UVA-F-DRAFT
COST OF CAPITAL: THE DOWNSIDE RISK APPROACH1
Emerging Markets and CAPM
CAPM is often the method of choice among practioners for estimating the cost of
equity for investment projects in developed economies. But for investment projects in
emerging economies, CAPM tends to give estimates that are considered too low by
experienced financial managers. Sometimes, CAPM estimates of the cost of equity for
projects in emerging markets are even lower than those for comparable projects in
developed economies. This observation calls into question whether the CAPM-based
model is applicable in emerging markets.
Conceptually, CAPM assumes fully integrated markets. Emerging markets are at
best partially integrated. Stock returns in many emerging markets are only weakly
correlated with stock returns in developed markets. In other words, emerging markets are
often low or even negative beta markets relative to the U.S. equity market. If we
maintain that beta is an appropriate measure of risk, many emerging markets would have
low investment risks.
Measuring Downside Risk
Modern portfolio theory measures the total risk of a portfolio by the standard
deviation of its returns. Although the standard deviation identifies risk associated with
the volatility of returns, it does not separate upside changes from downside changes in
returns. But the distinction of upsides and downsides is important in practice. Risk
averse investors will naturally be averse to downside volatility, but would welcome any
upside changes.
Porter, in discussing corporate risk management, states “[r]isk is a function of
how poorly a strategy will perform if the ‘wrong’ scenario occurs.” Therefore, it is
1
Prepared by Lourdes Alers under the direction of Professor Wei Li, Darden Graduate School of Businesss.
This technical note was written as a basis for class discussion rather than to illustrate effective or
ineffective handling of an administrative situation. Copyright  2002 by the University of Virginia Darden
School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to
dardencases@virginia.edu. No part of this publication may be reproduced, stored in a retrieval system,
used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying,
recording, or otherwise—without the permission of the Darden School Foundation.
common to hear practicing managers conceptualize risk in terms of an organization’s
failure to reach a performance target or an investment portfolio reaching a below-target
performance. In evaluating the performance of an investment, one could certainly
redefine risk as a function of how poorly an investment performs if the “wrong” scenario
occurs. More than anything, risk is really defined by the possibility of disaster (large bad
surprises).
As a measure of the downside risk, the semi-deviation of returns captures the
downside risk that investors want to avoid as opposed to the upside to which investors
want to be exposed. In other words, it gives a measure of the likely economic impact of
the downsides as defined by the inclusion of only the left-hand (downside) tail of the
returns distribution in the calculation.
The approach that has been proposed to quantify downside risk is simple, and
practitioners can apply it just as easily as the conventional CAPM. The downside risk
measure could be implemented using observed stock market returns. Among its
advantages are that it can be applied both at the market level and at the company level, it
is not based on subjective measures of risk, it can be fine-tuned to any desired benchmark
return, and it captures the downside risk that investors want to avoid.
The required rate of return can be decomposed into two parts: a risk-free rate and
a risk premium. The first part is the time value of money, which is demanded even from a
riskless asset. The second part is the extra compensation for bearing risk, which depends
on the asset considered. For an asset index by i, its required rate of return is
(1)
Ri = RF + M i * MRP
Here Ri is the required return for asset i, RF is the risk-free rate, MRP is the
world market risk premium, and M i is a measure of asset i’s risk. Under CAPM, M i is
simply the beta of asset i, measuring the systematic risk that a diversified investor would
be exposed to by holding asset i. Under the downside risk approach, we define the
downside risk of asset i as the downside volatility of the asset relative to the downside
volatility of the market portfolio. The downside volatility is in turn measured as the
semi-deviation on asset returns.
Calculating the Semi-Deviation of Returns
The semi-deviation on the returns of a particular asset is simply the standard
deviation of the asset’s downside returns. Here we define a downside return as one that
is less than the average return and an upside return as one that is at least as large as the
average return. By sorting returns into upside and downside bins, we can decompose the
variance of returns, rit , for asset i in period t = 1 to period t = T into two parts,
(2)
σ i2 =
1 T
1
1
(rit − ri )2 = ∑ (rit − ri )2 + ∑ (rit − ri )2
∑
T t =1
T rit < ri
T rit ≥ ri
14
4244
3 14
4244
3
downside variance
upside variance
1 T
∑ rit is the average return on asset i. Semi-deviation, as a measure of the
T t =1
downside deviation, is defined as the square root of the downside variance, or
where ri =
(3)
1
(rit − ri )2
∑
T rit <ri
semi-σ i =
The semi-deviation thus measures the downside volatility in the returns of asset i.
Similarly, we can also compute the semi-deviation of the world market returns as
(4)
semi-σ M =
1
(rMt − rM ) 2
∑
T rMt <rM
Calculating the Downside Risk Measure
The downside risk measure for asset i can now be calculated as the ratio of the semdeviation of the returns of asset i to the semi-deviation of the returns of the world market
portfolio. That is,
(5)
Mi =
semi-σ i
semi-σ M
The cost of capital for investing in asset i is thus
(6)
Ri = RF +
semi-σ i
*MRP
semi-σ M
An Example: Calculating the Semi-Deviation of Tsingtao Brewery’s Stock Returns
Exhibit 1 gives an example on the calculation of the semi-deviation of Tsingtao
Brewery’s stock returns. The data in column 1 is the monthly total return index of
Tsingtao Brewery listed on Hong Kong Stock Exchanges.
The first step here is to calculate the monthly total returns on the stock. The monthly
returns are alculated in two different ways: simple returns and continuously compounded
returns. In general, the continuously compounded returns (or log returns) are more
accurate than the simple returns. The reason is the following. In the worst case scenario,
the simple return would produce a monthly return of -1 or (0-P)/P. But using the log
return would give ln(0/P) = - ∞ . In statistical terms, the simple returns artificially
truncate the downside returns to -1, will therefore lead to an underestimation of the semideviation. To avoid biases in measuring the downside risk, log return is therefore
preferred.
Let Pit denote asset i’s total return index at time t. The log total return can be calculated
as
(7)
P 
rit = ln  i ,t +1 
 Pit 
Exhibit 1 shows the monthly total returns calculated first using simple returns and then
using log returns. A comparison of the two monthly total returns shows that whenever
the returns are negative, the simple returns are always smaller (in absolute terms) than the
log returns. So simple returns do underestimate downside returns.
Based on the month returns, I first compute the average return. Using the average return,
I then compute the downside deviation squared for each time period. The downside
deviation squared is simply, (rit − ri ) 2 if rit < ri or 0 if rit ≥ ri . The average of these
deviations squared is simply the downside variance in equation (2). The semi-deviation
is then simply the square-root of the downside variance. For Tsingtao, the semi-deviation
is 14.33% during the period between 8/31/93 and 8/31/95.
Exhibit 1
Calculation of semi-deviation of monthly returns on Tsingtao's H-shares
BIASED
Tsingtao H
total return
index
Date
HK$
8/ 31/ 93
152.8
9/ 30/ 93
162.5
10/ 29/ 93
165.3
11/ 30/ 93
198.6
12/ 31/ 93
270.8
1/ 31/ 94
219.4
2/ 28/ 94
229.2
3/ 31/ 94
190.3
4/ 29/ 94
208.3
5/ 31/ 94
208.8
6/ 30/ 94
165.3
7/ 29/ 94
175.5
8/ 31/ 94
189.3
9/ 30/ 94
174.0
10/ 31/ 94
168.2
11/ 30/ 94
162.4
12/ 30/ 94
123.3
1/ 31/ 95
101.5
2/ 28/ 95
108.8
3/ 31/ 95
120.4
4/ 28/ 95
79.8
5/ 31/ 95
95.3
6/ 30/ 95
90.1
7/ 31/ 95
96.8
8/ 31/ 95
75.2
Average
Standard deviation
Downside variance
Semi-deviation
Simple
Downside
monthly
deviation of
return
return squared
6.35%
0.00000
1.72%
0.00000
20.15%
0.00000
36.35%
0.00000
-18.98%
0.03035
4.47%
0.00000
-16.97%
0.02376
9.46%
0.00000
0.24%
0.00000
-20.83%
0.03715
6.17%
0.00000
7.86%
0.00000
-8.08%
0.00426
-3.33%
0.00031
-3.45%
0.00036
-24.08%
0.05070
-17.68%
0.02599
7.19%
0.00000
10.66%
0.00000
-33.72%
0.10344
19.42%
0.00000
-5.46%
0.00152
7.44%
0.00000
-22.31%
0.04308
-1.56%
16.48%
PREFERRED
Continuously
compounded
Downside
monthly return
deviation of
(log return)
return squared
6.15%
0.00000
1.71%
0.00000
18.35%
0.00000
31.01%
0.00000
-21.05%
0.04430
4.37%
0.00000
-18.60%
0.03459
9.04%
0.00000
0.24%
0.00000
-23.36%
0.05458
5.99%
0.00000
7.57%
0.00000
-8.43%
0.00710
-3.39%
0.00115
-3.51%
0.00123
-27.54%
0.07587
-19.46%
0.03785
6.95%
0.00000
10.13%
0.00000
-41.13%
0.16916
17.75%
0.00000
-5.61%
0.00315
7.17%
0.00000
-25.25%
0.06375
-2.95%
17.15%
0.01337
11.56%
0.02053
14.33%