Extract of question:
Finally, Remington and Montgomery discuss Isabelle Sebastian. During a recent conversation, Sebastian, a long-term client with a $2,900,000 investment portfolio, reminded Remington that she will soon turn age 65 and wants to update her investment goals as follows:
- Goal 1: Over the next 20 years, she needs to maintain her living expenditures, which are currently $120,000 per year (90% probability of success). Inflation is expected to average 2.5% annually over the time horizon, and withdrawals take place at the beginning of the year, starting immediately.
- Goal 2: In 10 years, she wants to donate $1,500,000 in nominal terms to a charitable foundation (85% probability of success).
Exhibit 2 provides the details of the two sub-portfolios, including Sebastian’s allocation to the sub-portfolios and the probabilities that they will exceed the expected minimum return.
Exhibit 2
Investment Sub-Portfolios & Minimum Expected Return for Success Rate
| Sub-Portfolio | BY | CZ |
|---|---|---|
| Expected return (%) | 5.70 | 7.10 |
| Expected volatility (%) | 5.10 | 7.40 |
| Current portfolio allocations (%) | 40 | 60 |
| Probability (%) | Minimum Expected Return (%) | |
| Time horizon: 10 years | ||
| 99 | 2.90 | 2.50 |
| 90 | 3.40 | 2.80 |
| 85 | 3.60 | 3.00 |
| Time horizon: 20 years | ||
| 95 | 5.10 | 5.40 |
| 90 | 5.20 | 5.70 |
| 85 | 5.60 | 5.90 |
Assume 0% correlation between the time horizon portfolios.
Question
Using Exhibit 2, which of the sub-portfolio allocations is most likely to meet both of Sebastian’s goals?
A. The current sub-portfolio allocation
B. A 43% allocation to sub-portfolio BY and a 57% allocation to sub-portfolio CZ
C. A 37% allocation to sub-portfolio BY and a 63% allocation to sub-portfolio CZ
Answer
C is correct. Sebastian needs to adjust the sub-portfolio allocation to achieve her goals. By adjusting the allocations to 37% × $2,900,000 = $1,073,000 in BY and 63% × $2,900,000 = $1,827,000 in CZ, she will be able to achieve both of her goals based on the confidence intervals.
Goal 1: Sebastian needs to maintain her current living expenditure of $120,000 per year over 20 years with a 90% probability of success. Inflation is expected to average 2.5% annually over the time horizon.
Sub-portfolio CZ should be selected because it has a higher expected return (5.70%) at the 90% probability for the 20-year horizon. Although sub-portfolio CZ has an expected annual return of 7.10%, based on the 90% probability of success requirement, the discount factor is 5.70%.
Goal 1: k = 5.70%; g = 2.50%.
Determine the inflation-adjusted annual cash flow generated by sub-portfolio CZ:
$1,827,000×(0.057−0.025)1−(1+0.0251+0.057)20=$120,432.04>$120,000
Goal 2: Sebastian wants to contribute $1,500,000 to a charitable foundation in 10 years with an 85% probability of success.
Sub-portfolio BY should be selected because it has a higher expected return (3.60%) at the 85% probability for the 10-year horizon. Although sub-portfolio BY has an expected annual return of 5.70%, based on the 85% probability of success requirement, the discount factor is 3.60%.
Goal 2: k = 3.60%.
Determine the amount needed today in sub-portfolio BY:
$1,500,000(1+0.036)10=$1,053,158.42<$1,073,000
A is incorrect: 40% × $2,900,000 = $1,160,000 in BY, and 60% × $2,900,000 = $1,740,000 in CZ.
I don’t quite get their solution. I did it another way:
For Goal 1: To find PV, N=20 , I/Y = 3.12195% [(1 + r / 1 + g) - 1], FV = 0 PV = $1,820,445.792
For Goal 2: 1,500,000/(1.036^10) = 1,053.422
Total (G1+G2) = 2,873,604.214
Allocation to CZ = 1,820,445.792/2,873,604.214 = 0.6335 (63%)
Allocation to BY = 1-0.63 = 0.37 (37%)
Would just like to check if this is a legit/sound way or if I just coincidentally got the same answer as the solutions? Many thanks.