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so that
rG [(1 0.10) (1 0.25) (1 0.20) (1 1 .0829, or 8.29%
The geometric return also is called a time-weighted average return because it ignores the
quarter-to-quarter variation in funds under management. In fact, an investor will obtain a
larger cumulative return if high returns are earned in those periods when additional sums have
been invested, while the lower returns are realized when less money is at risk. Here, the high-
est returns (25%) were achieved in quarters 2 and 4, when the fund managed $1,200,000 and
$800,000, respectively. The worst returns ( 20% and 10%) occurred when the fund managed
$2,000,000 and $1,000,000, respectively. In this case, better returns were earned when less
money was under management”an unfavorable combination.
The appeal of the time-weighted return is that in some cases we wish to ignore variation in
money under management. For example, published data on past returns earned by mutual
funds actually are required to be time-weighted returns. The rationale for this practice is that
since the fund manager does not have full control over the amount of assets under manage-
ment, we should not weight returns in one period more heavily than those in other periods
when assessing “typical” past performance.
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Essentials of Investments, and Prologue Companies, 2003
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134 Part TWO Portfolio Theory

Dollar-weighted return When we wish to account for the varying amounts under man-
agement, we treat the fund cash flows to investors as we would a capital budgeting problem in
corporate finance. The initial value of $1 million and the net cash inflows are treated as the cash
flows associated with an investment “project.” The final “liquidation value” of the project is the
ending value of the portfolio. In this case, therefore, investor net cash flows are as follows:


0 1 2 3 4
Net cash flow ($ million) 1.0 0.1 0.5 0.8 1.0

The entry for time 0 reflects the starting contribution of $1 million, while the entries for times
1, 2, and 3 represent net inflows at the end of the first three quarters. Finally, the entry for time
4 represents the value of the portfolio at the end of the fourth quarter. This is the value for
which the portfolio could have been liquidated by year-end based on the initial investment and
net additional investments earlier in the year.
dollar-weighted The dollar-weighted average return is the internal rate of return (IRR) of the project,
average return which is 4.17%. The IRR is the interest rate that sets the present value of the cash flows real-
ized on the portfolio (including the $1 million for which the portfolio can be liquidated at the
The internal rate of
end of the year) equal to the initial cost of establishing the portfolio. It therefore is the interest
return on an
investment. rate that satisfies the following equation:
0.8 1.0
0.1 0.5
(1 IRR)3 (1 IRR)4
(1 IRR)2
The dollar-weighted return in this example is less than the time-weighted return of 8.29% be-
cause, as we noted, the portfolio returns were higher when less money was under manage-
ment. The difference between the dollar- and time-weighted average return in this case is quite

> 1. A fund begins with $10 million and reports the following three-month results (with
negative figures in parentheses):

1 2 3
Net inflows (end of month, $ million) 3 5 0
HPR (%) 2 8 (4)

Compute the arithmetic, time-weighted, and dollar-weighted average returns.

Conventions for Quoting Rates of Return
We™ve seen that there are several ways to compute average rates of return. There also is some
variation in how the mutual fund in our example might annualize its quarterly returns.
Returns on assets with regular cash flows, such as mortgages (with monthly payments) and
bonds (with semiannual coupons), usually are quoted as annual percentage rates, or APRs,
which annualize per-period rates using a simple interest approach, ignoring compound inter-
est. The APR can be translated to an effective annual rate (EAR) by remembering that
APR Per-period rate Periods per year
Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition

5 Risk and Return: Past and Prologue

Therefore, to obtain the EAR if there are n compounding periods in the year, we first recover
the rate per period as APR/n and then compound that rate for the number of periods in a year.
(For example, n 12 for mortgages and n 2 for bonds making payments semiannually).

( )
1 EAR (1 Rate per period) 1
APR [(1 1] n (5.1)

The formula assumes that you can earn the APR each period. Therefore, after one year (when
n periods have passed), your cumulative return would be (1 APR/n)n. Note that one needs
to know the holding period when given an APR in order to convert it to an effective rate.
The EAR diverges from the APR as n becomes larger (that is, as we compound cash flows
more frequently). In the limit, we can envision continuous compounding when n becomes ex-
tremely large in Equation 5.1. With continuous compounding, the relationship between the
APR and EAR becomes
or equivalently,
APR ln(1 EAR) (5.2)

Suppose you buy a Treasury bill maturing in one month for $9,900. On the bill™s maturity date,
you collect the face value of $10,000. Since there are no other interest payments, the holding
period return for this one-month investment is:
Cash income Price change
HPR 0.0101 1.01% Treasury-Bill
Initial price
The APR on this investment is therefore 1.01% 12 12.12%. The effective annual rate is
1 EAR 1.1282
which implies that EAR .1282 12.82%

The difficulties in interpreting rates of return over time do not end here. Two thorny issues
remain: the uncertainty surrounding the investment in question and the effect of inflation.

Any investment involves some degree of uncertainty about future holding period returns, and
in most cases that uncertainty is considerable. Sources of investment risk range from macro-
economic fluctuations, to the changing fortunes of various industries, to asset-specific unex-
pected developments. Analysis of these multiple sources of risk is presented in Part Four on
Security Analysis.

Scenario Analysis and Probability Distributions
When we attempt to quantify risk, we begin with the question: What HPRs are possible, and how
likely are they? A good way to approach this question is to devise a list of possible economic
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Essentials of Investments, and Prologue Companies, 2003
Fifth Edition

136 Part TWO Portfolio Theory

State of the Economy Scenario, s Probability, p(s) HPR
TA B L E 5.2
Boom 1 0.25 44%
Normal growth 2 0.50 14
distribution of
HPR on the Recession 3 0.25 16
stock market

outcomes, or scenarios, and specify both the likelihood (i.e., the probability) of each scenario
and the HPR the asset will realize in that scenario. Therefore, this approach is called scenario
scenario analysis
analysis. The list of possible HPRs with associated probabilities is called the probability dis-
Process of devising
tribution of HPRs. Consider an investment in a broad portfolio of stocks, say an index fund,
a list of possible
which we will refer to as the “stock market.” A very simple scenario analysis for the stock mar-
economic scenarios
and specifying the ket (assuming only three possible scenarios) is illustrated in Table 5.2.
likelihood of each The probability distribution lets us derive measurements for both the reward and the risk of
one, as well as the
the investment. The reward from the investment is its expected return, which you can think
HPR that will be
of as the average HPR you would earn if you were to repeat an investment in the asset many
realized in each case.
times. The expected return also is called the mean of the distribution of HPRs and often is re-
ferred to as the mean return.
To compute the expected return from the data provided, we label scenarios by s and denote
the HPR in each scenario as r(s), with probability p(s). The expected return, denoted E(r), is
List of possible
then the weighted average of returns in all possible scenarios, s 1, . . . , S, with weights
outcomes with
equal to the probability of that particular scenario.
probabilities. S
E(r) p(s)r(s) (5.3)
s 1
expected return
We show in Example 5.3, which follows shortly, that the data in Table 5.2 imply E(r) 14%.
The mean value of the
Of course, there is risk to the investment, and the actual return may be more or less than
distribution of holding
period returns. 14%. If a “boom” materializes, the return will be better, 44%, but in a recession, the return will
be only 16%. How can we quantify the uncertainty of the investment?
The “surprise” return on the investment in any scenario is the difference between the
actual return and the expected return. For example, in a boom (scenario 1) the surprise is
30%: r(1) E(r) 44% 14% 30%. In a recession (scenario 3), the surprise is 30%:
r(3) E(r) 16% 14% 30%.
variance Uncertainty surrounding the investment is a function of the magnitudes of the possible sur-
prises. To summarize risk with a single number we first define the variance as the expected
The expected value of
value of the squared deviation from the mean (i.e., the expected value of the squared “sur-
the squared deviation
prise” across scenarios).
from the mean.

Var(r) p(s)[r(s) (5.4)
s 1

We square the deviations because if we did not, negative deviations would offset positive de-
viations, with the result that the expected deviation from the mean return would necessarily be
zero. Squared deviations are necessarily positive. Of course, squaring (a nonlinear transfor-
mation) exaggerates large (positive or negative) deviations and relatively deemphasizes small
Another result of squaring deviations is that the variance has a dimension of percent
squared. To give the measure of risk the same dimension as expected return (%), we use the
standard deviation, defined as the square root of the variance:
The square root of the
SD(r) Var(r) (5.5)
Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition

5 Risk and Return: Past and Prologue

A potential drawback to the use of variance and standard deviation as measures of risk is
that they treat positive deviations and negative deviations from the expected return symmetri-
cally. In practice, of course, investors welcome positive surprises, and a natural measure of
risk would focus only on bad outcomes. However, if the distribution of returns is symmetric
(meaning that the likelihood of negative surprises is roughly equal to the probability of posi-
tive surprises of the same magnitude), then standard deviation will approximate risk measures
that concentrate solely on negative deviations. In the special case that the distribution of re-
turns is approximately normal”represented by the well-known bell-shaped curve”the stan-
dard deviation will be perfectly adequate to measure risk. The evidence shows that for fairly
short holding periods, the returns of most diversified portfolios are well described by a normal

Applying Equation 5.3 to the data in Table 5.2, we find that the expected rate of return on the


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