To compare the returns from investing in $D$ and $e$, we need to calculate the expected returns based on the given interest rates, the initial investment of 6000 units, and the probability distribution of $e$.

Given:

• $d = 1$
• $e = \frac{1}{16}d = \frac{1}{16} \times 1 = \frac{1}{16}$
• Probability of $e$ being 16: 0.6
• Probability of $e$ going up to 32: 0.4
• Interest rate for $D$: 12% or 0.12
• Interest rate for $e$: 24% or 0.24
• Total investment: 6000 units, distributed equally in both $D$ and $e$, so 3000 units in $D$ and 3000 units in $e$.

Steps:

1. Return from Investment in $D$:

• The investment in $D$ is 3000 units.
• Return from $D$ at 12% interest: $\text{Return from } D = 3000 \times 0.12 = 360 \text{ units}$

2. Expected Return from Investment in $e$:

• The investment in $e$ is 3000 units.
• The value of $e$ can be either 16 or 32 with respective probabilities of 0.6 and 0.4.

Expected value of $e$: $E(e) = (16 \times 0.6) + (32 \times 0.4) = 9.6 + 12.8 = 22.4$

Expected Return from $e$: $\text{Expected Return from } e = E(e) \times 0.24 \times 3000$ $\text{Expected Return from } e = 22.4 \times 0.24 \times 3000 = 1612.8 \text{ units}$

3. Total Expected Returns:

• Total return from $D$ and $e$: $\text{Total Expected Return} = 360 + 1612.8 = 1972.8 \text{ units}$

Conclusion:

If you invest 6000 units, with 3000 in $D$ and 3000 in $e$, the total expected return would be 1972.8 units. The return from $e$ is much higher (1612.8 units) compared to $D$ (360 units), making $e$ the better option in terms of expected returns.

Would you like more details or have any questions?

Follow-up Questions:

1. How would the decision change if you could invest different amounts in $D$ and $e$?
2. What happens if the interest rates for $D$ and $e$ change?
3. How does the risk associated with $e$ compare to the guaranteed return from $D$?
4. What if you reinvest the