Corporate Finance, Module 6: “Risk, Return, and Cost of Capital”

Homework Assignment

(The attached PDF file has better formatting.)

An investor has a three stock portfolio.

    Percentage
Held    Expected
Return    Standard
Deviation    Correlations among Stocks
Stock                Stock A    Stock B    Stock C
A    50%    10%    20%    1.0    0.5    0.3
B    30%    15%    30%    0.5    1.0    0.1
C    20%    20%    40%    0.3    0.1    1.0


A.    What is the expected return for the three stock portfolio? (The expected return is the weighted average of the expected returns of each stock.)
B.    What is the variance of the return of the three stock portfolio? (We determine the variance of the three stock portfolio as the variance of a combination of three random variables. When computing the variance, make sure to use the square of the percentage held as the weights for the variance terms, and the product of the percentages held as the weights for the covariance terms. The correlations are used in the covariance terms. See the practice problems for this module for the formulas.)
C.    What is the standard deviation of the return of the three stock portfolio? (The standard deviation is the square root of the variance.)

Question: How do we determine the correlation between the returns on two stocks? In this homework assignment, we are given the correlations. In practice, how do we derive them?

Answer: The correlation ρa,b is defined as the covariance between the returns on stocks A and B divided by the product of the standard deviations of stocks A and B:

    ρa,b = covariance(A,B) / (σa × σb)

If we have a sample of returns, such as returns for the 252 trading days in the year, the correlation is

    

Question: Do we use this formula to derive correlations between stocks?

Answer: The high stochasticity of stock returns causes random fluctuations to overwhelm the true correlation. Many analysts assume the same correlation exists between any two firms in an industry. We examine the correlations of several large stock insurers, take the average, and assume the average holds for all insurers.

Question: The correlation between two auto insurers is greater than that between an auto insurer and a life insurer. Averages from all large insurers may not be correct.

Answer: We use averages of insurers of the same type for the expected correlation among insurers of the same type and averages of insurers of different types for the expected correlation among insurers of different types. But you are correct: every pair of insurers is has its own correlation, and averages are not always a good proxy. Much of the correlation estimate is judgment; we don’t always have good data.

Optional discussion forum posting

The distinction between systematic risk and unique risk affects the proper capitalization rate.

Question: The notion that only systematic risk warrants additional return is puzzling. Science advances by testing hypotheses. If we think that only systematic risk warrants additional return, we form experiments and test this hypothesis. Financial analysts have tried to test this hypothesis, and they get mixed results.

●    They can’t agree on the market return: the stock market, the market of traded assets, a global market, the market of all financial assets plus coins, stamps, art, and what not.
●    Empirical tests of the CAPM are not encouraging. Given the regard for the CAPM in the textbook, I expect the empirical tests to support the theory. Some tests support it partly, and some tests give it only lukewarm support.
●    Some tests so not support the CAPM. High beta stocks have actual returns below the CAPM indication, and low beta stocks have actual returns that are higher.

Despite all this, everyone seems to agree that only systematic risk warrants additional return. Where is the evidence for this?

Answer: This notion is supported by an arbitrage argument. The logic is compelling and few people argue with it. If the market return were proportional to total risk (both systematic risk and unique risk), arbitragers would make risk free profits.

Suppose the expected return on a security were the risk-free rate plus some constant times the total standard deviation of the return. An arbitrager would buy N different stocks with high unique risk, package them into a mutual fund, and sell the fund to other investors.

A.    Explain why the arbitrager earns risk-free profits. (The arbitrager buys stock with high risk, so the return is high. This means that the stock price is low. The arbitrager sells a mutual fund with low risk, so the return is low. This means that the market value of the mutual fund is low. The arbitrager earns the risk-free profits now, not later.)
B.    Explain how supply and demand eliminates the risk-free profit and eliminate the return on diversifiable risk. (Demand by arbitragers is high for the stocks with high total standard deviation. This drives up the price, reducing the return to arbitragers.)

This problem is not required homework. But you will use the concept of systematic risk many times in your actuarial career, and you will be asked to defend the thesis that only systematic risk warrants additional return.

I am having trouble extending the variance formula from two terms to three terms.  Can anyone help me with this?

I don't know if helping HW is a violation to the course. but I seemed to remember that working together is suggested.  [NEAS: That's fine]

Here is the hint to the variance. If you remember there is a picture in the book.
Figure 7.9 shows a 2-stocks, and next figure extend it to N-stocks.

Define some notation here.
V1 is the variance for stock 1
V2 is the variance for stock 2
x1 is the weight for stock 1
x2 is the weight for stock 2
Cov(1,2) is the covariance of Stock 1&2
= correlation coefficient * sqrt(V1) * sqrt(V2)
Vp is the portfolio variance

For 2-Stocks, Vp=(x1)^2 * V1 + (x2)^2 * V2 + 2*x1*x2*Cov(1,2)

extend this idea to N-stocks

Vp = (x1)^2 * V1 + (x2)^2 * V2 + (x3)^2 * V3 + ... + (xN)^2 * VN
+ 2*x1*x2*Cov(1,2) + 2*x1*x3*Cov(1,3) + .... + 2*x1*xN*Cov(1,N)
+ 2*x2+x3*Cov(2,3) + 2*x2*x4*Cov(2,4) + .... + 2*x2*xN*Cov(2,N)
+...+
+
( hope you can see the pattern here)

in particular, for 3-stocks

Vp = (x1)^2 * V1 + (x2)^2 * V2 + (x3)^2 * V3
+ 2*x1*x2*Cov(1,2) + 2*x1*x3*Cov(1,3) + 2*x2*x3*Cov(2,3)  [NEAS: Yes]

I think that it should be this:

Vp = (x1)^2 * V1 + (x2)^2 * V2 + (x3)^2 * V3 + 2*x1*x2*V1*V2*Cov(1,2) + 2*x1*x3*V1*V3*Cov(1,3) + 2*x2*x3*V2*V3*Cov(2,3)

In my above post I have COV but it should be the correlation coefficient.  I believe that my answer is still correct.  [NEAS: Use standard deviations for the cross terms, not variances]

Coviance(A,B) = correlation coefficient * Std Dev(A) * Std Dev(B)

p.s. thanks to Ohel Moed for pointing out

How can we find correlation coefficient ? According to text book correlation between returns on stock1 and 2 ? Is that mean (1.0+0.5+0.3)-(05+1+0.1)=0.2

Jacob: How do we determine the correlation between the returns on two stocks? In this homework assignment, we are given the correlations. In practice, how do we derive them?

Rachel: The correlation ρ_a,b is defined as the covariance between the returns on stocks A and B divided by the product of the standard deviations of stocks A and B:

ρ_a,b = covariance(A,B) / (σ_a × σ_b)

If we have a sample of returns, such as returns for the 252 trading days in the year, the correlation is

Jacob: Do we use this formula to derive correlations between stocks?

Rachel: The high stochasticity of stock returns causes random fluctuations to overwhelm the true correlation. Many analysts assume the same correlation exists between any two firms in an industry. We examine the correlations of several large stock insurers, take the average, and assume the average holds for all insurers.

Jacob: The correlation between two auto insurers is greater than that between an auto insurer and a life insurer.

Rachel: We use averages of insurers of the same type. But much of the correlation estimate is judgment; we don’t always have good data.

Quick one for NEAS:

Why are the Standard Deviations %ages? 

I ended up using .20 (for example) rather than 20, and got a very, very small variance.  Is that going to be acceptable, or should I be using 20 and the % sign is just an error or something?

[NEAS: The standard deviations are percentage points.  For Stock A, the expected return is 10% per annum.  A return 2 standard deviations above the mean is 50% per annum.  A return 2 standard deviations below the mean is -30% per annum.]

There was a fair bit of arithmetic on this homework so I just want to check with people to see if our answers line up for (B) and (C).

For (B) I got that the variance of portfolio is 397.4 (or .03974 if you use decimals instead of %'s)

for (C) I got a standard deviation of 19.934894% (or .19934894). Anyone else get those answers?