Asset Allocation - choosing the optimal portfolio

Emma Young, a 47-year-old single mother of two daughters, ages 7 and 10, recently sold a business for $5.5 million net of taxes and put the proceeds into a money market account. Her other assets include a tax-deferred retirement account worth $3.0 million, a $500,000 after-tax account designated for her daughters’ education, a $400,000 after-tax account for unexpected needs, and her home, which she owns outright.

Her living expenses are fully covered by her job. Young wants to retire in 15 years and to fund her retirement from existing assets. An orphan at eight who experienced childhood financial hardships, she places a high priority on retirement security and wants to avoid losing money in any of her three accounts.

A broker proposes to Young three portfolios, shown in Exhibit 1. The broker also provides Young with asset class estimated returns and portfolio standard deviations in Exhibit 2 and Exhibit 3, respectively. The broker notes that there is a $500,000 minimum investment requirement for alternative assets. Finally, because the funds in the money market account are readily investible, the broker suggests using that account only for this initial investment round.

Exhibit 1

Proposed Portfolios

Asset Class Portfolio 1 Portfolio 2 Portfolio 3
Municipal Bonds 5% 35% 30%
Small-Cap Equities 50% 10% 35%
Large-Cap Equities 35% 50% 35%
Private Equity 10% 5% 0%
Total 100% 100% 100%

Exhibit 2

Asset Class Pre-Tax Returns

Asset Class Pre-Tax Return
Municipal Bonds 3%
Small-Cap Equities 12%
Large-Cap Equities 10%
Private Equity 25%

Exhibit 3

Portfolio Standard Deviations

Proposed Portfolio Post-Tax Standard Deviation
Portfolio 1 28.2%
Portfolio 2 16.3%
Portfolio 3 15.5%

Young wants to earn at least 6.0% after tax per year, without taking on additional incremental risk. Young’s capital gains and overall tax rate is 25%.

According to CFAI, the right answer is portfolio 3. Portfolio 1, 2 and 3 have after-tax returns of 9.15%, 6.64% and 6.68% respectively. When you analyze risk-adjusted return using Roy Safety first ratio (Er-MAR/st deviation): for Portfolio 1, 2 and 3 you get 11%, 4% and 4% respectively which makes portfolio 1 the most efficient to me. Further, Portfolio 1 meets the minimum alternative investment (PE allocation of 10%). Can someone explain why portfolio 3 is still the right answer? Thanks!

Portfolio 3 has the greatest sharp ratio, more importantly the clients return requirement can be met (6% return) with the lowest amount of risk (standard deviation of returns) which is a high priority for this risk averse client.

Should also add that the 500k minimum is simply a minimum investment to get exposure to the PE asset class. This helps you rule out portfolio 2 because 5% of 5.5m doesn’t meet the .5m minimum investment.
The client doesn’t explicitly state they would like exposure to PE.

Thank you! How are you calculating a Sharpe without a given risk free rate? I thought the RSF ratio would be more appropriate here given a minimum threshold. How should we know which ratio is applicable when?

The ratios are essentially the same only the RSF ratio replaces the risk free rate with the minimum return requirement. You should review your RSF calculations as the answers don’t appear accurate.

Key point to remember here is the risk averse natural of the client

So the expected after-tax return #s are accurate as verified by the CFAI answers itself.

With the given standard deviations and MAR of 6%, I would assume the RSFs for pf 1,2 and 3 would be respectively : (9.15-6)/28.2= 11%, (6.64-6)/16.3=4% and (6.68-6)/15.5=4%…Would you plz mind pointing out any flaws in my RSF formulae? Thanks

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I stand corrected, the issue here then is the use of the RSF ration rather than the sharp ratio. I would always default to the sharp Vs the RSF to understand the return per unit of risk when figuring out which portfolio is more efficient.
The key point here is the clients risk aversion so if you can generate the required return by taking less risk it’s typically the most appropriate option even if that option isn’t the most efficient in terms of risk adjusted returns.
Not to over analyse the question but when you focus on efficiency e.g. ratios, you need be be aware of their flaws. The obvious one here is that PE exposure - much like property, can rely on appraisal pricing which smooths returns and makes them appear very efficient when in reality, there just isn’t a frequent mark price to accurately reflect the true deviation of returns.

Ok so just got a question wrong in the CFA Qbank that has cleared up when you would use the RSF ratio rather than the sharp. If the question ask which portfolio has the “highest probability” of enabling a client to meet their return objective, use the RSF ratio.
Sharp ratio is all about efficiency of returns on a risk adjusted basis

Ok very helpful and makes sense, thanks! I should have included that this question was framed as:

Q. Determine which proposed portfolio most closely meets Young’s desired objectives.

One last question on this sorry! What risk free rate exactly are we using to calculate to calculate the SR? Probably less relevant as you mentioned in this case, there are other factors into play including the PE issue, but just curious.

-Maisha

The sharp ratio tells you what level of return is per unit of risk taken. With this in mind if a risk free rate isn’t provided assuming zero or that the returns provided are above a risk free rate is fine as long as the approach is consistent across each of the portfolios you are relatively assessing.

You could answer this question by simply calculating the returns, discovering that they all meet the return requirement of 6% and then simply look for the portfolio that has the lowest standard deviation of returns. The more I look at this question the more I realise calculating a sharp or a RSF ratio isn’t a necessary step

Maisha did you asked to CFAI if there is any error? Do you have the complete answer from CFAI? Because it’s so confusing to me:

  1. I agree with Tez4715 that SR should be a possible way, but it’s so strange that they provide the minimum return (6%) and not the risk free…it is not aligned to other questions format and really misleading
  2. If the minimum 500k investment in alternative asset is a requirement, only PF1 respect this condition

The only way to be PF3 (and in this I agree with the last reply of Tex4715) is that “Young wants to earn at least 6.0% after tax per year, without taking on additional incremental risk
In this case the point could be to select the PF with lower volatility, given that it’s after tax return is at least 6% (make sense??)…but it’s still not clear the 500k of alternative investment requirement.

Bye

Ste

Hello there, yes I agree in that I find the question very misleading given no rf rate and only MAR.

I haven’t reached out to CFAI, not sure exactly who/which department to contact and how?

Here’s the below answer from CFAI which makes sense, but again not completely given how the question is framed and lack of rf rate.

Determine which proposed portfolio most closely meets Young’s desired objectives. (Select one.)

Q. Justify your response to the previous question.

Solution

Portfolio 3 comes closest to meeting Young’s desire to earn at least 6% after tax per year without taking on additional incremental risk. Portfolio 3 offers a lower standard deviation than Portfolio 2, as summarized in Exhibit 3, while producing approximately the same return. Portfolio 1 achieves the highest returns but at a much greater level of volatility than Portfolio 3, not satisfying Young’s risk criterion.
Given the $500,000 minimum investment requirement for alternative assets, at Young’s total portfolio size of $5.5 million, the suggested 5% allocation to private equity in Portfolio 2 results in only a $275,000 exposure, insufficient to invest in private equity. Thus, Portfolio 2, as presented, is not viable, whereas Portfolio 1, with a private equity investment of $550,000, meets the minimum requirement for alternative investments. This minimum investment requirement is not an issue for Portfolio 3 because it has no private equity component.

Asset Class Portfolio 3 Pre-Tax Return Post-Tax Return Resulting Return
Municipal Bonds 30% 3% 3.00% 0.90%
Small-Cap Equities 35% 12% 9.00% 3.15%
Large-Cap Equities 35% 10% 7.50% 2.63%
Private Equity 0% 25% 18.75% 0.00%
Total 100% 6.68%
Asset Class Portfolio 1 Pre-Tax Return Post-Tax Return Resulting Return
Municipal Bonds 5% 3% 3.00% 0.15%
Small-Cap Equities 50% 12% 9.00% 4.50%
Large-Cap Equities 35% 10% 7.50% 2.63%
Private Equity 10% 25% 18.75% 1.88%
Total 100% 9.15%
Asset Class Portfolio 2 Pre-Tax Return Post-Tax Return Resulting Return
Municipal Bonds 35% 3% 3.00% 1.05%
Small-Cap Equities 10% 12% 9.00% 0.90%
Large-Cap Equities 50% 10% 7.50% 3.75%
Private Equity 5% 25% 18.75% 0.94%
Total 100% 6.64%

From the answer I understand the the solution suggested by Tex4715 is the proper one: no need to calculate the RSF, but select the PF with lower volatility with a minimum return higher than 6%. I don’t understand the point about alternative assets, it’s like to say that it is minimum 500k if there is already some alternative asset

See the above comment, you don’t need to worry about PE exposure here