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Solving Two-Asset Portfolio Weights for a Target Return

Article Quant Q&A · Author: idknuttin
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Summary

The document explains how to determine the weights of two stocks when the portfolio must achieve a specified expected return. It uses scenario probabilities and stock returns to calculate each asset’s expected return, then combines the target-return equation with the requirement that the two weights sum to one. Solving those two equations determines the allocation; covariance is not needed to find weights when the return target is the only stated constraint.

The response also gives the portfolio variance formula, which incorporates each asset’s variance and their covariance. This helps assess risk once the weights are known and may support choosing a minimum-variance allocation for a specified expected return. The treatment assumes a fully invested portfolio and does not discuss constraints such as short selling, transaction costs, or additional assets.

Key ideas

  • Calculate each stock’s expected return by weighting scenario returns by their probabilities.
  • Set the weighted average of the two expected returns equal to the portfolio return target.
  • Use the condition that the two portfolio weights sum to one to solve for both weights.
  • Use variances and covariance to calculate portfolio variance after determining the weights.

Tags

EquitiesPortfolio constructionStatistics
Full text
# how to find the weights in a portfolio?


# how to find the weights in a portfolio?












Compute the weights in a portfolio consisting of two kinds of stocks if the expected return on the portfolio is to be $E(K_v)=10\%$, given the following information on the returns on stock 1 and 2: $$ \begin{matrix} Scenerio & probability & return K_1 & return K_2 \\ \omega_1 & 0.1 & -10\% & 10\% \\ \omega_2 & 0.3 & 0\% & -5\% \\ \omega_3 & 0.6 & 15\% & 20\% \\ \end{matrix} $$

I found $E(K_1)=8\%$ and $(E_2)=11.5\%$ so $0.08\omega_1 + 0.115\omega_2 = 0.1$ But I don't know how to find the weights? I think the covariance will help me so I found that to equal $0.0109$ but I am not sure if it is correct and I don't know how to find the weights?

## Answer by Neeraj (score 1)

https://quant.stackexchange.com/a/24436

In case of 2 securities, each and every combination of portfolio lies on efficient frontier. In your question, you have given to achieve expected return of exactly 10%. So, we have $$E(R_p)=w_1E(K_1) + w_2E(K_2)=0.10 \tag{1}$$ subject to: $$w_1 + w_2=1 \tag{2}$$ Solve your equation 1 and 2 to get $w_1$ and $w_2$. Resulting weights would lead to minimum variance for given expected return. Variance of portfolio: $$var(R_p)= w_1^2 \sigma_{K_1}^2 + w_2^2 \sigma_{K_2}^2 + 2 w_1 w_2\, cov(K_1, K_2) $$ where, $\sigma_{K_1}$ and $\sigma_{K_2}$ are standard deviation of $K_1$ and $K_2$ respectively.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.