4
M
ost investors don’t hold all their wealth in just one asset; virtually all of them hold portfolios of securities with different degrees of diversification. This makes it necessary to estimate the risk and return of portfolios (as opposed to those of an individual security), which is the issue we’ll discuss in this chapter. We’ll also discuss a few related concepts, such as feasible sets, efficient sets, and the minimum variance portfolio.Two assets: Risk and return
It’s usually convenient to start with the simplest possible scenario, and in this case that means a two-asset portfolio. Consider then the returns of Bank of America (BoA) and IBM between the years 1994 and 2003, displayed in panel A of Table 4.1. As the table shows, IBM delivered a higher mean return (26.7%
versus 18.8%) with higher volatility (33.1% versus 23.4%) than BoA.
TABLE 4.1
Panel A Panel B
Year BoA IBM x1 x2 Risk Return
(%) (%) (%) (%) (%) (%)
1994 –4.4 32.2 100.0 0.0 23.4 18.8
1995 59.8 25.7 90.0 10.0 22.2 19.6
1996 44.4 67.7 80.0 20.0 21.5 20.4
1997 27.9 39.3 70.0 30.0 21.4 21.2
1998 1.3 77.5 60.0 40.0 21.8 22.0
1999 –14.0 17.6 50.0 50.0 22.8 22.7
2000 –4.5 –20.8 40.0 60.0 24.2 23.5
2001 42.7 43.0 30.0 70.0 26.0 24.3
2002 14.5 –35.5 20.0 80.0 28.2 25.1
2003 20.1 20.5 10.0 90.0 30.5 25.9
AM 18.8% 26.7% 0.0 100.0 33.1 26.7
SD 23.4% 33.1%
Let’s consider first the calculation of the annual return of a portfolio containing these two markets. Obviously, the returns of such a portfolio would depend on how much we invest in each market. Let’s call the proportion of money invested xi(that is, the amount of money invested in asset idivided by the total amount of money invested in the portfolio), and Rithe return of asset i.
Then, the return of the portfolio (Rp)in any given period would be given by
Rp= x1· R1+ x2· R2 (4.1)
where x1+ x2= 1. (This implies that, given the amount of money to be invested in a portfolio, we invest all of it in the two assets considered. This assumption extends to all portfolios regardless of the number of assets; that is, given a portfolio of n assets, and weights x1 . . . xn, the usual assumption is that x1+ . . . + xn= 1.)
For example, in 2003, the return of a portfolio invested 60% in BoA and 40%
in IBM delivered a (0.60)(0.201) + (0.40)(0.205) = 20.3% return. Simple enough. If we had held this 60/40 portfolio during the years 1994 to 2003, then we would have obtained a (0.60)(0.188) + (0.40)(0.267) = 22.0% mean annual return. Again simple enough. There is really no mystery in how to calculate the return of a two-asset portfolio.
Now, what about its risk? That’s a bit more complicated, but not too bad in the two-asset case. The standard deviation of a portfolio (SDp), which is a measure of its risk, is given by
SDp= {(x1)2(SD1)2+ (x2)2(SD2)2+ 2x1x2SD1SD2Corr12}1/2 (4.2)
where SDpis the standard deviation (risk) of the portfolio, SDiis the standard deviation of asset i, and Corr12 is the correlation between assets 1 and 2 (BoA and IBM, in our case). Note that because, by definition, SD1 · SD2· Corr12 = Cov12, where Cov12 is the covariance between assets 1 and 2, then you may occasionally find the third term of the right-hand side of equation (4.2) written as 2x1x2Cov12. (If your knowledge of covariances and correlations is a bit rusty, you may want to read Chapter 27 before continuing.)
Back to the 60/40 portfolio, note that we know at this point all the numbers in equation (4.2) except for one, the correlation between BoA and IBM, which is a rather low 0.28. (You could try to calculate it for yourself from the data in the table.) With this number, the 60/40 weights, the standard deviations in Table 4.1, and equation (4.2), we get that the volatility of the 60/40 portfolio over the 1994–2003 period was
SDp= {(0.60)2(0.234)2+ (0.40)2(0.331)2+ 2(0.60)(0.40)(0.234)(0.331)(0.28)}1/2= 21.8%
In short, calculating the risk and return of a two-asset portfolio is simple, even using a handheld calculator. However, as we will see below, the computational burden increases exponentially with the number of assets, which means that for portfolios larger than three or four assets, spreadsheets become essential. Before we discuss portfolios of more than two assets, however, let’s take a look at a few useful definitions in the two-asset case.
Two assets: Other concepts
Now that we know how to compute the risk and return of a two-asset portfolio, let’s take a look at panel B of Table 4.1. Using equations (4.1) and (4.2), you should have no difficulty replicating the numbers in this panel, which shows the risk and return of several combinations of BoA and IBM. Note that if we invest 100% of our money in either stock, the portfolio reflects the risk and return of that stock. Note, also, that although the numbers in the ‘Return’ column are the weighted average of the returns of BoA and IBM, the numbers in the ‘Risk’
column are notthe weighted average of the risks of these two stocks. (This is due to the diversification effect, which we’ll discuss in the next chapter.)
The last two columns of Table 4.1 are depicted in Exhibit 4.1. This line is called the feasible set, and it’s simply the set of all the possible combinations (portfolios) between BoA and IBM. The points labeled BoA and IBM indicate a 100% investment in each of these stocks, and all the points in between indicate (infinite) other combinations between these two stocks. Point A, for example, indicates a portfolio invested 90% in BoA and 10% in IBM, and point B indicates a portfolio invested 90% in IBM and 10% in BoA.
Note that each point along the feasible set is a portfolio, and each of these portfolios has a different risk–return combination. Note, also, that the feasible set could go beyond the points labeled BoA and IBM in the presence of short- selling (that is, allowing an investor to borrow one asset, to sell it, and to invest more than 100% of his capital in the other asset).
For all the obvious reasons, the point of the feasible set farthest to the left is called the minimum variance portfolio (MVP). Of all the possible combinations between BoA and IBM, this is the one that minimizes the risk of the portfolio. In the two-asset case, in fact, the equation to find it is not too difficult and is given by
(4.3) x1= (SD2)2– Cov12
(SD1)2+ (SD2)2– 2Cov12
EXHIBIT 4.1
IBM v. BoA, feasible set
Note that, in our case, Cov12= SD1· SD2· Corr12= (0.234)(0.331)(0.28) = 0.0216. Therefore, the proportion of money to be invested in BoA (x1) in order to minimize the risk of the portfolio is equal to 72.6%, leaving 27.4% to be invested in IBM. You should have no difficulty calculating that this portfolio has a risk of 21.4% and a return of 21.0%. (Actually, the risk is 21.38%, just slightly lower than the risk of the 70/30 portfolio, which is 21.40%.)
Finally, the efficient setis the upper half of the feasible set, beginning at the MVP. Take another look at Exhibit 4.1. Would you choose a portfolio in the lower branch of the feasible set (that is, the branch that goes down from the MVP)? Of course not. For each portfolio in the lower branch, you could choose one with the same level of risk but higher return in the upper branch. That’s why it’s called the efficient set: because it’s the set of portfolios that, for any chosen level of risk, offers the highest possible return.
Three assets
Before considering the general n-asset case, let’s take a quick look at a three- asset portfolio. The return of this portfolio is straightforward; it is (again) the
28.0
26.0
24.0
22.0
20.0
18.0
16.0
20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0
IBM B
A BoA
MVP
Risk (%)
Return (%)
weighted average of the returns of all the assets in the portfolio (three in this case). That is,
Rp= x1· R1+ x2· R2+ x3 · R3 (4.4)
Now for the bad news. The inclusion of just one more asset complicates the calculation of the risk of the portfolio quite a bit. It’s not difficult, just messy.
Let’s start with the expression, which is given by
SDp= {(x1)2(SD1)2+ (x2)2(SD2)2+ (x3)2(SD3)2+ 2x1x2Cov12+ 2x1x3Cov13+ 2x2x3Cov23}1/2 (4.5)
It looks a bit scary but there’s really nothing to it. Let’s compare it with equation (4.2) and think about both a bit.
Note, first, that equation (4.2) has four terms (the third term is multiplied by 2, so it’s actually two identical terms) and equation (4.5) has nine terms (again, the last three terms are multiplied by 2 and each is made up of two identical terms). Can you see the pattern? The expression for the risk of a portfolio has as many terms as the square of the number of assets in the portfolio; that is, 22= 4 in the two-asset case and 32= 9 in the three-asset case.
Note, also, that for each asset in the portfolio we’ll have a ‘variance term’
that consists of a weight multiplied by a standard deviation, both squared; these are the (xi)2(SDi)2terms. To determine the number of ‘covariance terms’
(xixjCovij), we just count all the different combinations of assets and multiply this number by 2. In the two-asset portfolio, we find only one combination (1-2), so there should be two covariance terms (2x1x2Cov12). In the three-asset portfolio, we find three combinations (1-2, 1-3, and 2-3), so there should be six covariance terms (2x1x2Cov12, 2x1x3Cov13, and 2x2x3Cov23).
In the three-asset case, the feasible set is no longer a line as it is in the two- asset case. In fact, it is a bullet-shaped surface, as shown in Exhibit 4.2. The MVP is still the point farthest to the left of this feasible set, and the efficient set is the upper border of the feasible set, beginning at the MVP.
EXHIBIT 4.2
Feasible set, efficient set and MVP
n assets
Now for the general case. Regardless of the number of assets, the return of a portfolio is always equal to the weighted average of returns of all the assets in the portfolio. That is,
Rp= x1· R1+ x2· R2+ . . . + xn· Rn (4.6)
No trouble there. Perhaps it’s convenient to add at this point that, in order to calculate the expected return of a portfolio, we simply replace in (4.6) the observed returns of the assets by their respective expected returns. (This doesn’t mean that estimating expected returns is simple. We briefly discuss this issue in Chapter 11.)
The risk of an n-asset portfolio, however, is much more difficult to estimate, particularly when the number of assets is large. As we’ve seen above, even for a very small portfolio of three assets the expression to estimate its risk is not all that simple. Formally, the standard deviation of an n-asset portfolio can be written as
Return
Risk Efficient set
MVP Feasible set
(4.7)
It doesn’t look that scary, but that may be simply because the two sum signs are hiding the burden. In a relatively small portfolio of 20 assets, equation (4.7) implies that we have to come up with 400 terms. For all practical purposes, we may as well forget this expression, which is just another way of saying that, when calculating the standard deviation (risk) of a portfolio, we’d better have a spreadsheet at hand.
However, even with a spreadsheet, we need to know what to do. The Excel program to optimize portfolios discussed in Chapter 11 provides a simple way to estimate both the risk and return of a portfolio for any number of assets. In any case, when calculating the standard deviation of a portfolio, it’s important to keep in mind the following. First, we write down as many ‘variance terms,’
(xi)2(SDi)2, as we have assets in the portfolio. Second, we determine every possible combination of assets (1-2, 1-3, . . . 1-n, 2-3, 2-4, . . . 2-n, . . .) and write down two ‘covariance terms’ for each, that is, 2x1x2Cov12, . . . 2x1xnCov1n, 2x2x3Cov23, . . . 2x2xnCov2n, . . . Third, we add up all the terms. And fourth, we take the square root of the sum.
Sometimes it may help to visualize the variance–covariance matrix, including all the relevant weights. In the general, n-asset case, this matrix looks like the one displayed in Table 4.2.
TABLE 4.2
1 2 3 n
1 (x1)2(SD1)2 x1x2Cov12 x1x3Cov13 . . . x1xnCov1n 2 x2x1Cov21 (x2)2(SD2)2 x2x3Cov23 . . . x2xnCov2n
. . . . . . . . . . . . . . . . . .
n xnx1Covn1 xnx2Covn2 xnx3Covn3 . . . (xn)2(SDn)2
Note that, at the end of the day and regardless of the number of assets, the variance of a portfolio is given by the sum of all the elements in this matrix (and the standard deviation simply by the square root of this variance). Think about this matrix a bit and relate it to the discussion above. If you were able to follow the discussion, you should have no trouble writing down this matrix for any number of assets.
SDp=
{ S
n n xixjCovij}
1/2i=1 j=1
S
Finally, note that the feasible set, efficient set, and MVP of an n-asset portfolio, when n is larger than 2, look just like those in Exhibit 4.2. In other words, the feasible set is a bullet-shaped surface, the MVP is the point farthest to the left of the feasible set, and the efficient set is the upper border of the feasible set beginning at the MVP.
The big picture
Calculating the risk and return of a portfolio may be time consuming without a spreadsheet or other software package. However, the intuition behind the calculations is relatively simple. The same applies to some portfolios in which investors may be particularly interested, such as those in the efficient set or the minimum variance portfolio. In Chapter 11 we discuss an Excel program that quickly and easily estimates all the magnitudes and portfolios we have just discussed.
Note that this chapter is mostly about mechanics, that is, about how to calculate the risk and return of different portfolios. But we still haven’t discussed why investors may want to form portfolios. That is the issue we discuss in the next chapter.
Excel section
There are two new concepts to implement in Excel in this chapter, covariance and correlation. Both are very easy to deal with. Suppose you have two series of ten returns each, the first in cells A1 through A10 and the second in cells B1 through B10. Then you do the following:
■ To calculate the covariance between the assets, type
‘=covar(A1:A10,B1:B10)’ in cell A11 and then hit ‘Enter.’
■ To calculate the correlation coefficient between the assets simply type
‘=correl(A1:A10,B1:B10)’ in cell A12 and then hit ‘Enter.’
You may also find it useful to know that in Excel you can not only sum numbers along a row or a column but also over a whole matrix (such as the variance–covariance matrix discussed above). Suppose you have a 3 3 variance–covariance matrix in the range A1:B3. Then you do the following:
■ To sum all the elements in the matrix, type ‘=sum(A1:B3)’ in cell D4 and then hit ‘Enter.’
Challenge section
1 Consider the annual returns of Pepsi and Hewlett-Packard (HP) during the years 1994 to 2003 in panel A of Table 4.3. Then calculate:
(a) The mean annual return of both companies.
(b) The annual standard deviation of returns of both companies.
(c) The correlation of returns between the two companies. (Is it high?
Low? What do you make of it?)
TABLE 4.3
Panel A Panel B
Year Pepsi HP xP xHP Risk Return
(%) (%) (%) (%) (%) (%)
1994 –9.5 28.1 100.0 0.0
1995 56.7 69.4 90.0 10.0
1996 6.2 21.1 80.0 20.0
1997 36.6 25.3 70.0 30.0
1998 14.3 10.7 60.0 40.0
1999 –12.5 67.7 50.0 50.0
2000 42.6 –28.6 40.0 60.0
2001 –0.5 –34.0 30.0 70.0
2002 –12.1 –13.9 20.0 80.0
2003 12.0 34.5 10.0 90.0
0.0 100.0
2 Given the weights for Pepsi and HP in panel B of Table 4.3, calculate the risk and return of those ten portfolios. Then:
(a) Make a graph of the feasible set.
(b) Of the ten portfolios calculated, which is the one with the lowest risk?
(c) Calculate now the MVP using equation (4.3). Is it too different from the portfolio you found in the previous question?
3 Consider, finally, a four-asset portfolio and a five-asset portfolio. In both cases, write the expressions for the risk and the return of each portfolio.