Measuring Idiosyncratic Risks in Leveraged Buyout Transactions

Measuring Idiosyncratic Risks in Leveraged Buyout
Transactions


Feb. 2007

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Measuring Idiosyncratic Risks in Leveraged Buyout
Transactions

Abstract:
We use a contingent claims analysis model to calculate the idiosyncratic risk in leveraged buyout
transactions. A decisive feature of the model is the consideration of amortization. From the
model, asset value volatility and equity value volatility can be derived via a numerical procedure.
For a sample of 40 leveraged buyout transactions we determine the necessary model
parameters and calculate the implied idiosyncratic risks. We verify the expected model
sensitivities by varying the input parameters. From the reported returns to the equity investors
we calculate Sharpe Ratios for individual transaction levels, thereby fully incorporating the
superimposed leverage risks.

JEL classifications: G13, G24, G32
Keywords: Idiosyncratic Risk, LBO, Private Equity, Benchmarking, CCA

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1. Introduction
Leveraged Buyouts (LBOs) are transactions in which a financial investor takes over a
company via a special purpose vehicle. The funding of the special purpose vehicle is typically
composed of several layers of debt and non-traded equity claims. In most of the cases the
debt/equity ratios of LBO transactions are above what is considered normal. These two
properties - the illiquidity of the private equity market and the leverage ratio make LBOs high-risk
investments. No adequate methods have been developed so far to successfully determine the
risks implied by particular transactions; consequently, risk-adjusted performance ratios that
benchmark individual transactions are non-existent. In this paper we use contingent claims
analysis (CCA) to provide a measure of inherent risks in LBO transactions. Typical LBO deal
structures mean, however, that standard CCA models cannot be used, and so adjustments need
to be made.
Green (1984), Sick (1989), and Arzac (1996) point out that when debt/equity ratios in LBO
transactions are high, the similarity of the equity valuation to a call option valuation becomes
obvious. Shareholders can exercise the option, take over the company and redeem the debt if
the enterprise value is above the liabilities. Default is triggered if the company value reaches the
level of liabilities. Leaving aside agency costs, debt financing thus induces more speculative
behavior of the shareholders, according to Myers (1977), because they have unlimited earnings
potential but only limited risk. This makes a CCA-approach a promising method for estimating
the implied volatilities of transactions in which shareholders and lenders bear a certain amount
of idiosyncratic risk.
Underlying the CCA approach is the assumption, that the arrangers of a transaction will
model future company development and potential scenarios. In accordance with their models,
they agree a purchase price with the seller and structure the transaction with several equity and
debt layers. At this moment, prices for the different equity and debt strips are fixed. The LBO

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sponsors have a finite investment horizon and plan to sell their stakes after a certain holding
period. The holding period, which is reflected in their transaction model, determines a sponsor’s
expected internal rate of return on the investment. The transaction model is based on free cash
flow estimates and considers due redemption of the debt and adherence to the debt covenants.
Lenders usually ask for an immediate reduction of their exposure in large scales after the closing
of a LBO transaction. Hence, an appropriate leverage ratio must be applied that secures the
desired return on equity and that fulfills the lender requirements regarding redemption and
covenants. In a simplified model, where there is just a single layer of common equity and a
single layer of risky debt with one amortization payment, the CCA approach determines both, the
transaction’s and the equity’s idiosyncratic risks. These risks depend on the deployed capital
structure at closing, the planned investment horizon, the assumed debt redemption capabilities
of the target company, the risk-free interest rate, and the debt-credit spread. In other words, for a
given transaction, the arrangers assume a certain volatility in the target company’s asset and
equity values when they agree on debt and equity prices, bearing in mind the amortization
schedule over the investment horizon – which is the moment of closing the transaction.
To a manually collected sample of 40 LBO transactions we apply the Ho and Singer
(1984) model in order to price risky corporate debt (with one amortization payment). We
implement a numerical approach to calculating the implied asset value volatilities, which range
from 13.6% to 106.4% p.a. (mean 35.3% and median 30.4% p.a.). From the asset value
volatilities we derive the idiosyncratic equity risks for the same transactions, obtaining high
values ranging from 57.8% to 182.5% p.a. (mean 94.1% and median 93.5% p.a.). This is the
first time that idiosyncratic risks are calculated for individual LBO transactions. Using data on the
returns received by the LBO sponsors, we can – also for the first time - calculate Sharpe ratios
for the individual LBO transactions.
It is not the purpose of this paper to present a mean idiosyncratic risk for the asset class,
but to propose an approach to calculate risk-adjusted returns, bearing in mind different degrees

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of leverage in the transactions. The model can be applied to the benchmarking of current and
future transactions, which will lead to improved understanding of, and transparency in, the asset
class. Comprehensive analyses of the mean risk-adjusted return of the asset class wil be
possible with the presented approach if more data on individual transactions becomes available
for academic research.
2. Literature
Overview
2.1.
Literature on Financial Risk Measures for the Private Equity
Market Segment
Several papers deal with associated risks in private equity markets, but do not usually
differentiate between different market segments, such as venture capital (VC) and LBOs, and do
not pay sufficient attention to their particularities, especially when referring to LBO transactions.
For instance, Cochrane (2005) reports a mean volatility of 86% p.a. for a sample of 16,638
private equity transactions – calculated via maximum likelihood estimates and sample bias
correction for unobservable returns but does not differentiate between VC and LBOs; more
importantly, he does take account of the degrees of leverage deployed in the LBOs. Although
Kaplan and Schoar (2005) analyze the performance of private equity investments and create a
sub sample of LBOs, rather than consider idiosyncratic risk, they consider systematic risk -
which they assume to be both equal for every transaction, and equal to the systematic risk of the
S&P 500 Index. Their approach also implies, therefore, that leverage in public and private
markets is identical. Quigley and Woodward (2003) create a VC index similar to that of Peng
(2001), and report a mean annual standard deviation of 14.6% of the asset class, while Peng
(2001) reports annual standard deviations between 9.5% and 70.3% for the period from 1987 to
1999. Both papers focus on correcting missing values and selection bias and fail to either create
a LBO sub-sample or to consider individual LBO capital structures in their approach. Ljungqvist

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and Richardson (2003) distinguish between VC and LBO market segments and analyze LBO
performance while controlling for systematic risk. However, because they do not have access to
exact data for individual deals, they assume industry averages for the debt/equity in their
calculations of LBO beta factors. These authors report an average beta factor of 1.08 for their
LBO sample. Groh and Gottschalg (2006) investigate LBO performance and focus on systematic
risk for transactions using detailed information on debt/equity ratios. For several scenarios based
on differentiated assumptions about the risk of debt and of debt tax shields, they calculate
average equity beta factors ranging from 0.78 to 2.57 at transaction closing. However, it is not
clear which scenario is the “right” one. Furthermore, their focus on systematic risk does not
enable individual transactions or undiversified portfolios to be benchmarked.
This paper differs from the existing literature regarding two important aspects. First of all,
we focus on idiosyncratic risk and exclusively on LBO transactions. Secondly, our proposed
model is derived from CCA, adapted for the characteristics of LBOs and transferred to the asset
class. This represents a unique and promising approach to the calculation of implied transaction
risks that account for the risk superimposed by the debt deployed to financing the transaction.
This approach will enable comprehensive risk-adjusted performance analyses to be performed
when more data (e.g. on current and future transactions) become available.
The next section describes the origin of our CCA model and related literature.
2.2.
The Origin of our CCA Model
Black and Scholes (1972) and Merton (1973 and 1974) devise models to calculate the
equity value of a company, given the value - and fluctuations in value – of its assets, and
including the notion of a pure discount bond, time to maturity of the bond, and the risk-free
interest rate. Assuming that all the other parameters are given, the models can be solved
numerically for the asset value fluctuation. This asset value fluctuation is called implied volatility,
and represents a measure of the expected fluctuation in company value. It can be assumed that

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originators of a LBO transaction set a price for equity according to that risk. All the other
parameters are usually known on closing a LBO transaction and the Black and Scholes
approach can be directly extended to calculating the implied volatility of a LBO if this is financed
with default-free zero bonds.
However, since LBO transactions are typically characterized by a large proportion of debt
at closing and substantial subsequent debt redemptions, decreasing debt levels have to be
taken into account in the CCA valuation model. Furthermore, some deployed debt instruments
allow flexible amortization payments. This complicates the CCA approach because the exercise
price of the option is uncertain, as discussed in Fisher (1978). Moreover, standard boundary
conditions are no longer valid and interest due dates convert the approach into a compound
option, namely, a path-dependent valuation problem. Black and Cox (1976), and Ho and Singer
(1982) expand the Merton (1974) pricing model and introduce several bond indenture
provisions, such as safety covenants, subordination arrangements, and restrictions on the
financing of interest and dividend payments.
Geske (1977 and 1979), and Brockman and Turtle (2003) deal with the path-dependent
option problem. Unfortunately, the models cited are not suitable for our purposes, both because
they include many parameters that are not observable in our sample data and because they
require assumptions which cannot be taken as given in LBO transactions. The Geske (1977)
approach, for example, requires equal sinking fund payments for each coupon date in order to
completely retire the face value of the debt by maturity. In LBO transactions debt is not usually
fully redeemed within the financial investor’s planned holding period.
Jones, Mason, and Rosenfeld (1983 and 1984) formulate a partial differential equation and
boundary conditions that price risky bonds with sinking fund provisions. They empirically test
their model and conclude that their approach is more appropriate than a naive model based on
the assumption of risk-less debt, especially when valuing low grade bonds issued by companies
with high debt/equity ratios.

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Based on the Cox-Ross-Rubinstein (1979) binomial model, Ho and Singer (1984) derive a
closed form solution to pricing a risky coupon bond with a single redemption payment either
made by open market repurchase of the bond or by calling the bond at par value. In LBO
transactions debt is usually redeemed according to an agreed schedule. Payments are made at
discrete points and typically in different amounts. The Ho and Singer (1984) model does not
reflect different kinds of amortization schedules, but reduces the complexity by assuming a
single payment in the lifetime of the bond. However, this model perfectly matches one important
typical feature of LBOs, by allowing for a bullet payment to redeem the outstanding principal at
maturity. We adapt this model to LBO transactions and, from the observable price of the debt
deployed in the transactions, conclude the implied volatility of the target company’s asset value.
From the asset value volatility we can derive the subsequent implied equity risk.
We describe the Ho and Singer (1984) model and the assumptions necessary to be able
to transfer it to our LBO sample in the next section.
3. The LBO CCA model
The approach refers to Black and Scholes (1973), who emphasize that common stock of a
corporation that has outstanding coupon bonds can be considered as a compound option. On
each coupon date, shareholders have the option of buying the next option by paying the coupon,
or of forfeiting the firm to the bondholders. Their final option is to repurchase lender claims by
paying off the principal at maturity. If bond indentures require amortizations the shareholders
have the additional options of either buying the next option at the redemption date or of forfeiting
the company. To our knowledge, the literature does not provide a closed form solution for
valuing risky coupon debt redeemed at discrete points in time (possibly in different amounts) for
a non-zero principal at maturity. However, the Ho and Singer (1984) model comes close to that
desired feature. The authors provide a closed form solution for a discount bond with one

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amortization payment and a final payoff at maturity. The modifications necessary for transfer of
this model to typical LBO transactions are as follows:1
1. The value of the firm’s assets is independent of its capital structure.
2. The firm’s capital structure consists of a single equity and a single debt layer.
3. The yield curve is flat and non-stochastic.
4. Until the maturity of the debt, the firm’s investment decisions are known.
5. The firm does not pay dividends and does not make any other distributions to
shareholders.
6. Default occurs when the firm fails to satisfy the bond indentures. If defaulted, the
shareholders forfeit the assets to the lenders costlessly.
7. The amortization payment is financed with new equity.
8. The amortization payment is fixed in the bond indentures as a proportion of the
debt outstanding at a given time.
We employ the following notation:
V(t)
is the value of the firm at t, when the amortization payment falls due.
F
is the value of outstanding debt at the exit date of the LBO transaction.
sF
is the value of debt to be amortized as a proportion of the value of outstanding debt
at the exit date of the LBO transaction.
T
is the holding period of the LBO transaction.
τ
is the time from the amortization payment to the exit date, i.e. τ = T – t.

1 Compare to Ho and Singer (1984) p. 317.

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B[V(t);F,τ] is the value of the debt remaining after the amortization, i.e. the value of a discount
bond with future Value F and time to maturity τ.
c
is the constant yield of the debt over the holding period.
sFe-cτ
is the amortization payment.
According to Merton (1974), the value of the company’s equity just after the amortization
payment can be expressed as the value of a call option on the underlying value of the firm, with
exercise price F and time to expiration τ. Default is triggered when the asst value becomes
smaller than the amortization payment. It is further assumed that trading can take place
continuously and that the value of the company’s assets follows an Itô diffusion process
dV / V = α dt + σ dw, where α is the constant instantaneous expected growth of the asset value
of the firm, σ is the instantaneous standard deviation, and dw is the increment of a standard
Gauss-Wiener process. The standard deviation of the underlying assets σ is the parameter in
question, which is finally solved numerically. Ho and Singer (1984) formulate a solution that can
be transferred for valuing the debt of a company on closing a LBO transaction, using our
notation and the necessary assumptions:
∞

D = VN (x −
t )
−rt
+ e
f x B x dx + N − x sFe− −
b
( ) ( )
(
∫
τ
σ
, (1)
b )
c
rt
xb
where xb is the solution of
sFe−cτ + B(xb )
ln
rt
1

Ve
x =
+ σ t , (2)
b
σ t
2
with
− 1σ 2t σ
+
tx +rt

B(x) = N (y −σ t )Ve 2
+ N(− y) − τr
Fe
(3)
and

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