To determine which option is better between $D$ and $e$, we need to compare the expected returns based on their respective interest rates and the probabilities associated with $e$.

Given:

• $d = 1$
• $e = \frac{1}{16}d = \frac{1}{16} \times 1 = \frac{1}{16}$
• The value of $e$ can either be 16 with a probability of 0.6 or 32 with a probability of 0.4.
• Interest rate for $D$: 12% or 0.12
• Interest rate for $e$: 24% or 0.24

Steps:

1. Calculate the expected value of $e$: $E(e) = (16 \times 0.6) + (32 \times 0.4)$ $E(e) = 9.6 + 12.8 = 22.4$

2. Calculate the expected return for $e$: $\text{Expected Return for } e = E(e) \times \text{interest rate of } e = 22.4 \times 0.24 = 5.376$

3. Calculate the return for $D$: Since $D$ has a value of $d = 1$ with an interest rate of 12%, the return for $D$ is: $\text{Return for } D = 1 \times 0.12 = 0.12$

Conclusion:

The expected return for $e$ is 5.376, while the return for $D$ is 0.12. Clearly, the expected return for $e$ is much higher, making $e$ the better option based on expected returns.

Would you like more details or have any questions?

Follow-up Questions:

1. How would the scenario change if the interest rates were different?
2. What is the impact on the decision if the probability distribution for $e$ changes?
3. What happens if the value of $d$ is not 1 but another number?
4. How would inflation affect the comparison between $D$ and $e$?
5. What would be the decision if you consider risk-adjusted returns instead of expected returns?

Tip: When comparing investments, always consider both the expected returns and the associated risks.