Risk Investment(1):ValuationFramework
keymaterial
A. and Return
Systematic/Specific risk: diversification
Asset pricing models: CAPM / APT / Empirical CAPM
B. Capital Budgeting
Principle
Risk measurement
Value drivers
FMP 1 - 1
Financial Management and Policy
A.Riskandreturnrelationship
A1. Systematic versus specific risk : diversification
Question: how should risk be measured ?
Naive answer: the variance of each security
it captures deviations from expectations
it is a measure for a single security
it is relevant for the Normal distribution
Example:
E[RA ] = EA = 0.13 ; 2A = 0.04
Investments
Risk
Returns
variance gives total risk
FMP 1 - 2
FIRM
Financial Management and Policy
Smarter answer...: the contribution of each security to the
variance of the most diversified portfolio
it still captures a deviation from expectations
it gives a measure related to the market
it is relevant for the multivariate Normal distribution
Example:
E[Ri ] = Ei = 0.13 ; 2i = 2j = 0.04, ij = 0.3
with two securities, equal holdings:
2p = E[(0.5 (Ri-Ei) + 0.5 (Rj-Ej))2]
= 0.52 2i + 0.52 2j
+2 (0.5) (0.5) E[(Ri-Ei) (Rj-Ej)]
= 2 x 0.25 2i + 2 x 0.25 x ij i j
= 0.026 !!
FMP 1 - 3
Financial Management and Policy
with N securities, same principle:
2p = 1/N 2i + (1-1/N) ij 2i ij 2i as N
Thus, the portfolio variance can be reduced to 0.3 x 0.04
= 0.12 simply because of
diversification
2 i
ij 2i
1
FMP 1 - 4
N
Financial Management and Policy
A2. Asset Pricing Models: CAPM / APT /Multi-factor
Question: how should risk and return be matched ?
Systematic risk should be priced only
Investors can mix risky assets with the riskless asset
Diversification reduces risk: everybody should hold the most
diversified portfolio
FMP 1 - 5
Financial Management and Policy
1. CAPM
start from: RF , RA, RB
investors try to maximize expected return and minimize risk
Ei
CML
m
*
RF
A
B
All choose the same portfolio (m) and mix it with the riskless asset
The market portfolio is the most diversified
Risk is measured by i = cov( Ri , Rm )
2
m
FMP 1 - 6
Financial Management and Policy
Risk premium should reward systematic risk (SML)
Example:
Ei = RF + i [ Em RF ]
Market with three securities A, B and M
EA = 0.14 ; 2A = 0.04
EB = 0.1 ; 2B = 0.02
EM = 0.13; 2M = 0.0225
RF = 0.07
imagine the investor wants a return of 10%
buy security B, whose beta is 0.5 ? no (exposed to total risk)
should diversify first ! Buy 0.5 M and 0.5 F
then, holds security B through the holding of M
FMP 1 - 7
Financial Management and Policy
2. Arbitrage Pricing Theory
Starts from the difficulty to identify M; rather, start from a
simple equation :
Ri = expected part + surprise
= Ei + U
= Ei + m +
with
is the specific risk, uncorrelated from one firm
to another:
m is the systematic, market-related risk. It is made of several
independent factors (inflation, GDP, interest rates...) and each firm has
a different responsiveness coefficient wrt each of them:
Ri = E i + i , INFL FINFL + i ,GDP FGDP + i , r Fr + ... +
with E ( F j ) = 0 as factors represent surprises
FMP 1 - 8
Financial Management and Policy
Compute risk premiums by finding portfolios exposed to a single
source of risk
Diversify to infinity: Var(1/N i) 0 !
Specific risk can be eliminated; if it were priced, there would be
arbitrages
This gives something like the SML:
Ei = RF + i1 [ E1 RF ] + i 2 [ E2 RF ] + ...
APT versus CAPM:
Pedagogy
Application
FMP 1 - 9
Financial Management and Policy
3. Empirical CAPM
Some stock characteristics seem to influence expected
returns:
Size (small caps receive a premium)
Book-to-market value (value stocks are, ceteris paribus,
riskier)
Momentum effect (past winners are riskier)
Introduce them in a market model:
Ei = RF + i [ Em RF ] + iS [ ES EL ] + iBTM [ EH EL ] + iMom [ Ew El ]
FMP 1 - 10
Financial Management and Policy
4. BARRA Approach
A given market may provide cross-sectional risk premia
associated to given characteristics:
Size (small caps receive a premium)
Book-to-market value (value stocks are, ceteris paribus,
riskier)
Operating Leverage (indicates business risk)
Financial Leverage (indicates financial risk)
Introduce them in a cross-sectional regression:
Ri = RF + sizei RPsize + BTM i RPBTM + ... +
Then use fitted values to compute cost of capital:
^
^
ki = RF + sizei RP size + BTM i RP BTM + ...
FMP 1 - 11
Financial Management and Policy
B. CAPITAL BUDGETING UNDER RISK
The NPV equation for a riskless project:
Ct
t
t =1 (1 + r )
where r is the riskless rate
FIRM
T
NPV = C 0 +
Risky project introduction of risk
a) Certainty equivalent method
CE t
t
t =1 (1 + r )
where CE is the certainty equivalent
Investments
Risk
T
NPV = C 0 +
b) Risk-adjusted discount rate method
E (C t )
NPV = C + 0
t
t =1 (1 + radj )
T
where radj is the required return
FMP 1 - 12
Financial Management and Policy
Returns
B1. Principle
Cash
Dividend
Investment
Shareholder
Project
Financial
Market
No
Diversification
Diversified
Portfolio
Security
Risk: i
FMP 1 - 13
Firm with
same project
Cash flows
Variance: j2
Risk: j
Financial Management and Policy
Project Level versus Firm Level:
Firm Level (without taxes):
The cost of capital (and the beta) is a weighted average of the required
return for each project (asset side), and rewards debt and equity
proportionally (liabilities side):
n
WACC = Rassets = wproject i Rproject i
i =1
=
D
E
Rdebt +
R
D+E
D + E equity
Project Level:
The required return is a linear function of the projects systematic risk:
Rproject i = R f + project i ( Rm R f )
SML
*
WACC
*
i
FMP 1 - 14
Financial Management and Policy
B2. Risk measurement
1) Finding the total risk of a project
Scenario analysis: what would be the NPV under
better/worse than expected situation
Proportional variations (quick & dirty)
Variations related to the quality of estimation (for high
expertise)
Equivalent probability variations (to obtain a better picture)
Sensitivity analysis
Select sensible parameters, refine estimated variability
FMP 1 - 15
Financial Management and Policy
Example: Company basecase
Estimation of
Scenarios: price 10000 +/- 5% ; inflation 2,5% +/- 20%; etc... ;
tax rate 34 or 40%
Find that market share, growth and tax rate are most
uncertain
Sensitivity based on 5 scenarii
price, inflation
market size, market share
market growth
fixed costs, variable costs
discount rate, tax rate
very optimistic: high market share and growth (proba 10%)
optimistic: high market share, same growth (proba 20%)
base case (proba 30%)...
Tax rate can be 34% or 40% with proba 50%
Discount at the risk free rate to find the reference NPV
FMP 1 - 16
Financial Management and Policy
Result:
(can be done for IRR or for NPVrisk free):
tends to
normal distribution
proba
NPV or IRR
and CV= NPV/NPV
Scn. FMP 1 - 17
0
base
Financial Management and Policy
Scn. + NPV
2) Determining the relevant risk measure
A. Inevitable corporate risk:
When undiversified shareholders of small business
adjust for the projects contribution to corporate risk
use the CML with for the corporate cost of capital
1. With IRR: direct volatility of a rate of return
ln(1 + CV 2 )
2 =
2. With NPV: use transformation
H
3. With cash flows: compute variance of yearly rates of returns
compute a beta as if the market was the firm ( standalone beta)
FMP 1 - 18
Financial Management and Policy
Cost of capital for the undiversified investor
Imagine that an entrepreneur has a very risky project p which
cannot be broken apart
*p
EP
kP
RF
CML
*
chosen
portfolio q
P
Cost of capital kp is on the investors indifference curve corresponding to her
chosen portfolio q
2
2
U q = Eq A q U q = k p A p with A = risk aversion coefficient
FMP 1 - 19
Financial Management and Policy
Cost of capital for the undiversified investor: explanation
Because the entrepreneur cannot break the project, he has to choose either p or
q;
The criterion for this choice is related to the expected utility extracted from each
portfolio;
The required return of the project is the one that is on the same Indifference Curve
(IC) as the chosen portfolio but with the risk of the project;
The greater the risk aversion coefficient, the greater the slope of the IC: the
EP
required return increases then.
k'P
*p
CML
RF
FMP 1 - 20
*
chosen
portfolio q
Financial Management and Policy
P
B) Measurement of
1) Comparison method:
find comparable companies in same business
Compute leverage of the firm and the comparables
Un-leverage observed betas for leverage by Hamadas
L
formula
U =
1+(1Tc)B
S
Solve for the most reliable asset beta for the activities
(system of equations)
Reconstruct the beta of the company
Re-leverage the beta of the company with the reversed
Hamada formula
FMP 1 - 21
Financial Management and Policy
2) Accounting method:
Compute EBIT/A for the company and for the market
Run a regression
3) Fundamental method:
Start from the average = 1
Adjust for all determinants of :
industry: each industry has a particular intrinsic risk, linked to its cyclicality and
exposure to market risk
financial leverage: balance business/ financial risk
operating leverage: ratio fixed / variable costs
growth: high growth prospects mean higher risk
size: small caps earn a premium
Example: Fundacom
Chemicals sector (+0.3), higher fixed costs than average (+0,15), higher leverage
(+0,25), much smaller cap than competitors (+0,2), growth-oriented (+0,1)
More judgment, biases (hopefully)
tend to cancel out
FMP 1 - 22
Financial Management and Policy