bre44380_ch07_162-191.indd 178 09/02/15 04:11 PM
178 Part Two RiskThe portfolio standard deviation is, of course, the square root of the variance.
Now you can try putting in some figures for JNJ and Ford. We said earlier that if the two
stocks were perfectly correlated, the standard deviation of the portfolio would lie 40% of the
way between the standard deviations of the two stocks. Let us check this out by filling in the
boxes with ρ 12 = +1.The variance of your portfolio is the sum of these entries:Portfolio variance = (^) [ (.6)^2 × (13.2)^2 ] + (^) [ (.4)^2 × (31.0)^2 ] + 2(.6 × .4 × 1 × 13.2 × 31.0)
= 412.9
The standard deviation is √
412.9 = 20.3% or 40% of the way between 13.2 and 31.0.
JNJ and Ford do not move in perfect lockstep. If recent experience is any guide, the cor-
relation between the two stocks is about .19. If we go through the same exercise again with
ρ 12 = .19, we find
Portfolio variance = (^) [ (.6)^2 × (13.2)^2 ] + (^) [ (.4)^2 × (31.0)^2 ]
- 2(.6 × .4 × .19 × 13.2 × 31.0) = 253.8
The standard deviation is √
253.8 = 15.9%. The risk is now less than 40% of the way between
13.2 and 31.0. In fact, it is not much more than the risk of investing in JNJ alone.
The greatest payoff to diversification comes when the two stocks are negatively correlated.
Unfortunately, this almost never occurs with real stocks, but just for illustration, let us assume
it for JNJ and Ford. And as long as we are being unrealistic, we might as well go whole hog
and assume perfect negative correlation (ρ 12 = –1). In this case,
Portfolio variance = (^) [ (.6)^2 × (13.2)^2 ] + (^) [ (.4)^2 × (31.0)^2 ]
- 2[.6 × .4 × (−1) × 13.2 × 31.0] = 20.1
The standard deviation is √
20.1 = 4.5%. Risk is almost eliminated. But you can still do better
by putting 70% of your investment in JNJ and 30% in Ford.^30 In that case, the standard devia-
tion is almost exactly zero. (Check the calculation yourself.)
When there is perfect negative correlation, there is always a portfolio strategy (represented
by a particular set of portfolio weights) that will completely eliminate risk. It’s too bad perfect
negative correlation doesn’t really occur between common stocks.
General Formula for Computing Portfolio Risk
The method for calculating portfolio risk can easily be extended to portfolios of three or more
securities. We just have to fill in a larger number of boxes. Each of those down the diagonal—
the red boxes in Figure 7.13—contains the variance weighted by the square of the proportion
JNJ Ford
JNJ x^21 σ 12 = (.6)^2 × (13.2)^2 x 1 x 2 ρ 12 σ 1 σ 2 = (.6) × (.4) × 1 × (13.2) × (31.0)
Ford x 1 x 2 ρ 12 σ 1 σ 2 = (.6) × (.4) × 1 × (13.2) × (31.0) x 22 σ 22 = (.4)^2 × (31.0)^2
(^30) The standard deviation of Ford is 31.0/13.2 = 2.348 times the standard deviation of JNJ. Therefore you have to invest 2.348 times
more in JNJ than in Ford to eliminate all risk in a two-stock portfolio. The portfolio weights that exactly eliminate risk are .7014 for
JNJ and .2986 for Ford.