inherent risks
– Can’t observe the correlations
Need to use proxy data or subjective estimates
Correlation Matrix, ΛWeighted Inherent Risks, σ
σσσ ..Λ= Tportfolio
Intuition for the large portfolio case
Applies to a large homogeneous portfolio of long positions
σ is the weighted average Inherent Risk
ρ is the average correlation between positions
Ignores: value of the put option and discounting see later
Approximate estimate for portfolio economic capital
Z1-α is a distributional multiplier taking us to the α-quantile of portfolio loss (eg. for the Normal distribution Z1-α = 3.09 at 99.9% confidence)
ρσα ⋅⋅≈ −1ZK
Inherent Risk Factor
Diversification Factor
ρσσ ⋅≈portfolio
Ways of characterising Inherent Risk
By Inherent Risk we mean the risk of an individual position on a stand-alone basis
We will focus on the Probability of Total Loss, which is defined as the probability of losing the entire equity stake in a position
Probability of Total Loss can be assigned via a rating scale, for example a credit rating scale (for today’s presentation we will use the S&P credit ratings scale with idealised default probabilities)
Characterisation of Inherent Risk Pros and Cons
Equity volatility OK, but then need a distributional assumption to take it to the “tail” (99.9% confidence level)
Reasonable Maximum Loss
Intuitive, and already a “tail” figure, but:• Confidence level not explicit• Loses discriminatory power at high confidence levels (at 99.9% confidence, Reasonable Maximum Loss should be 100% for most equity positions)
Probability of Total Loss
Already a “tail” figure, and retains discriminatory power, but not intuitive so would need to be implemented via a “ratings scale”
Variation of correlations with industry
Correlation of 1200+ individual listed and rated equity positions with a broad market index (S&P Global 1200) over an 18 year period (Source: Bloomberg, Macquarie analysis)
Empirical Asset-Market Correlations
0%
10%
20%
30%
40%
50%
60%
Utilitie
s
Real E
state
Inves
tmen
t Trus
ts
Health
Care
Equ
ipmen
t & S
ervice
s
Food
Bev
erage
& Tob
acco
House
hold
& Pers
onal
Produc
tsInf
rastru
cture
Commerc
ial S
ervice
s & S
uppli
esTr
ansp
ortati
on
Pharm
aceu
ticals
, Biot
echn
ology
Energy
Real E
state
Manag
emen
t & D
evelo
pmen
t
Food &
Stap
les R
etailin
g
Telec
ommun
icatio
n Serv
ices
Consu
mer Serv
ices
Retailin
gMate
rials
Miscell
aneo
us
Automob
iles &
Com
pone
nts
Software
& S
ervice
sBan
ks
Semico
nduc
tors &
Sem
icond
uctor
Consu
mer Dura
bles &
App
arel
Media
Tech
nolog
y Hard
ware &
Equ
ipmen
tCap
ital G
oods
Insura
nce
Diversi
fied F
inanc
ials
Cor
rela
tion
Variation of correlations with size & rating
AAAABBBBBCCC+
0.0
0.2
0.4
0.6
AAAABBBBBCCC+
Distance to Default
Corr
elat
ion
Distribution of Asset-Market Correlation by Rating
0.0
0.2
0.4
0.6
Log of Market Capitalisation HAU$mL
Corr
elat
ion
Distribution of Asset-Market Correlation by Size
Variation of correlations with time
Correlations vary with time, however even in periods of stress remain below 100%Correlation of 5000+ individual global large and small cap equities with the S&P 500 Index
Annual correlations with the S&P 500 Index over time
0%
10%
20%
30%
40%
50%
60%
1980 1985 1990 1995 2000 2005Year
Cor
rela
tion
Annual Average (S&P 500) All Years Average (S&P 500) 75th Percentile (S&P 500)
1990 Nikkei Crash
1987 Wall Street Crash
The story so far: rough answer
The story so far: rough answer
AAA4.59
AA+4.22
AA3.79
AA-3.63
A+3.57
A3.52
A-3.41
BBB+2.89
BBB2.75
BBB-2.41
BB+2.17
BB2.00
BB-1.82
B+1.74
B1.62
0.00
0.05
0.10
0.15
RatingDis tance to Default
%of
Inde
xby
Mkt
Cap
Distribution of S&P 500 by Rating
45.0
48.028.309.3
≈
⋅≈
⋅≈
⋅⋅≈
ρ
ρσ
α
α
DDZZK
Distance to Default
Ignores: value of the put option, discounting & equity recovery at default see later
Approximate capital ratio for an equityportfolio (e.g. S&P 500):
Modelling the “insolvency put option”• Merton Model: equity is a geared asset
position plus an ‘insolvency put option’
• Put option ensures that equity holders can't lose more than they put in
• Loss on any equity exposure is capped at 100%
– …but the probability of losing 100% is greater for more volatile assets
– More volatile assets make a bigger contribution to portfolio risk
• Measure Inherent Risk by Probability of Total Loss
Equity Payoff
Debt Payoff
Face Value of Debt
Asset Value at End of Period
BB-RatedS&P Idealised PD = 2.30%
More volatile assets make a bigger contribution to portfolio risk, despite loss being capped
B
BBB
Under Basel 2 assumptions we derive an analytical formula for the equity portfolio loss
jjjj UXY ⋅−+⋅= ρρ 1
Single Factor model
Portfolio risk contribution of an equity position at α-quantile
From this we can compute the unexpected loss ratio
Combine with valuation model to produce a current capital requirement
[ ]( )[ ]jjjjjj
jj UXdE
dX
dE ρρ
ρααα −−−−++= 1,0Max11,
Ignoring Put Option HNormalLWith Put Option
0.0 0.5 1.0 1.50.0
0.5
1.0
1.5
0.04 0.30
0.31
Theoretical Quantile
Sim
ulat
edQ
uant
ile
QQ-Plot
Capital ratio by industry and ratingApplying the analytical formula to the S&P500 gives a capital ratio of 42% (cf. 45% under classical portfolio model approximation)
Using S&P idealised PD’s; assuming no equity recovery at default
AAA AA A BBB BB B CCCUtilities 0.19 0.27 0.28 0.32 0.37 0.41 0.54Infrastructure 0.23 0.31 0.32 0.37 0.43 0.47 0.60Transportation 0.27 0.37 0.38 0.44 0.51 0.54 0.68Materials 0.30 0.42 0.43 0.49 0.57 0.61 0.74Miscellaneous 0.30 0.40 0.42 0.48 0.55 0.59 0.72Banks 0.32 0.42 0.44 0.50 0.57 0.61 0.74Media 0.33 0.46 0.48 0.55 0.63 0.67 0.78Diversified Financials 0.43 0.57 0.59 0.66 0.74 0.78 0.8795% Asset−Asset Correlation 0.63 0.78 0.84 0.99 1.00 1.00 1.00
LBO PortfolioBBB infrastructure investment
First-loss piece of securitisationB-rated media firm
Summary and further issuesEquity risk and credit risk require different economic capital treatments
An analytical formula for the equity portfolio distribution can be derived under Basel 2 assumptions
Inherent Risk can be characterised by Probability of Total Loss, mapped from a rating scale
Correlations exhibit variation by industry, rating and over time
Equity positions require more capital than credit positions, but substantially less than 100%
• Further Issues– Unrealised gains / ’latent gains’– Economic Capital Ratio vs. Unexpected Loss Ratio– Required Capital vs. Required Return– Equity recovered at default– Single-name and sector concentration– Non-normal distributions / non-normal dependence– Inter-risk diversification (or lack thereof)– Time Horizon