Assume N securities. The expected returns on all the securities are equal to 0.0
ID: 2714308 • Letter: A
Question
Assume N securities. The expected returns on all the securities are equal to 0.01 and the variances of their returns are all equal to 0.01. The covariances of the returns between two securities are all equal to 0.005.
1.What are the expected return and the variance of the return on an equally weighted portfolio of all N securities?
2.What value will the variance approach as N gets large?
3.What characteristic of the securities is most important when determining the variance of a well-diversified portfolio?
Explanation / Answer
Answer (a) & (b)
Expected return = 0.01
Variance = 0.01
Covariance between two securities =0.005
Let the number of securities in the portfolio =2 with equal weights ie., 0.50
Expected return = weight of stock 1 * return on stock 1 + weight of stock 2 * return on stock 2
= 0.5 * 0.01 + 0.5 * 0.01 = 0.005 + 0.005 = 0.05
When we consider an equally weighted 3 stock portfolio weight of each stock would be 1/3
Expected return = 1/3*0.01 +1/3 * 0.01 + //3 * 0.01 = 0.00333+0.00333 + 0.00333 = 0.01
The same analogy can be extended to any number of securities in a portfolio. When all the securities in the portfolio have equal weights and same return of 0.01, then
Expected return of a portfolio with N equally weighted securities = (1/N * 0.01) = 0.01
Similarly for a two security portfolio the value of variance can be calculated as
Variance = weight1^2*Variance1+weight2^2 * variance2 + 2 * weight 1 * weight 2 * covariance(1,2)
Taking values of given portfolio Variance = 0.01 and covariance of two securities = 0.005
Variance of a two equally weighted securities portfolio is
Variance = 0.5^2 * 0.01 + 0.5^2 * 0.01 + 2 * 0.5 * 0.5 * 0.005
= 0.25 * 0.01 + 0.25 * 0.01 + 0.0025
= 0.0025 + 0.0025 + 0.0025
This can also be written as
= 2* (0.5^2 * 0.1) + 2*0.5*0.5*0.005
= 0.005 + 0.0025
= 0.0075
Variance of a three securities portfolio would be
Variance = w1^2*var1 + w2^2*var 2 + w3^2*var3 + 2*w1*w2*covariance(1,2) + 2*w1*w3*covariance(1,3)+ 2*w2*w3*covariance(2,3)
For the portfolio given in question the variance a three equally weighted security portfolio would be
Variance = (1/3)^2 * 0.01 + (1/3)^2 * 0.01 + (1/3)^2 * 0.01 + 2*1/3*1/3*0.005 + 2*1/3*1/3*0.005 +2*1/3*1/3*0.005
Variance = 0.1111*0.01 + 0.1111*0.01+ 0.1111*0.01 + 0.001111 + 0.001111 + 0.001111
= 0.00111 + 0.001111 + 0.001111 + 0.001111 + 0.001111 + 0.001111
= 0.0066666
The same can be written in the following manner
Variance = 3 * (1/3)^2 * 0.01 + 3* (2*1/3*1/3*0.005) = 3 * 0.001111 + 3 * 0.001111
= 0.003333 + 0.003333 = 0.006666
Similarly for a equally weight 4 security portfolio variance would be
Variance = 4 * (1/4)^2 * 0.01 + 6 *(2*1/4*1/4*0.005)
= 4 * 0.0625 * 0.01 + 6 * 0.00625
= 0.0025 + 0.00375 = 0.00625
We can see the first term which accounts for the multiplication square of weight with variance changed as 0.005, 0.0033, 0.0025 for a two security, three security and four security portfolio with equal weights and for N securities this term would be N * (1/N)^2 * 0.01 and approaches zero with addition of securities with same variance and equal weight.
We can see that the second term accounting for covariances changed as 0.0025, 0.0333,0.00375 for a two security, three security and four security portfolio. That is this term is approaching the covariance term of 0.005 as we add further securities of same variance to the portfolio and rebalance it in such a way that all securities have equal weights.
Thus for N securities , the first term will become smaller and smaller as the number of securities increases and approaches zero, while the second becomes larger and larger and approaches 0.005 which is the covariance.
Answer (c)
When determining the variance of a well diversified portfolio, we shall be looking at the systematic risk which is the second term above covariance of securities with one another which can also expressed as a correlation coefficient which can be derived by covariance of two securities divided by the standard deviation of the two securities. However when we are considering a well diversified portfolio, the variance by addition of a stock can be determined by
Risk of stock = Change in portfolio variance / weight of stock
= 2 * covariance (stock, Market)
In the given case this would be
Risk of stock = 2 * 0.005 = 0.01 which is the variance of the stock.
And covariance of a well diversified portfolio divided by the standard deviation of returns of a stock denotes Beta which is a measure of systematic risk which cannot be further diversified away. Hence for a well diversified portfolio, beta of individual stocks would be an important measure.
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