To calculate the expected rate of return for Bronc Gerson's portfolio, we need to use the formula for the expected return based on the weights of each investment and their respective expected returns.

Current Portfolio Composition:

• Treasury Bills: $290,000 with an expected return of 4.5%
• S&P 500 Index Fund: $437,000 with an expected return of 8.8%
• Emerging Market Fund: $273,000 with an expected return of 12.2%

Total Portfolio Value:

$\text{Total Portfolio Value} = 290,000 + 437,000 + 273,000 = 1,000,000$

Expected Return Calculation:

The expected return $E(R)$ is calculated as: $E(R) = \left( \frac{V_1}{V_T} \times R_1 \right) + \left( \frac{V_2}{V_T} \times R_2 \right) + \left( \frac{V_3}{V_T} \times R_3 \right)$ Where:

• $V_1, V_2, V_3$ are the values of the investments
• $V_T$ is the total portfolio value
• $R_1, R_2, R_3$ are the expected returns for each investment

Plugging in the values: $E(R) = \left( \frac{290,000}{1,000,000} \times 0.045 \right) + \left( \frac{437,000}{1,000,000} \times 0.088 \right) + \left( \frac{273,000}{1,000,000} \times 0.122 \right)$

Calculating each term:

1. Treasury Bills: $\frac{290,000}{1,000,000} \times 0.045 = 0.01305$
2. S&P 500 Index Fund: $\frac{437,000}{1,000,000} \times 0.088 = 0.038536$
3. Emerging Market Fund: $\frac{273,000}{1,000,000} \times 0.122 = 0.033366$

Now, summing these results: $E(R) = 0.01305 + 0.038536 + 0.033366 = 0.085952$

Final Expected Rate of Return:

$E(R) \approx 0.085952 \times 100 \approx 8.60\%$

Thus, the expected rate of return for Bronc's portfolio is approximately 8.60% (rounded to two decimal places).

Part (b):

If Bronc moves all of his money from the Emerging Market Fund into Treasury Bills, the new portfolio composition will be:

• Treasury Bills: $290,000 + $273,000 = $563,000
• S&P 500 Index Fund: $437,000
• Emerging Market Fund: $0

New Total Portfolio Value: $1,000,000

New Expected Return Calculation:

$E(R) = \left( \frac{563,000}{1,000,000} \times 0.045 \right) + \left( \frac{437,000}{1,000,000} \times 0.088 \right) + \left( \frac{0}{1,000,000} \times 0.122 \right)$

Calculating each term:

1. Treasury Bills: $\frac{563,000}{1,000,000} \times 0.045 = 0.025335$
2. S&P 500 Index Fund: $\frac{437,000}{1,000,000} \times 0.088 = 0.038536$
3. Emerging Market Fund: $0 = 0$

Summing these results: $E(R) = 0.025335 + 0.038536 + 0 = 0.063871$

Final New Expected Rate of Return:

$E(R) \approx 0.063871 \times 100 \approx 6.39\%$

So, the expected rate of return on the resulting portfolio will be approximately 6.39%.

Would you like more details or do you have any questions? Here are some related questions to consider:

1. How does changing the allocation affect risk?
2. What are the implications of a lower expected return?
3. What are the benefits of investing in emerging markets?
4. How can Bronc balance risk and return in his portfolio?
5. What factors should Bronc consider before reallocating his investments?

Tip: Always consider your risk tolerance when reallocating investments, especially close to retirement.