|
The Reward-to-Volatility (Sharpe) Ratio
Finally, it is worth noting that investors presumably are interested in the expected
excess return they can earn by replacing T-bills with a risky portfolio, as well as the
risk they would thereby incur. While the T-bill rate is not constant over the entire
period, we still know with certainty what nominal rate we will earn if we purchase a
bill and hold it to maturity. Other investments typically entail accepting some risk in
return for the prospect of earning more than the safe T-bill rate. Investors price risky
assets so that the risk premium will be commensurate with the risk of that expected
excess return, and hence it’s best to measure risk by the standard deviation of excess,
not total, returns.
The importance of the trade-off between reward (the risk premium) and risk (as mea-
sured by standard deviation or SD) suggests that we measure the attraction of a portfolio
by the ratio of risk premium to SD of excess returns.
Sharpe
ratio 5
Risk premium
SD of excess return
(5.18)
This reward-to-volatility measure (first proposed by William Sharpe and hence called the
Sharpe ratio ) is widely used to evaluate the performance of investment managers.
Notice that the Sharpe ratio divides the risk premium (which rises in direct proportion
to time) by the standard deviation (which rises in direct proportion to square root of unit of
time). Therefore, the Sharpe ratio will be higher when annualized from higher frequency
returns. For example, to annualize the Sharpe ratio (SR) from monthly rates, we multiply
the numerator by 12 and the denominator by
"12. Hence the annualized Sharpe ratio is
SR
A
5 SR
M
"12 . In general, the Sharpe ratio of a long-term investment over T years will
increase by a factor of
"T when T -period rates replace annual rates.
10
When returns are uncorrelated, we do not have to worry about covariances among them. Therefore, the variance
of the sum of 12 monthly returns (i.e., the variance of the annual return) is the sum of the 12 monthly variances.
If returns are correlated across months, annualizing is more involved and requires adjusting for the structure of
serial correlation.
bod61671_ch05_117-167.indd 134
bod61671_ch05_117-167.indd 134
6/18/13 8:03 PM
6/18/13 8:03 PM
Final PDF to printer
C H A P T E R
5
Risk, Return, and the Historical Record
135
5.6
The Normal Distribution
The bell-shaped normal distribution appears naturally in many applications. For exam-
ple, heights and weights of newborns are well described by the normal distribution. In fact,
many variables that are the end result of multiple random influences will exhibit a normal
distribution, for example, the error of a machine that aims to fill containers with exactly
1 gallon of liquid. By the same logic, if return expectations implicit in asset prices are
rational, actual rates of return should be normally distributed around these expectations.
To see why the normal curve is “normal,” consider a newspaper stand that turns a profit
of $100 on a good day and breaks even on a bad day, with equal probabilities of .5. Thus,
the mean daily profit is $50 dollars. We can build a tree that compiles all the possible
outcomes at the end of any period. Here is an event tree showing outcomes after 2 days:
Two good days, profit
= $200
Two bad days, profit
= 0
One good and one bad day, profit
= $100
Notice that 2 days can produce three different outcomes and, in general, n days can
produce n 1 1 possible outcomes. The most likely 2-day outcome is “one good and
Take another look at Spreadsheet 5.1 . The scenario analysis for the proposed investment
in the stock-index fund resulted in a risk premium of 5.76%, and standard deviation
of excess returns of 19.49%. This implies a Sharpe ratio of .30, a value in line with the
historical performance of stock-index funds. We will see that while the Sharpe ratio is
an adequate measure of the risk–return trade-off for diversified portfolios (the subject of
this chapter), it is inadequate when applied to individual assets such as shares of stock.
Do'stlaringiz bilan baham: | 
