How to calculate portfolio return and risk based on given scenario [closed]

Question

I am interested to invest Rs. 1 lacks in security market. I have securities A and B for this purpose.

Here is the details:

Security    Risk    Expected_return
A           10%         12%
B           18%         20%

Coefficient of correlation between A and B is 0.15.

If I decide to invest 50% of his fund in A and 50% in B.

What if I decide to invest 75% of his fund in A and 25% in B. Will risk and return change.

How can I calculate Portfolio risk and return in botch cases?

I was not able to use Coefficient of correlation in formula mentioned in textbooks.

Answer 1

Good afternoon! Since you did not specify the measure of risk, I will asume that it's standard deviation of return. So for the portfolio 50/50 ($w_A$ - weight of asset A, $w_B$ - weight of asset B, $\sigma$ - standard deviation, $Var$ - variance, $r$ - return, $r_p$ - portfolio return): $$ corr(r_A, r_B) = \frac{cov(r_A, r_B)}{\sqrt{Var(r_A) Var(r_B)}} \\[20pt] \Leftrightarrow cov(r_A, r_B) = corr(r_A, r_B) \sqrt{Var(r_A) Var(r_B)} = \\[14pt] = corr(r_A, r_B) \sigma_A \sigma_B\\[14pt] E(r_p) = w_A E(r_A) + w_B E(r_B) = \mathbf{16\%} \\[8pt] \sigma(r_p) = \sqrt{Var[w_A E(r_a) + w_B E(r_B)]} \\[8pt] Var[w_A E(r_a) + w_B E(r_B)] = w_A^2 Var(r_A) + w_B^2 Var(r_B) + 2 w_A w_B cov(r_A, r_B) = \\[8pt] = 0.25 \times 0.01 + 0.25 \times 0.0324 + 2 \times 0.5 \times 0.5 \times 0.15 \times 0.1 \times 0.18 = 0.01195 \\[8pt] \Leftrightarrow \sigma_p = \sqrt{Var[r_p]} = \sqrt{0.01195} = \mathbf{10.93\%} $$

About the intuition: since there is little correlation between two assets, we can exploit some benefits from diversification, with portfolio that obtains higher expected returns, than asset $A$ itself, but with risk close to risk of $A$. Now you can plug in different weights in the formula and calculate what happens when weights change. Hint: shifting weights towards less risky asset $A$ shall both decrease expected portfolio risk and return. Expected return will be 14%, standard deviation 9.31%.