probability ratio

Remington Wealth Partners Case Scenario

Preston Remington is the managing partner of Remington Wealth Partners. The firm manages high-net-worth private client investment portfolios using various asset allocation strategies. Analyst Hannah Montgomery assists Remington.

Remington and Montgomery’s first meeting of the day is with a new client, Spencer Shipman, who recently won $900,000 in the lottery. Shipman wants to fund a comfortable retirement. Earning a return on his investment portfolio that outpaces inflation over the long term is critical to him. He plans to withdraw $54,000 from the lottery winnings investment portfolio in one year to help fund the purchase of a vacation home and states that it is important that he be able to withdraw the $54,000 without reducing the initial $900,000 principal. Montgomery suggests they use a risk-adjusted expected return approach in selecting one of the portfolios provided in Exhibit 1.

Exhibit 1

Investment Portfolio One-Year Projections

ReturnStandard DeviationPortfolio 110.50%20.0%Portfolio 29.0013.0Portfolio 37.7510.0

All data are tax adjusted.

Remington and Montgomery discuss the importance of strategic asset allocation with Shipman. Remington states that the firm’s practice is to establish targeted asset allocations and a corridor around the target. Movements of the asset allocations outside the corridor trigger a rebalancing of the portfolio. Remington explains that for a given asset class, the higher the transaction costs and the higher the correlation with the rest of the portfolio, the wider the rebalancing corridor. Montgomery adds that the higher the volatility of the rest of the portfolio, excluding the asset class being considered, the wider the corridor.

Remington and Montgomery next meet with client Katherine Winfield. The firm had established Winfield’s current asset allocation on the basis of reverse optimization using the investable global market portfolio weights with further adjustments to reflect Winfield’s views on expected returns.

Remington and Montgomery discuss with Winfield some alternative asset allocation models that she may wish to consider, including resampled mean–variance optimization (resampling). Remington explains that resampling combines mean–variance optimization (MVO) with Monte Carlo simulation, leading to more diversified asset allocations. Montgomery comments that resampling, like other asset allocation models, is subject to criticisms, including that risker asset allocations tend to be under-diversified and the asset allocations inherit the estimation errors in the original inputs.

Montgomery inquires whether asset allocation models based on heuristics or other techniques might be of interest to Winfield and makes the following comments:

1. The 60/40 stock/bond heuristic optimizes the growth benefits of equity and the risk reduction benefits of bonds.
2. The Norway model is a variation of the endowment model that actively invests in publicly traded securities while giving consideration to environmental, social, and governance issues.
3. The 1/N heuristic allocates assets equally across asset classes with regular rebalancing without regard to return, volatility, or correlation.

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-PortfolioBYCZExpected return (%)5.707.10Expected volatility (%)5.107.40Current portfolio allocations (%)4060Probability (%)Minimum Expected Return (%)Time horizon: 10 years   992.902.50   903.402.80   853.603.00Time horizon: 20 years   955.105.40   905.205.70   855.605.90

Assume 0% correlation between the time horizon portfolios.

Question

Which of the portfolios provided in Exhibit 1 has the highest probability of enabling Shipman to meet his goal for the vacation home?

1. A.
2. Portfolio 1
3. B.
4. Portfolio 2
5. C.
6. Portfolio 3

Solution

B is correct. Portfolio 2 has the highest probability of enabling Shipman to meet his goal for the vacation home. All three of the portfolios’ expected returns over the next year exceed the 6.0% (see calculations below) required return threshold to avoid reducing the portfolio. However, on a risk-adjusted basis, Portfolio 2 (probability ratio of 0.231) has a higher probability of meeting and surpassing the threshold than either Portfolio 1 (probability ratio of 0.175) or Portfolio 3 (probability ratio of 0.225).

Step 1Calculate the required return threshold: 54,000 ÷ 900,000 = 0.06 = 6.0%.

Step 2To decide which allocation is best for Shipman, calculate the probability ratio:

[E(R_P) – R_L] ÷ σ_P, where

R_P = The return for the portfolio

R_L = The required return threshold

σ_P = The standard deviation of the portfolio

Portfolio 1: (10.50% – 6.0%) ÷ 20.0% = 4.50% ÷ 20.0% = 0.225.

Portfolio 2: (9.00% – 6.0%) ÷ 13.0% = 3.00% ÷ 13.0% = 0.231. (Highest)

Portfolio 3: (7.75% – 6.0%) ÷ 10.0% = 1.75% ÷ 10.0% = 0.175.

A is incorrect. Portfolio 1 was chosen because it has the highest projected return.

C is incorrect. Portfolio 3 was chosen because it has the lowest projected standard deviation.

Principles of Asset Allocation Learning Outcome

1. Recommend and justify an asset allocation using mean–variance optimization

老师我想问一下这里的考点是Model1里面的还是Model3里面的衡量SAA用什么的方法？算得这个probability ratio和在SAA里面衡量好坏的information ratio 的区别是什么