Comparing House to Buy Using the Sharpe
Ratio
Hrishikesh D. Vinod
∗
April 9, 2014
Abstract
We discuss a first approximation to choosing between competing
decision options using objective probabilistic data to aid the judgment.
Roughly speaking we simply compare the Sharpe Ratios defined as the
ratio of the average return to the standard deviation of returns.
1
A Story Behind Expected Values
Mathematicians from 17th century developed the notion of expected value
by using probability theory. There is an important story behind this concept
of “expected value”. Before the start of the Industrial Revolution in Europe,
the feudal landlords had money to spend and capitalists were eger to get
capital. A place called Monte Carlo located on the French riviera had one of
first casino gambling establishments in the world.
The idea of a casino is simple. It offers gambling opportunities. If the
gambler wins, the casino loses and vice versa. Obviously, then, if a casino
wants to survive, it must offer gamblers only those options cleverly designed
in such way that the casino itself (known as the house) never loses. Enter probability experts. The development of probability theory was greatly
helped by the need of casino establishments in creating games that can make
sure that the casino house never loses.
∗
Professor of Economics, Fordham University, Bronx, New York, USA 10458. E-mail:
[email protected].
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The Expected value of a gamble is precisely what the gambler can “expect
to win” in the long run if the game is played several times. The way all casino
gambling works is by making sure that the expected value of each gamble is
negative.
2
Risk Adjusted Comparison of Returns
Certain types of games of chance are called zero sum games and should not
be confused with the stock market, where investors in “productive” establishments win when profits are earned.
We assume that past returns form a random variable X, and that the
return data on X are classified into n class intervals. Each class interval has
a ‘within class midpoint’ return of xi . We assume that we have an exhaustive
set of n intervals with midpoint values: ‘xi , i = 1, . . . , n. The returns fall
in the i-th class with probability
P (xi ). An important check on whether we
P
have an exhaustive set is: ni=1 P (xi ) = 1.
Now the expected value of the payoff or expected payoff is:
E(x) =
n
X
xi P (xi ).
(1)
i=1
Prof. William Sharpe (Nobel prize winner, 1990) and Prof. Marlowitz
amongh many others pointed out that it is not enough to compare the expected values E(x) of the competing uncertain prospects, since different
prospects are associated with different risk profiles. For example, investment in a portfolio of municipal bonds may have lesser risk (fluctuations)
than investment in a portfolio of stocks.
At the elementary level, it is customary to measure the ‘risk’ associated
with an uncertain prospect (e.g. investment) by the historical standard deviation of past returns from that prospect. This is only a first approximation
to the risk, valid only when the probability distribution of returns is Normal.
In reality it often has fat tails (similar to Student’s t density) and positive
or negative skewness. We compute the standard deviation of the set of n
payoff awards by using the following square root of the variance formula for
classified data:
2
V (x) =
n
X
P (xi ) (xi − E(x))2 .
(2)
i=1
A popular summary measure of the “risk adjusted return” is named after
Prof. Sharpe and is defined next. A risky prospect (stock market investment)
generally has a larger standard deviation than a less risky prospect (municipal
bonds). Sharpe’s idea is to divide E(x) by standard deviation of returns.
The ratio is designed to reduce the apparent attractiveness of risky prospects
despite higher E(x).
2.1
Sharpe Ratio defined
In Finance, the Sharpe ratio is defined as a portfolio’s mean return in excess of
the riskless return divided by the portfolio’s standard deviation. The Sharpe
Ratio represents a measure of the portfolio’s risk-adjusted (excess) return.
Since we expect to compare two or more uncertain prospects (portfolios) we will need k = 1, . . . , K estimates of Sharpe ratios identified by the
subscript k in the definition here.
Ek (x)
,
Shak = √
(Vk (x))
(3)
where Ek (x) denotes expectation based on eq. (1) and Vk (x) denotes variance
of the k-th prospect based on eq. (2).
If we have only two uncertain prospects A and B, we shall use k = (A, B),
respectively, to determine both ShaA and ShaB . If ShaA > ShaB , we say
that A has a higher risk-adjusted return. That is, A has the larger Sharpe
Ratio, and prefer A over B. Conversely, if ShaB > ShaA , we prefer B.
Sharpe Ratios have at least three flaws: (a) It ignores the estimation risk.
(b) It can be misleading when comparing prospects with negative returns.
(c) The true downside risk should exclude the positively skew portion of
the probability distribution of returns. After all when xi > E(x), (in the
positively skew range) that xi represents a gain, not a loss. An appropriate
measure of risk should only focus on the downside of the distribution. One
should compute the downside standard deviation (DSD) described in Vinod
and Reagle (2005). Hence one should divide E(x) by DSD(x), instead of
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dividing by the simple standard deviation used in eq. (3). These flaws have
been stated in Vinod and Morey (B2001), Vinod (2008), and elsewhere.
Despite the flaws, known by now for some years, Sharpe Ratio is a good
first approximation to risk adjusted returns valid when the distribution of
returns is Normal. Wall Street “experts” define the “risk adjusted return”
from Sharpe Ratios without adjusting for its known flaws. After all, they are
often dealing with “other people’s money” (OPM).
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House Buying Problem
You have a choice buying house A or B in two different neighborhoods, which
are otherwise equally suitable to you and your spouse. House A has a chance
that it will go down in value by 10K with probability 0.2 or go up by 15K
with probability 0.4 or up 6K with probability 0.4. House B has similar house
value changes: down 15K, up 30K or up 36K with corresponding probabilities
(0.4, 0.3, 0.3) respectively. Which house should you buy?
3.1
Quick Hints for a Solution
Compute the Sharpe ratios for both A and B and choose the one with the
larger Sharpe ratio.
We must first translate “House A has a chance that it will go down in
value by 10K with probability 0.2 or go up by 15K with probability 0.4 or
up 6K with prob. 0.4.” into R code. We denote ‘a’ as the value in thousands
and ‘pa’ as corresponding probabilities.
#The # symbol in R means comment.
#everything after the symbol on that line is ignored by R
rm(list=ls()) #this cleans R memory
options(prompt = " ", continue = " ", width = 68,
useFancyQuotes = FALSE)
a=c(-10,15,6) #this loads variable a
pa=c(.2,.4,.4) # this loads corresponding probabilities
ea=sum(a*pa);ea #compute and report E(A)=6.4
[1] 6.4
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va=sum(((a-ea)^2)*pa);va #compute and report V(A)=83.44
[1] 83.44
sha=ea/sqrt(va) #compute Sharpe Ratio for A
sha #[1] 0.7006366
[1] 0.7006366
This completes Sharpe Ratio for A. Now we translate the sentence House B
has similar house value changes: down 15K, up 30K or up 36K with probabilities 0.4, 0.3 or 0.3 respectively. We denote ‘b’ as the value in thousands
and ‘pb’ as corresponding probabilities.
b=c(-15, 30, 36)
pb= c(0.4, 0.3, 0.3)
eb=sum(b*pb);eb #compute E(B)=13.8
[1] 13.8
vb=sum(((b-eb)^2)*pb);vb #compute V(B)=558.36
[1] 558.36
shb=eb/sqrt(vb) #compute Sharpe Ratio for A
shb #[1] 0.5840122
[1] 0.5840122
Sharpe ratio for option A (=0.7001) is larger than that for B (=0.584),
so choose A.
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Table 1: Setup of Table for hand calculation of expected value and variance
of grouped data
xi P (xi ) xi P (xi ) xi − E(x)
1 -10
0.2
-2
-16.4
2 15
0.4
6
8.6
0.4
2.4
-0.4
3 6
4
E(a)=
6.4
3.2
[xi − E(x)]2
268.96
73.96
0.16
V(x)=
P (xi )[xi − E(x)]2
53.792
29.584
0.064
83.44
Solution without using R
If one wants to do these calculations without using R it is necessary to set up
a table. See Table 1, where all additions are in columns, so that one is less
likely to make arithmetic errors. Begin by making sure that the column sum
for probabilities in column P (ai ) is 1. The row numbered 4 identifies column
sums when they are the expected value, E(x), and the variance V (x). Here
is how to set up a table designed to compute equations 1 and 2:
The next step in hand calculation is to compute the Sharpe Ratio for
option A by dividing E(x) by the square root of V (x) as defined in eq. 3.
We repeat this set up for each (decision /investment) option and compare
their Sharpe Ratios. Finally we choose the option with the highest Sharpe
Ratio.
We remind the reader that we have already mentioned some flaws associated with comparing Sharpe ratios, such as when some pathological cases
arise in comparisons of strong loss situations. Vinod and Morey (B2001)
describes an extension of Sharpe Ratio to allow for estimation risk called
‘double Sharpe Ratio.’ Another paper Vinod and Morey (B2000) describes
confidence intervals on Sharpe Ratios.
References
Vinod, H. D. (2008), Hands-on Intermediate Econometrics Using R: Templates for Extending Dozens of Practical Examples, Hackensack, NJ: World
Scientific, iSBN 10-981-281-885-5, URL http://www.worldscibooks.
com/economics/6895.html.
Vinod, H. D. and Morey, H. R. (B2000), “Confidence intervals and hypothesis
testing for the sharpe and treynor performance measures: A bootstrap
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approach,” in “Computational Finance 1999,” , eds. Abu-Mostafa, Y. S.,
LeBaron, B., Lo, A. W., and Weigend, A. S., Cambridge, MA: MIT Press,
chap. 3, pp. 25–39.
Vinod, H. D. and Morey, M. R. (B2001), “A double sharpe ratio,” in “Advances in Investment Analysis and Portfolio Management,” , ed. Lee, C. F.,
New York: JAI-Elsevier Science, vol. 8, pp. 57–65.
Vinod, H. D. and Reagle, D. (2005), Preparing for the Worst: Incorporating Downside Risk in Stock Market Investments (Monograph), New York:
Wiley.
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