Practice Problem – Portfolio Std Dev Allocation

Practice Question

Intro to Finance
Portfolio Theory
Standard Deviation
Short Answer

Assume that you have decided to invest in a combination of the Apple Inc. stocks with an expected return of 5 percent and the Risk-Free asset. The standard deviation of Apple Inc. stocks’ returns is 10%. What fraction of your wealth should be invested in the risk-free asset, so that your portfolio will have a target standard deviation of 7 percent? The risk-free rate is 2 percent.

Answer +
0.300
Explanation +
Step 1: Define the Variables
  • w = fraction of wealth in Apple Inc. stock
  • σ_A = 10%
  • σ_RF = 0%
  • σ_P = 7% (target portfolio standard deviation)
Step 2: Use the Standard Deviation Formula

Since the risk-free asset has zero volatility, the portfolio standard deviation simplifies to:

σ_P = w × σ_A
=> 0.07 = w × 0.10
=> w = 0.07 / 0.10 = 0.7
Step 3: Find the Risk-Free Allocation
1 - w = 1 - 0.7 = 0.3
Final Answer

30% of your wealth should be invested in the risk-free asset to achieve a portfolio standard deviation of 7%.