In Chapter 4, we saw that it is generally possible to reduce the risk of a portfolio by diversification. In this section, we look at the implications of the market model on diversification.
Consider two assets, with returnsRi,tandRj,t, respectively, at timet. Let σ2i = Var(Ri,t),σ2j = Var(Rj,t),and letρij denote the correlation ofRi,t, Rj,t. Suppose that the market model holds for each asset so that
Ri,t−Rf,t=αi+βi(Rm,t−Rf,t) +i,t, t= 1,2, . . . , T
and
Rj,t−Rf,t =αj+βj(Rm,t−Rf,t) +j,t t= 1,2, . . . , T . Now consider a portfolio of assetsiandj, with return of the form
Rp,t=wRi,t+ (1−w)Rj,t, where 0< w <1. Letσ2p= Var(Rp,t); then
σ2p=w2σ2i+ (1−w)2σ2j+ 2w(1−w)ρijσiσj. Suppose, for simplicity, thatσ2i =σ2j. Then
σ2p=
w2+ (1−w)2+ 2w(1−w)ρij
σ2i
= (1−2(1−ρij)w(1−w))σ2j.
Because w(1−w)>0 for 0< w <1 and ρij <1, it follows that σ2p <σ2j for any 0< w <1; see Section 4.2 for further details. That is, for any 0< w <1, the risk of the portfolio is less than that of either of the two assets used to form it. However, diversification has very different effects on the market and nonmarket components of risk.
As discussed in the previous section, the market model holds for the portfolio:
Rp,t−Rf,t=αp+βp(Rm,t−Rf,t) +p,t, t= 1,2, . . . , T where
αp=wαi+ (1−w)αj and βp=wβi+ (1−w)βj.
Then the market component of the variance of the portfolio return is given byβ2pσ2m.
Suppose that, without loss of generality,βi≤βj. Then βi ≤βp≤βj.
Therefore, provided thatβi>0, as is typically the case, β2iσ2m≤β2pσ2m≤β2jσ2m
so that the market component of variance for the portfolio return lies between the market components of return variance for the two assets. Thus, the market component of return variance for the portfolio cannot be reduced below the smaller of the market components of return variance for the two assets.
In particular, ifβi=βj thenβp=βi and, hence, the market component of the variance ofRp,t isβ2iσ2m, the same as the market components of variance for returns on each of assets i and j. That is, in this case, diversification does not reduce the market component of risk. Therefore, the reduction in the variance of the portfolio return, as compared to the return variances of
assetsiandj, is entirely because of a reduction in the nonmarket component of variance.
Example 8.12 Consider two assets, asset i and asset j; assume that σi= σj= 0.5, βi=βj= 0.8, and ρij, the correlation of the returns of the two assets, is 0.2. Suppose that the market portfolio has return standard deviation of 0.4. Then the equally weighted portfolio of these assets has return variance
1 2
2 σ2i +
1 2
2
σ2j+ 21 2 1
2ρijσiσj = 0.15.
That is, the risk of 0.5 for each asset is reduced to √ 0.15 .
= 0.39 for the portfolio.
The market component of the variance of each of the two asset returns is (0.8)2(0.4)2= 0.1024,
and for each asset, the nonmarket component of the return variance is 0.25−0.1024 = 0.1476.
The equally weighted portfolio hasβp= 0.8; hence, its market component of return variance is also 0.1024. It follows that the nonmarket component of the return variance for the portfolio is
0.15−0.1024 = 0.0476.
Thus, each asset has a return variance of 0.25, with a market component of 0.1024. The return variance of the portfolio is 0.15, but the market component of that variance is the same as the market component of the return variance for the two assets, 0.1024. The nonmarket component of return variance for each of the two assets is 0.1476, while the nonmarket component of return
variance for the portfolio is only 0.0476.
Recall that, according to the CAPM, it is the market component of risk that is rewarded by a higher expected return, as discussed in Section 7.3. Thus, not only does diversification tend to reduce risk, the reduction is greater for the nonmarket component of risk, the component of risk that is not rewarded with a higher expected return.
Portfolios of Several Assets
Similar considerations apply to a portfolio ofN assets. LetR1,t, R2,t, . . . , RN,t denote the asset returns at time t and suppose that the market model holds for each asset, that is, for eachi= 1,2, . . . , N,
Ri,t−Rf,t =αi+βi(Rm,t−Rf,t) +i,t, t= 1,2, . . . , T .
Let
α=
⎛
⎜⎜
⎜⎝ α1
α2
... αN
⎞
⎟⎟
⎟⎠, β=
⎛
⎜⎜
⎜⎝ β1
β2
... βN
⎞
⎟⎟
⎟⎠, and t=
⎛
⎜⎜
⎜⎝ 1,t
2,t
... N,t
⎞
⎟⎟
⎟⎠
denoteN×1 vectors. Consider a portfolio based on a weight vectorw∈ N, with returnRp,t at timet. Then
Rp,t−Rf,t =αp+βp(Rm,t−Rf,t) +p,t
whereαp=wTα,βp=wTβ, andp,t=wTt.
LetΣ denote the covariance matrix oft. Then the market component of Var(Rp,t) isβ2pσ2m and the nonmarket component is
σ2,p≡Var(p,t) = Var(wTt) =wTΣw.
Because of the benefits of diversification, the variance of the return on the portfolio tends to be small relative to the variances of the returns on the indi- vidual assets; as in the two-asset case, this is generally because of a reduction in the nonmarket components of return variance.
Example 8.13 Consider the equally weighted portfolio of eight stocks ana- lyzed in Example 8.11, with returns inbig8. For the eight individual stocks, the standard deviations are given by
> apply(big8, 2, sd)
AAPL BAX KO CVS XOM IBM JNJ DIS
0.0739 0.0556 0.0412 0.0578 0.0459 0.0458 0.0386 0.0579
Now consider estimation of the nonmarket component of risk for each of the stocks. Note that ˆσ,i for asset i is available from the results of the lm function by extracting thesigmacomponent of the results from thesummary function; for example,
> summary(lm(ibm~sp500))$sigma [1] 0.0398
Define a function f.sighatby
> f.sighat<-function(y){summary(lm(y~sp500))$sigma}
Note that
> f.sighat(ibm) [1] 0.0398
The nonmarket components of risk—that is, the standard deviations cor- responding to the nonmarket components of variance—for the eight stocks may now be calculated using theapplyfunction in the usual way as follows:
> apply(big8, 2, f.sighat)
AAPL BAX KO CVS XOM IBM JNJ DIS
0.0659 0.0490 0.0372 0.0418 0.0322 0.0398 0.0331 0.0370
Now consider the equally weighted portfolio of the stocks represented in big8; recall that the returns for this portfolio are stored in the variable big8.port and that the estimate of β for the portfolio is ˆβp= 0.803. The total risk and the nonmarket risk for this portfolio may be calculated using the functionssdandf.sighat, respectively.
> sd(big8.port) [1] 0.0340
> f.sighat(big8.port) [1] 0.0158
Note that the standard deviation of the portfolio return is about two-thirds as large as the average return standard deviation for the eight stocks, given by
> mean(apply(big8, 2, sd)) [1] 0.0521
However, the standard deviation corresponding to the nonmarket component of variance for the portfolio is only about one-third as large as the aver- age nonmarket component of return standard deviation for the eight stocks, given by
> mean(apply(big8, 2, f.sighat)) [1] 0.042
Recall that, for the observed returns on the S&P 500 index,Sm2 = 0.00141;
it follows that the market component of risk for the “big8” portfolio is βˆpSm= (0.803)√
0.00141 = 0.0302.
This value is similar to the market components of risk for the eight individual stocks:
> big8.mm$coefficients[2,]*sd(sp500)
AAPL BAX KO CVS XOM IBM JNJ DIS
0.0346 0.0269 0.0182 0.0403 0.0330 0.0232 0.0203 0.0448
Therefore, the total risk of the portfolio, 0.0340, is smaller than the total risks of the individuals stocks, which range from 0.0386 to 0.0739; this difference is attributable primarily to a decrease in the nonmarket risk.
Because the total variance of the portfolio return consists of the market component, which is similar to the market components of return variance for the individual assets in the portfolio, and the nonmarket component, which tends to be much less than the individual nonmarket components of return variance, the proportion of return variance explained by the market return tends to be higher for the portfolio than for the individual assets. That is, R-squared for the portfolio tends to be larger thanR-squared for the individual stocks.
Example 8.14 Consider the returns for the eight stocks stored in the variable big8and analyzed in the previous example. Recall that theR-squared values for these stocks are given by
> apply(big8, 2, f.rsq)
AAPL BAX KO CVS XOM IBM JNJ DIS
0.219 0.234 0.196 0.485 0.516 0.257 0.276 0.599
TheR-squared value for the equally weighted portfolio of the eight stocks may be calculated usingf.rsqas well:
> f.rsq(big8.port) [1] 0.787
Therefore, R-squared for the portfolio is considerably larger than R-squared
for the individual stocks.
Note that a relatively large value of R-squared for a portfolio indicates that it is well diversified, in the sense that most of its risk is because of its relationship with the market portfolio which, by definition, is diversified.
Some Further Results on Portfolio Risk
The properties of the portfolios in Examples 8.13 and 8.14 hold in general, at least to some degree. As noted previously, let R1,t, R2,t, . . . , RN,t denote the asset returns at timetand suppose that the market model holds for each asset so that
Ri,t−Rf,t=αi,t+βi(Rm,t−Rf,t) +i,t, t= 1,2, . . . , T
for i= 1,2, . . . , N. Let t denote the vector (1,t,2,t, . . . ,N,t)T and let Σ denote the covariance matrix oft.
Suppose that for any i and j, the residual returns i,t and j,t are uncorrelated. Then
Σ=
⎛
⎜⎜
⎜⎜
⎝
σ2,1 0 ã ã ã 0 0 . .. . .. ... ... . .. . .. 0 0 . . . 0 σ2,N
⎞
⎟⎟
⎟⎟
⎠ (8.10)
where σ2,j= Var(j,t). The assumption that Σ is a diagonal matrix is a strong assumption that leads to the single-index model for the returns
R1,t, R2,t, . . . , RN,t. The single-index model and its implications for portfolio theory are covered in detail in Chapter 9.
Consider the equally weighted portfolio with weight vector w= (1/N)1.
Then
σ2,p= 1
N21TΣ1= 1 N
⎛
⎝1 N
N j=1
σ2,j
⎞
⎠= 1 Nσ¯2 where
¯σ2 = 1 N
N j=1
σ2,j
is the average nonmarket component of the return variance of the assets.
Therefore, ifN is large and the residual returns for different assets are uncor- related, then the nonmarket component of return variance for an equally weighted portfolio tends to be small.
It is important to note that such a conclusion depends on the assumption that Σ is a diagonal matrix, that is, on the assumption that i,t and j,t
are uncorrelated for i=j. For a general covariance matrixΣ the minimum possible nonmarket return variance can be found by choosingw to minimize wTΣw subject towT1= 1; note that this is the same as finding the weight vector for the minimum variance portfolio, but with Σ replacing Σ, the covariance matrix of the returns.
Therefore, according to Proposition 5.3,wTΣw is minimized by w˜ = Σ−1 1
1TΣ−1 1
and the minimum nonmarket component of return variance is given by w˜TΣw˜ = 1
1TΣ−1 1.
For instance, suppose thatΣ is of the form σ2Mρ where
Mρ=
⎛
⎜⎜
⎜⎜
⎝
1 ρ ã ã ã ρ ρ . .. . .. ... ... . .. . .. ρ ρ . . . ρ 1
⎞
⎟⎟
⎟⎟
⎠ (8.11)
andσ2 >0; see Example 5.7. Then all residual returns have standard deviation σand any pair of residual returns for different assets in the same time period has correlation ρ. In Example 5.7, it was shown that the equally weighted portfolio is the minimum variance portfolio in this case and it has variance
σ2
N21TMρ1σ2 = N+N(N−1)ρ
N2 =
ρ+1−ρ N
σ2.
Therefore, the nonmarket component of return variance whenΣ=σ2Mρ
is never less thanρσ2. Thus, even for a large number of assets and a diversified
portfolio, the nonmarket component of the variance is not negligible unless ρσ is close to 0.
It is worth noting that, even when it is not negligible, the nonmarket component of risk for the portfolio, which is approximately√ρσfor largeN, is still less than that of the individual assets, which is given byσ.