Paul Glasserman∗
The main objective of this chapter is to explain how statistical parameters of portfolio returns
— mean and variance — are determined by statistical parameters of the assets in the portfolio
— their means, variances and correlations. In particular, this chapter emphasizes how corre-lations between the assets influence the variance of the portfolio’s returns. The implications of this relationship are illustrated through various applications, including
• determining risk-return tradeoffs in alternative portfolio mixes;
• understanding how diversification reduces risk;
• finding the minimum-variance hedge ratio in hedging one asset with another;
• understanding how serial correlation in returns affects risk over different time horizons;
• calculating the value-at-risk in a portfolio;
• finding the probability that one portfolio outperforms another.
I.A.2.1 Means and Variances of Past Returns
Means and variances of asset returns can be understood in two different ways:
• as statistical descriptions of returns experienced in the past;
• as probabilistic predictions of returns to be experienced in the future.
In the first case, means and variances are calculated from historical data and serve to help summarize this data. In the second case, the mean (or expected value) and variance apply to random variables, not data. They help summarize the probability distribution of returns to be experienced in the future, rather than to summarize a record of past observations. Of course, we often use past data as a way to estimate parameters of the distribution of future returns, and in doing so we are connecting the two perspectives. It is nevertheless useful to keep the distinction in mind.
We begin by discussing means and variances of past returns because this case does not involve any probabilistic concepts or assumptions.
I.A.2.1.1 Returns
We use the notation x1, x2, . . ., xn to denote n consecutive returns on some asset — e.g., monthly returns on a stock. Suppose, for example, that we have a record of stock prices S0, S1, . . ., Sn; then the corresponding returns are the percentage price changes
xi=Si− Si−1
Si−1
, i = 1, . . ., n. (I.A.2.1)
∗Columbia University, Graduate School of Business c
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Returns are sometimes computed on a continuously compounded basis, in which case
xi= ln(Si/Si−1). (I.A.2.2)
This simply states that
Si= exiSi−1.
If the prices S0, S1, . . ., Sn are recorded at times t0, t1, . . . , tn, then this means that the stock grew at a continuously compounded rate of xi/(ti+1− ti) over the period of time from ti−1to ti. With the first-order approximations
ln(u) ≈ u − 1, ex≈ 1 + x,
(I.A.2.1) and (I.A.2.2) become equivalent, so the two will be close, especially over short time horizons.
These equations assume that the stock pays no dividends. For a stock that pays dividends, the dividends should be included in the return calculation. Suppose the stock pays a dividend diat time ti; then (I.A.2.1) becomes
xi=Si+ di− Si−1 Si−1
, i = 1, . . ., n.
If the dividend is paid sometime between ti−1and ti, then the diin this formula should include not only the dividend itself but also the income earned by reinvesting the dividend between the time it is paid and the date ti.
Some assets may generate storage costs or other carrying costs; this is often true of commodi-ties, for example. A carrying cost may be interpreted as a negative dividend for purposes of the return calculation.
I.A.2.1.2 Mean, Variance and Standard Deviation
From now on we assume that the returns x1, . . ., xn have been calculated over equally spaced intervals (e.g., daily returns or monthly returns) with proper accounting for dividends and any carrying costs.
The mean return, denoted by ¯x, is the arithmetic average
¯
The variance of the returns is the average squared difference from the mean, s2x= 1
An alternative but algebraically equivalent formula is s2x= 1
Equation (I.A.2.4) defines what is usually called the population variance. The symbol s2x is often reserved for the sample variance, given by
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The sample variance (I.A.2.5) is appropriate when we interpret x1, . . . , xnas a sample from a larger population of returns. The distinction between dividing by n and dividing by n − 1 is minor if n is even moderately large, so we will not dwell on this point.
In Microsoft Excel, the function VAR calculates (I.A.2.5). The function VARP calculations the population variance (I.A.2.4).
This is the population standard deviation; the sample standard deviation is the square root of the sample variance (I.A.2.5). The Excel function STDEVP calculates (I.A.2.6), whereas STDEV calculates the sample standard deviation.
Variance and standard deviation are both measures of variability in a set of data: the more the individual values xideviate from the mean ¯x, the greater the values of sx and s2x. As a consequence of the square root in (I.A.2.6), the standard deviation always has the same units as the underlying data x1, . . . , xn. If the data are percentages (as is often the case for returns), then the standard deviation is a percentage. If x1, . . . , xn were in dollars, then sx
would be in dollars, whereas the variance would be in dollars squared. This property often makes the standard deviation easier to interpret than the variance.
Example: Calculate the mean and standard deviation of the monthly returns listed in the first column of Table I.A.2.1.
Table I.A.2.1: Calculation of mean, variance and standard deviation
The calculation is illustrated in the table. At the bottom of the first column, we have the mean 3.21%. The second column lists the deviations from the mean xi− ¯x, and the third column lists the squared deviations (xi− ¯x)2. The average of these squared deviations yields a variance of s2x= 0.00393, and taking the square root yields a standard deviation of sx= 6.27%.
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I.A.2.1.3 Portfolio Mean, Variance and Standard Deviation
Consider, now, a portfolio with holdings in two assets, XXX and YYY. Let (x1, y1), . . . , (xn, yn) be pairs of past returns on the two assets. For example, if the returns are calculated on a monthly basis, then (x1, y1) are the returns on the two assets in the first month, (x2, y2) are the returns in the second month, and so on.
Consider a portfolio with weight w on XXX and weight 1 − w on YYY; e.g., w = 0.6 for a 60–40 weighting. What are the returns on the portfolio? Assuming the portfolio is rebalanced at the end of each period to maintain the same weights, the portfolio returns are given by the weighted asset returns,
πi= wxi+ (1 − w)yi, i = 1, . . ., n. (I.A.2.7) For example, if XXX increases by 5% and YYY increases by 10%, then the portfolio return is
(0.6)(5%) + (0.4)(10%) = 7%.
How are the mean, variance and standard deviation of the portfolio returns related to those of the assets in the portfolio? The mean portfolio return is just the weighted average of the individual asset returns:
¯
π = w¯x + (1 − w)¯y.
This can be verified by noting that the mean portfolio return
¯
This is illustrated in Table I.A.2.2 for a portfolio holding 60% XXX and 40% YYY. The monthly portfolio returns in the last column are calculated month by month from returns on the two assets. The average of the portfolio returns in the last column is 2.73%. This is the same as the value obtained by taking the weighted average
0.6¯x + 0.4¯y = (0.6)(3.21%) + (0.4)(2.02%) = 2.73%
of the mean returns on the two assets.
In contrast, the variance and standard deviation of the portfolio are not quite as simply related to those of the assets in the portfolio. This is illustrated by the example in Table I.A.2.2, which shows the variances and standard deviations calculated from each column of numbers using the formulas (I.A.2.4) and (I.A.2.6). The portfolio standard deviation is not given by the weighted average of the asset standard deviations,
sπ6= wsx+ (1 − w)sy,
nor is the portfolio variance given by the weighted average of the asset variances, s2π6= ws2x+ (1 − w)s2y.
To relate the portfolio variance and standard deviation to the individual assets, we need to introduce the correlation between the asset returns.
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Table I.A.2.2: Calculation of portfolio returns and their mean, variance and standard deviation
I.A.2.1.4 Correlation
Correlation measures the direction and strength of the relationship between two sets of obser-vations x1, . . . , xnand y1, . . . , yn. The sign of the correlation determines the direction, with a positive correlation meaning that above average values of one variable tend to be paired with above average values of the other, and negative correlation meaning that above average values of one variable tend to be paired with below average values of the other. The magnitude of the correlation measures the strength of the association — more precisely, it measures the strength of the linear relationship between the two variables, with perfect correlation meaning that the points (xi, yi) lie exactly on a straight line.
To define correlation precisely, we first introduce the covariance between the xi and the yi, given by The Excel function COVAR applied to two ranges containing the xi and yi executes this calculation.
Observe that sxy will be large and positive if those i for which xi> ¯x coincide with those for which yi> ¯y. Conversely, sxy will be large and negative when values xi> ¯x are paired with values yi< ¯y.
A shortcoming of covariance as a measure of association is that it has no natural scale: what counts as a “large” covariance depends on context and even on the units of the data. If the xi are measurements in meters, for example, then changing their units to centimeters would increase the covariance by a factor of 100 without in any way changing the relationship between the xi and yi.
The correlation coefficient gets around this shortcoming by using the standard deviations of the observations to normalize the covariance:
rxy= sxy
sxsy
. (I.A.2.9)
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-15%
-5%
5%
15%
-15% -5% 5% 15%
Figure I.A.2.1: Scatter plot of the monthly returns (xi, yi) from Table I.A.2.2 and the regression line through the data.
Here, sx is the standard deviation in (I.A.2.6) and sy is the standard deviation of y1, . . . , yn. The Excel function CORREL applied to two ranges containing the xi and yi calculates (I.A.2.9).
The correlation coefficient rxyhas the same sign as the covariance sxy, so the interpretation of the sign as an indicator of positive or negative association between two variables is the same as before. But the correlation coefficient satisfies
−1 ≤ rxy≤ 1,
and is thus on a universal scale, regardless of the underlying observations (xi, yi) or their units.
The extreme cases rxy = ±1 occur if and only if the points fall on a straight line, meaning that
yi= a + bxi, i = 1, . . . , n,
for some intercept a and slope b. If b > 0, then rxy= 1 and if b < 0 then rxy= −1.
At intermediate values of rxy, the magnitude |rxy| gives an indication of how well the points (xi, yi) are approximated by a straight line. If the points (xi, yi) form a cloud, with no evident pattern, the correlation coefficient will be close to zero.
However, a correlation of zero does not necessarily indicate the absence of a relation between the two variables — it indicates the absence of a linear relation. For example, if we take for the xi the integers −5, −4, . . ., 4, 5 and set yi = x2i, then rxy= 0 though there is evidently a very strong relation between the xi and yi. This strong relation just happens not to be a linear relation.
Figure I.A.2.1 shows a scatter plot of the monthly returns (xi, yi) in Table I.A.2.2: each pair (xi, yi) provides the coordinates of one point in the figure. The figure also shows the regression line through the data, which is the best linear fit in the least-squares sense. Using (I.A.2.9), we can calculate a correlation of rxy= 0.7287 for the data in Table I.A.2.2. Consistent with this value, the figure indicates a positive association between the xiand yi, and only a moderately strong linear relation between the two.
The magnitude of the correlation coefficient measures how closely the regression line fits the data, but it should not be confused with the slope of the regression line, which is given by
b = sy
sxrxy. The line in Figure I.A.2.1 has a slope of 0.51.
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I.A.2.1.5 Correlation and Portfolio Variance
We now return to the portfolio returns in Table I.A.2.2 and the question we posed earlier about the connection between the portfolio variance s2π and the variances of the individual assets.
Recall that the portfolio returns πiare related to the asset returns xi, yithrough the weights w and 1 − w as in (I.A.2.7). Having introduced correlation, we can now present one of the most important identities of this chapter:
s2π= w2s2x+ (1 − w)2s2y+ 2w(1 − w)sxsyrxy. (I.A.2.10) Thus, the variance of the portfolio returns depends not only on the variances s2xand s2y of the individual assets but also on the correlation between them.
Using (I.A.2.9), we can alternatively write (I.A.2.10) as
s2π= w2s2x+ (1 − w)2s2y+ 2w(1 − w)sxy (I.A.2.11) which uses the covariance sxy in place of sxsyrxy.
Equation (I.A.2.10) is an algebraic identity that results from the definitions in (I.A.2.4), (I.A.2.7) and (I.A.2.9). It is useful because it decomposes the portfolio variance into sim-pler quantities — asset variances and correlation. In so doing, it reveals which features of the asset returns are relevant to the variance of the portfolio returns.
In (I.A.2.10) we see that a positive correlation rxy> 0 will lead to a larger value of the portfolio return s2π, and a negative correlation will lead to a smaller value.
This fits with intuition. If the two asset returns are negatively correlated, then losses in one asset will tend to be offset by gains in the other, resulting in lower variability in the overall portfolio returns.
But it must be stressed that diversification — splitting investments across more than one asset — reduces variance even if the assets are positively correlated. We return to this point in Section I.A.2.2.
Example: Apply (I.A.2.10) to the returns in Table I.A.2.2. The correlation between the monthly returns xi and yi in the table is 0.7287. This can be calculated using (I.A.2.9) or, more easily, by applying the Excel function CORREL to the data. Using the values sx= 6.27%
and sy= 4.39% calculated previously (see the bottom of the table) and recalling the weights w = 0.6, 1 − w = 0.4, the right side of (I.A.2.10) becomes
(0.6)2(0.0627)2+ (0.4)2(0.0439)2+ 2(0.6)(0.4)(0.0627)(0.0439)(0.7287) = 0.00268 and thus coincides with the value computed in the table by taking the variance of the individual returns π1, . . . , πn.
I.A.2.1.6 Portfolio Standard Deviation
By taking the square root of each side of (I.A.2.10), we find that the portfolio standard deviation sπ is given by
sπ=q
w2s2x+ (1 − w)2s2y+ 2w(1 − w)sxsyrxy. (I.A.2.12) For any weight w, 0 < w < 1, the portfolio standard deviation satisfies
sπ≤ wsx+ (1 − w)sy,
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with strict inequality unless rxy = 1. Thus, diversification reduces the portfolio standard deviation (compared to the weighted average of the asset standard deviations) except in the extreme case rxy = 1. When rxy = 1, the returns of the two assets are in a straight line relation with each other, so investing in one is equivalent to investing in the other.
In Section I.A.2.6, we will see that correlation is also relevant to the problem of annualizing standard deviations calculated from daily or monthly returns. The relevant correlation param-eter in that setting is the correlation between returns of a single asset (or portfolio) in different time periods, rather than the correlation between returns of different assets in the same time period. We will see, for example, that in the absence of serial correlation an annual standard deviation is√
12 times larger than a monthly standard deviation — not 12 times larger. This may be interpreted as the result of diversification over time.
I.A.2.2 Mean and Variance of Future Returns
The discussion in the previous section applies to data (xi, yi), i = 1, . . . , n, such as observations of past returns. In particular, the relationships (I.A.2.10) and (I.A.2.12) relating the portfolio variance and standard deviation to properties of the assets in the portfolio are purely algebraic and do not rely on any probabilistic or statistical assumptions. As such, these relationships are merely descriptive of historical data.
To turn from a historical perspective to a forward-looking one, we need to introduce a proba-bilistic formulation. We will model future returns as random variables and show that analogous relationships between the portfolio and individual assets hold in this case as well.
Most of the formulas in this section are counterparts of those in the previous section. The key difference is that in the previous section we gave equal weight to all returns, but now in calculating a forward-looking mean or variance we will weight each possible return by its probability.
I.A.2.2.1 Single Asset
Denote the (unknown and uncertain) return on an asset over the next month by the random variable X. The random variable is characterized by its distribution. If X can take only a finite number of possible values — as in a model that considers a finite number of possible scenarios — its distribution can be specified through a probability mass function pX, for which
pX(x) = P (X = x).
Here, P (·) denotes the probability of the event in parentheses, so pX(x) is the probability that the random variable X takes the value x. Alternatively, the distribution of X may be specified through a probability density fX (as in the case of normally distributed returns), in which case the probability that the random variable X is less than or equal to an arbitrary value x is given by
P (X ≤ x) = Z x
−∞
fX(u) du.
We denote the expected value of the random variable by E[X] or µX and also refer to it as the mean of the random variable. For a random variable taking only the values x1, . . . , xn, the expected value is the probability-weighted average of these values and is given by
E[X] = µX =
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For a random variable with a probability density, the expected value is given by E[X] = µX =
Z ∞
−∞
xfX(x) dx.
We denote the variance of the random variable by Var[X] or σ2X. It is given by
Var[X] = σ2X =
n
X
i=1
(xi− µX)2pX(xi)
in the case of a mass function, and by Var[X] = σX2 =
Z ∞
−∞(x − µX)2fX(x) dx
in the case of a probability density. In both cases, the standard deviation, StdDev[X] or σX, is given by the square root of the variance.
The expressions in (I.A.2.3) and (I.A.2.4) coincide with what one obtains for the mean and variance of a random variable that is equally likely to take any of the values x1, . . ., xn.
I.A.2.2.2 Covariance and Correlation
Now let the random variables X and Y denote the unknown returns of two assets over the next period. Their joint distribution may be specified through a mass function
pXY(x, y) = P (X = x and Y = y)
giving the joint probability that X takes the value x and Y takes the value y. The joint distribution may alternatively be given by a joint density fXY. In this case, the probability that X ≤ x and Y ≤ y is or if fXY(x, y) = fX(x)fY(y) in the second case. Intuitively, independence means that the value of one random variable has no information about the value of the other.
The covariance between X and Y is given by
Cov[X, Y ] = E[(X − µX)(Y − µY)], (I.A.2.13) the expectation calculated using the joint distribution of X and Y .
The correlation between X and Y , denoted by ρXY, is given by ρXY = Cov[X, Y ]
σXσY
.
The correlation coefficient ρXY for random variables should be interpreted in much the same way as the coefficient rxy for data points: it measures the strength of the linear relationship between X and Y .
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I.A.2.2.3 Mean and Variance of a Linear Combination
Let the random variables X and Y denote the unknown returns of two assets over the next period. The return on a portfolio with weight w on the first asset and weight 1 − w on the second is given by the random variable
Π = wX + (1 − w)Y.
In order to present expressions for the mean and variance of the portfolio return Π, we first consider the more general case of an arbitrary linear combination aX + bY , with constants a and b not necessarily summing to 1. For the mean of the linear combination, we have simply E[aX + bY ] = aE[X] + bE[Y ] = aµX + bµY, (I.A.2.14) a linear combination of the means.
For the variance, the counterpart to (I.A.2.10) is
Var[aX + bY ] = a2Var[X] + b2Var[Y ] + 2abCov[X, Y ].
Equivalently,
Var[aX + bY ] = a2σ2X+ b2σY2 + 2abσXσYρXY. (I.A.2.15) The standard deviation is obtained by taking the square root.
In the special case a = w and b = 1 − w, we get the portfolio mean
µΠ= wµX + (1 − w)µY, (I.A.2.16)
the portfolio variance
σ2Π= w2σ2X+ (1 − w)2σY2 + 2w(1 − w)σXσYρXY, (I.A.2.17) and the portfolio standard deviation is given by the square root of (I.A.2.17).
σ2Π= w2σ2X+ (1 − w)2σY2 + 2w(1 − w)σXσYρXY, (I.A.2.17) and the portfolio standard deviation is given by the square root of (I.A.2.17).