Suppose you held a well-diversified portfolio with a very large number of securities,and that the single index model holds.If the β of your portfolio was 0.18 and β_Mwas 0.22,the β of the portfolio would be approximately ________. A)0.64 B)1.19 C)0.82 D)1.56 E)none of the above

Question

Quizplus · Accepted Answer

The single index model states that the beta of a security/portfolio can be calculated as the product of its correlation with the market (represented by beta on the x-axis) and its standard deviation divided by the market's standard deviation (represented by alpha on the y-axis).

Given that betaS1U1B1M = 0.22, we know that the correlation between the security and the market is 0.22 times their respective standard deviations. Similarly, betaS1U1B0 represents the beta of a "typical" security, which is equal to 1.

Using these values, we can rearrange the formula for a security's beta to solve for its correlation:
correlation = beta * (market's standard deviation / security's standard deviation)

Plugging in the values given for betaS1U1B1M and betaS1U1B0, we get:
0.22 = 1 * (market's standard deviation / security's standard deviation)
market's standard deviation / security's standard deviation = 0.22

We also know that the beta of our portfolio (call it betaP) is equal to the weighted average of each security's beta, where the weights are based on the proportion of the portfolio invested in each security:
betaP = (w1 * beta1) + (w2 * beta2) + ...

Since we have a well-diversified portfolio with a large number of securities, we can assume that each security has an equal weight of 1/n, where n is the number of securities. Therefore:
betaP = (1/n * beta1) + (1/n * beta2) + ... = (1/n) * (beta1 + beta2 + ...)

We're given that the beta of our portfolio is 0.18, so we can plug in these values and solve for the average beta of our securities:
0.18 = (1/n) * (beta1 + beta2 + ...)
beta1 + beta2 + ... = 0.18n

Now we can use the formula we derived earlier that relates a security's beta to its correlation with the market and its standard deviation. We'll use betaS1U1B0 as a reference point, since its beta is equal to 1:
1 = correlation(S1,M) * (market's standard deviation / security's standard deviation)
correlation(S1,M) = security's standard deviation / market's standard deviation

Substituting in the values we have:
1 = correlation(S1,M) * 1/0.22
correlation(S1,M) = 0.22

This tells us that a security with a beta of 1 has a correlation of 0.22 with the market. We can now use this correlation to solve for the beta of our portfolio:
0.18 = (1/n) * (betaS1U1B1M + betaS1U1B0 + ...)
0.18 = (1/n) * (0.22 + 1 + ...)
0.18 = (1/n) * (1.22 + ...)
0.18 = (1/n) * (betaP + 1.22)
betaP = 0.18n - 1.22

Since our portfolio has a very large number of securities, we can assume that n is very large and approach infinity. Therefore, betaP approaches 0.18n. Plugging in a large number like 100 or 1000 for n confirms that our portfolio beta is approximately equal to 0.18.

Note that while we were given betaS1U1B0 as equal to 1, this is not always the case. In general, betaS1U1B0 represents the beta of a "typical" security, but the value will depend on the specific characteristics of the market and the security universe being considered.

Suppose You Held a Well-Diversified Portfolio with a Very Large