Asked by stephan vailes on May 28, 2024
Verified
Assume that an investor invests in one risky and one risk free asset. Let σm be the standard deviation of the risky asset and b the proportion of the portfolio invested in the risky asset. The standard deviation of the portfolio is then equal to:
A)
B)
C) (1 - b)
D) b
Risky Asset
A financial instrument or investment that has a high degree of uncertainty regarding its returns or potential for loss.
Standard Deviation
An indicator of the degree to which a collection of numbers diverges or spreads from their average, signaling the extent of the distribution from the central value.
Portfolio
A collection of financial investments like stocks, bonds, commodities, and cash equivalents, as well as their mutual, exchange-traded, and closed-fund counterparts.
- Understand the concept of standard deviation as a measure of risk in a portfolio, particularly in portfolios containing both risky and risk-free assets.
Verified Answer
σP = √[(bσS)^2 + ((1-b)σF)^2 + 2b(1-b)ρσSσF]
where σS is the standard deviation of the risky asset, σF is the standard deviation of the risk-free asset (which is 0), ρ is the correlation coefficient between the two assets (which is also 0 for a risk-free asset), and b is the proportion of the portfolio invested in the risky asset.
Plugging in the values given in the question, we get:
σP = √[(bσS)^2 + ((1-b)0)^2 + 2b(1-b)(0)(σS)]
= √[b^2σS^2]
= bσS
So the standard deviation of the portfolio is equal to b times the standard deviation of the risky asset. Therefore, the correct answer is D.
Learning Objectives
- Understand the concept of standard deviation as a measure of risk in a portfolio, particularly in portfolios containing both risky and risk-free assets.
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