To solve this problem, let's go through each part carefully based on the given information.

Given Data:

1. Outcomes and Payoffs:

   • Massive success $h$: Probability $\pi = 0.1$, Return $r = 300\% = 3.0$
   • Moderate success $m$: Probability $\pi = 0.2$, Return $r = 100\% = 1.0$
   • Failure $f$: Probability $\pi = 0.7$, Return $r = -50\% = -0.5$

2. Risk-Free Rate:

   • $r_0 = 7.0\% = 0.07$
   • This gives $R_0 = 1 + r_0 = 1.07$.

3. Excess Return:

   • Defined as $\hat{r} = r - r_0$.

Part (a): Expected Return of the Project

The expected return of the project, $E(r)$, can be calculated by taking the weighted average of the returns based on their probabilities:

$E(r) = \pi_h \cdot r_h + \pi_m \cdot r_m + \pi_f \cdot r_f$

Substitute the values:

$E(r) = (0.1 \cdot 3.0) + (0.2 \cdot 1.0) + (0.7 \cdot -0.5)$

Calculating each term:

• $0.1 \cdot 3.0 = 0.3$
• $0.2 \cdot 1.0 = 0.2$
• $0.7 \cdot -0.5 = -0.35$

Adding these:

$E(r) = 0.3 + 0.2 - 0.35 = 0.15$

So, the expected return of the project is 15%.

Part (b): Variance of the Returns of the Project

The variance of the returns, $\text{Var}(r)$, is calculated by taking the weighted average of the squared deviations from the expected return:

$\text{Var}(r) = \pi_h \cdot (r_h - E(r))^2 + \pi_m \cdot (r_m - E(r))^2 + \pi_f \cdot (r_f - E(r))^2$

Substitute $E(r) = 0.15$ and calculate each term:

1. For $r_h = 3.0$: $(r_h - E(r))^2 = (3.0 - 0.15)^2 = 2.85^2 = 8.1225$ $\pi_h \cdot (r_h - E(r))^2 = 0.1 \cdot 8.1225 = 0.81225$

2. For $r_m = 1.0$: $(r_m - E(r))^2 = (1.0 - 0.15)^2 = 0.85^2 = 0.7225$ $\pi_m \cdot (r_m - E(r))^2 = 0.2 \cdot 0.7225 = 0.1445$

3. For $r_f = -0.5$: $(r_f - E(r))^2 = (-0.5 - 0.15)^2 = (-0.65)^2 = 0.4225$ $\pi_f \cdot (r_f - E(r))^2 = 0.7 \cdot 0.4225 = 0.29575$

Adding these values together:

$\text{Var}(r) = 0.81225 + 0.1445 + 0.29575 = 1.2525$

Thus, the variance of the returns of the project is approximately 1.2525.

Would you like a breakdown of any of these steps or further assistance with interpreting the results?

Related Questions:

1. How does the standard deviation of the returns compare to the expected return?
2. What would be the impact on the expected return if the probabilities of each outcome changed?
3. How does investing more or less (changing $\alpha$) impact Frankie’s expected consumption?
4. What is the role of the risk-free rate in this model?
5. How would the analysis change if there was an additional outcome?

Tip:

When calculating variance, always remember to square each deviation before multiplying by the probability—it’s a common error to miss squaring or applying the probability factor only once.