Question

A moderately risk-averse investor has 50 percent of her portfolio invested in stocks and 50 percent in riskfree Treasury bills. Show how each of the following events will affect the investor's budget line and the proportion of stocks in her portfolio: a. The standard deviation of the return on the stock market increases, but the expected return on the stock market remains the same. b. The expected return on the stock market increases, but the standard deviation of the stock market remains the same c. The return on risk-free Treasury bills increases.

Short answer

In summary, an increase in standard deviation (or risk) results in a steepened budget line and possibly a decreased proportion in stocks, an increase in expected return results in an upward shift of the budget line and possibly an increased proportion in stocks, and an increase in risk-free return results in a higher starting point of the budget line and possibly a decreased proportion in stocks.

Step-by-step solution

Effect of Increase in Standard Deviation

In financial terms, standard deviation represents risk. Therefore, if the standard deviation of the return on the stock market increases with the expected return remaining the same, this implies that the risk associated with stock investment increases while the potential return is unchanged. In response to this change, the budget line of the investor steepens because the potential risk (i.e., the cost) associated with each additional unit of potential return increases. This might cause the investor to reduce the proportion of stocks in her portfolio due to her risk-averse nature.

Effect of Increase in Expected Return

When the expected return on the stock market increases and the standard deviation remains the same, this means that the potential gain from stock investment becomes greater while the risk remains unchanged. Consequently, the budget line of the investor shifts upwards to reflect higher potential returns at each level of risk. Given this change, the investor might increase the proportion of stocks in her portfolio due to higher expected returns.

Effect of Increase in Risk-Free Return

If the return on risk-free treasury bills increases, this implies that the investor can earn a higher return without taking on any additional risk. This shifts the starting point of the budget line upwards, while the slope (representing the trade-off between risk and return) does not change. Given her risk-averse nature, the investor would likely increase the proportion of her portfolio invested in risk-free asset and decrease the proportion in stocks.

Key concepts

Risk Aversion

Risk aversion is the degree to which an investor is willing to take on risk in search of higher returns. A risk-averse investor prefers to avoid uncertainty and is inclined to choose safer investments over riskier ones. In the example given, the investor is described as moderately risk-averse, indicating she is cautious but not completely against taking risks.

This cautious nature influences her investment decisions. She is inclined to maintain a balanced portfolio that mitigates potential risks. If the risk associated with stocks increases, she may adjust her allocation toward safer assets, like Treasury bills. Conversely, if the potential returns increase significantly, a risk-averse investor might rethink her risk exposure to benefit from enhanced gains without drastically increasing her portfolio's risk.

Investment Risk

Investment risk relates to the likelihood of losing value in an investment. It encompasses the uncertainty regarding the returns expected from an investment.

In an investment portfolio, stocks are generally considered riskier compared to fixed-income securities like Treasury bills. This is due to market fluctuations that can affect stock prices. When the standard deviation of the return on stocks rises, it signals increased volatility and hence greater investment risk. A risk-averse investor, who values stability, may react by decreasing her holdings in stocks to reduce exposure to this increased risk.

Expected Return

Expected return is the estimate of the amount an investor anticipates earning from an investment over a specific period. It provides a forecast of potential gains without guaranteeing them, necessitating consideration of accompanying risks.

In the original exercise, when the expected return on stocks rises while the risk level remains stable, the investor's opportunity to gain increases. This change might make stocks more appealing despite the risk, convincing a moderately risk-averse investor to allocate a higher percentage of her portfolio towards stocks to take advantage of the higher expected returns.

Standard Deviation

Standard deviation in finance measures the amount of variation or volatility in an investment's returns. Greater standard deviation signifies larger swings in price, indicating higher risk.

An increase in the stock market's standard deviation, with no change in expected return, implies heightened uncertainty, making investments more unpredictable. Investors might see such situations as risky because they could lead to significant losses. Therefore, someone like a moderately risk-averse investor might reduce her stock investments in favor of securities with lower standard deviations, like Treasury bills, to conserve her capital.