so that

0.25)]1/4

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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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:

Time

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

1.0

(1 IRR)3 (1 IRR)4

(1 IRR)2

1 IRR

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

large.

> 1. A fund begins with $10 million and reports the following three-month results (with

Concept

negative figures in parentheses):

CHECK

Month

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

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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).

n

( )

APR

n

1 EAR (1 Rate per period) 1

n

Rearranging,

EAR)1/n

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

eAPR

EAR 1

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

EXAMPLE 5.2

period return for this one-month investment is:

Annualizing

$100

Cash income Price change

HPR 0.0101 1.01% Treasury-Bill

$9,900

Initial price

Returns

The APR on this investment is therefore 1.01% 12 12.12%. The effective annual rate is

higher:

(1.0101)12

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.

5.2 RISK AND RISK PREMIUMS

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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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%

Probability

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.

probability

To compute the expected return from the data provided, we label scenarios by s and denote

distribution

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.

associated

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.

S

2

E(r)]2

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

deviations.

Another result of squaring deviations is that the variance has a dimension of percent

standard

squared. To give the measure of risk the same dimension as expected return (%), we use the

deviation

standard deviation, defined as the square root of the variance:

The square root of the

SD(r) Var(r) (5.5)

variance.

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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

distribution.

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