# Portfolio returns, volatility and weights of capital

I would just like to check if I've done these questions right, I feel like I might have used the complete wrong methods to get my answers.

I've been given information on 3 stocks:

I've filled in the missing values in the correlation matrix and I've been asked the following questions:

1. "An asset manager forms a portfolio by investing 10% of capital in Stock A, 40% of capital in Stock B, and the rest in Stock C. Calculate the expected return and volatility of this portfolio."
1. " If you are a portfolio manager, your client requests to invest in the global minimum variance portfolio using Stock B and Stock C. Calculate the weights of capital for this portfolio."

So for Part 1 this is what I did:

I took the expected returns and multiplied them by their percentage in the portfolio; so I had $$A = 8 * 0.1 = 0.8$$% , $$B = 10 * 0.4 = 4$$%, $$C = 15*0.5=7.5$$% and the total expected return of the portfolio would be the sum of these values, which is $$12.3$$%

For the volatility I used the same method, multiplying the volatility in the table by the percentage of the stock in the portfolio and I got the total volatility as $$14.5$$%

I'm not sure if my method is correct so please let me know if what I've done is ok or completely invalid.

For Part 2:

I wasn't entirely sure what to do but I added the expected returns of $$B$$ and $$C$$ and got $$25$$%, $$100/25 = 4$$ so I thought the weights of each stock would be $$Weight B= 10*4 = 40$$% and $$Weight C=15*4=60$$% but I'm not sure if this is correct, should I have used the volatility instead since the global minimum variance portfolio cares more about volatility than return so the weights would be $$50/50$$ ? Or is there a completely different method I should use?

These questions might be very simple for this site but I would just really like to know if my methods are ok or if I'm just making up methods that aren't valid, any help would be appreciated.

Hi: I'm not sure how the question defines volatility but let's assume it's standard devation so $$\sigma_{i}$$ for the ith asset.

So, if you want the volatility of a portfolio with weights, $$w_1$$,$$w_2$$ and $$w_3$$, you need to calculate the portfolio variance and then take the square root.

To get the portfolio variance, it's

$$var(w_1 X_1 + w_2 X_2 + w_3 X_3) =$$

= $$w_1^2 var(X_1) + w_2^2 var(X_2) + w_3^2 var(X_3) + 2 w_1 w_2 cov(X_1,X_2) + 2 w_2 w_3 cov(X_2,X_3) + 2 w_1 w_3 cov(X_1,X_3)$$

( check the formula because it's been a while and my memory fails me sometimes ).

Note that, in the expression above,

A) the var terms are the squares of the volatilities given.

B) the cov terms are the respective correlations in the matrix multiplied by the the given volatilities.

So, cov( $$X_1,X_2) = \rho_{1,2} \sigma_1 \sigma_2$$.

C) The $$w_{i}$$ are the weights so 0.10, 0.40 and 0.50

Then, when you put those in, take the square root of the result and that's the volatility of the portfolio.

For part 2), here's a hint. If you had a portfolio of stocks with equal volatility of 15 percent, then the variance would be:

$$var(w_2 X_2 + w_3 X_3) = w_2^2 Var(X_2) + w_3^2 Var(X_3) + 2 w_2 w_3 cov(X_2 X_3)$$.

So, part 2 is asking you to find the $$w_2$$ and $$w_3$$ that minimize the expression above, given that the var terms are 0.15^2 and cov term is known ( for cov take correlation of B and C and multiple by their volatilities ). Oddly enough, I was once asked the same question on an interview so I found this question interesting.

• Thanks a lot, these explanations are really great, one question I have is that you've mentioned how to obtain the volatility and weights, does that mean what I've done for the expected return is ok? Or do I need to change my method there as well? May 20 at 14:40
• For part 2, I've got up to this: $Var(w_2X_2 +w_3X_3) = 0.0225w_2^2+0.0225w_3^2$ but I'm not sure where to go from here to get $w_2$ and $w_3$ May 20 at 14:59
• @Charlie P: Your answer for the expected return sounded correct to me. The expected return of the portfolio is $w_1 E(X_1) + w_2 E(X_2) + w_3 E(X_3)$ and it seemed like you did that but I didn't check it. May 20 at 20:00
• @Charlie P: It sounds like the cov was zero which definitely could be the case. I didn't calculate it. But, given that what you have is correct, and you know that $w_2$ and $w_3$ have to sum to 1, how would one minimize the sum of their squares ? You can use substition method ( include constraint to get rid of one of the 2 unknowns and then take first derivative and set to zero ) or think about what would make the sum of their squares smallest given that they need to sum to 1. If you still have problems, write back and I'll write it out. But it's good to try these things to get better. May 20 at 20:09
• Thanks for the clue, Using what you said about how the weights must sum to 1 I decided to try this: $0.0225w_2^2 + 0.0225(1-w_2)^2 = 0$ I simplified this until I got a quadratic equation: $w_2^2 - w_2 +0.5 = 0$ but then when I try to solve this i get $w= 1/2 +i(1/2)$ which I don't think is right... May 21 at 12:29