Problem 22 - A First Course in Probability

Question

Would the results of Example \(5 \mathrm{f}\) change if the investor were allowed to divide her money and invest the fraction \(\alpha, 0<\alpha<1,\) in the risky proposition and invest the remainder in the risk-free venture? Her return for such a split investment would be \(R=\alpha X+(1-\alpha) m\)

Step-by-Step Solution

Answer

In summary, when the investor divides her money between the risky and risk-free ventures with a fraction \(0 < \alpha < 1\), the expected return of the split investment \(E(R) = \alpha E(X) + (1 - \alpha)m\) will be less than the expected return of solely investing in the risky proposition. The standard deviation of her split investment, \(SD(R) = \sqrt{\alpha^2 * Var(X)}\), will also be reduced, indicating a reduction in risk.

Step 1: Calculate the Expected Return

To calculate the expected return, we will determine the expected value of R, which can be written as: E(R) = E(αX + (1 - α)m) Since m is a constant (the return of the risk-free venture), we can separate the expectation: E(R) = αE(X) + (1 - α)m

Step 2: Calculate the Standard Deviation of the Return

Next, we will calculate the standard deviation of R. The variance of R can be expressed as follows: Var(R) = Var(αX + (1 - α)m) Since m is a constant, its variance is 0. Thus, we can write: Var(R) = α^2 * Var(X) Now, to find the standard deviation, we take the square root of the variance: SD(R) = √(α^2 * Var(X))

Step 3: Compare the Results

Now that we have determined the expressions for the expected return and standard deviation of the split investment, we can compare these to the values from the case when the investor only invests in the risky proposition. In that case, the expected return is E(X) and the standard deviation is SD(X). If we compare the expected returns, we have: E(R) = αE(X) + (1 - α)m When α = 1, we get E(R) = E(X), which represents the case where the entire investment is in the risky proposition. For 0 < α < 1, E(R) will be a weighted average of the risky and risk-free returns. Since the risk-free return (m) is generally less than the expected return of the risky proposition (E(X)), E(R) will be less than E(X) for 0 < α < 1. In terms of the standard deviation, we found: SD(R) = √(α^2 * Var(X)) When α = 1, SD(R) = SD(X), representing the full investment in the risky proposition. As α decreases (0 < α < 1), the standard deviation of the split investment will decrease, indicating a reduction in risk. In summary, when the investor is allowed to divide her money between the risky and risk-free ventures, the expected return of her split investment will be less than the expected return of solely investing in the risky proposition, while the standard deviation (risk) will also be reduced.

Key Concepts

Variance

Variance is a crucial concept in finance, particularly when discussing investments. It measures how much the returns on an investment are spread out over time. Essentially, variance gives you an idea of the investment's risk. Higher variance indicates that the returns on the investment are more spread out, suggesting higher risk. Conversely, a lower variance indicates less spread and lower risk.

In mathematical terms, variance is calculated by averaging the squared deviations from the mean return. This means if you have returns data for an investment, you first find the average return, subtract this average from each return, square the result, and then average those squared differences. In symbols, it's expressed as:

• The variance for a random variable X is given by: o \( \text{Var}(X) \ = \ \sum (X_i - \bar{X})^2 / N \), where \( \bar{X} \) is the mean and N is the number of observations.
• For our case, because \( m \) (the risk-free return) is a constant, its variance is zero, simplifying the variance expression to \( \alpha^2 \cdot \text{Var}(X) \).

Figuring the variance of an investment allows investors to understand the volatility and helps in making informed decisions about the level of risk they are willing to take.

Standard Deviation

Standard deviation is closely related to variance, and it's another crucial measure of risk in an investment. While variance provides a numerical value indicating how spread out the returns are, the standard deviation offers a more intuitive scale since it is expressed in the same units as the data (for example, percentage points).

To find the standard deviation, you take the square root of the variance:

• The formula for standard deviation (SD) is: o \( \text{SD}(X) = \sqrt{\text{Var}(X)} \).
• In the context of the given problem, when mixing risky and risk-free investments, standard deviation helps to understand the mix's total risk via this formula: \( \text{SD}(R) = \sqrt{\alpha^2 \cdot \text{Var}(X)} \).

By combining investments with different standard deviations, such as a risky and a risk-free one, you can effectively reduce the overall investment risk. As a result, standard deviation is a vital tool for evaluating and comparing the risk of various investment choices. A lower standard deviation in your investment means more predictable returns, which is generally preferable to investors seeking stability.

Risk-Free Investment

A risk-free investment is a critical part of understanding returns and investment decisions. Although no investment is entirely free of risk, the term typically refers to investments considered extremely stable, such as government bonds from stable economies. For investors, a risk-free investment provides a baseline for expected return with almost no volatility.

Risk-free investments are significant for several reasons:

• They provide safety and reliability, contributing to a balanced portfolio when mixed with riskier investments.
• The return from a risk-free investment, often denoted as "m" in calculations, does not change, leading to zero variance and standard deviation.
• In our context, the risk-free rate is used as part of a broader investment strategy, providing a stable income component when combined with a more volatile, risky investment.

When including a risk-free investment as part of your portfolio, you hedge against the uncertainties associated with riskier investments. The risk-free component minimizes overall risk, reducing the portfolio's standard deviation and variance, making it an important tool for investors to manage their risk levels and stabilize returns.