Calculating Portfolio Return and Risk with Correlated Assets
Summary
The document explains how portfolio weights, asset returns, volatilities, and correlation determine expected return and standard deviation for a two-asset portfolio. Expected portfolio return is calculated as the weighted average of the assets’ expected returns. To calculate risk, correlation is first converted into covariance by multiplying it by the two asset volatilities; portfolio variance then combines each weighted asset variance with a covariance term.
The answer assumes that “risk” means standard deviation of returns and works through the equal-weight case before giving the result for a portfolio weighted more heavily toward the lower-volatility asset. The example illustrates that imperfect correlation can reduce portfolio risk through diversification, while shifting weight toward the less volatile asset also lowers expected return in this scenario. These calculations depend on the supplied return, volatility, and correlation estimates; they do not address estimation error, changing correlations, or other definitions of risk.
Key ideas
- Expected portfolio return is the sum of asset expected returns weighted by their portfolio shares.
- Asset correlation can be converted to covariance using both assets’ standard deviations.
- Portfolio variance includes weighted individual variances and a cross-asset covariance term.
- Imperfect correlation can lower portfolio risk through diversification.
- The example assumes standard deviation is the intended measure of risk.
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Full text
# How to calculate portfolio return and risk based on given scenario
# How to calculate portfolio return and risk based on given scenario
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 by rsx (score 1)
https://quant.stackexchange.com/a/70178
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%.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.