## Abstract

The authors provide an operating manual for some useful, beyond-the-basics private wealth management tools. In this article, they have assembled a set of practical quantitative tools that private wealth managers can benefit from knowing and using. Believing wealth managers must embrace the tools of mathematics and statistics, they focus on five quantitative tools. Their presumed audience is a private wealth manager who has mastered the elementary quantitative toolkit. While the more-specialized tools presented in this article are developed in more detail elsewhere, the authors offer a curated selection of the large literature of private wealth management and attempt to convey the underlying intuition in plain English. Further, they offer shortcuts to implementing the calculations in Excel.

**TOPICS:** Wealth management, quantitative methods

Private wealth management is the subfield of investment management focused on high-net-worth individuals and families. It includes both taxable investment management and complex personal financial planning. Private wealth managers face a number of daunting prospects in designing appropriate portfolios and sound financial advice for their clients. In this article, we have assembled a set of five quantitative tools that we believe private wealth managers can benefit from knowing and using. We believe that wealth managers must embrace the tools of mathematics and statistics. Only when portfolio design and financial planning insights are built on an analytically sound foundation can the important interpersonal client relationship aspect of private wealth management begin.

Our presumed audience is private wealth managers who have mastered the elementary quantitative toolkit, perhaps for their CFA, CFP, CAIA, or CPM exam. They know the time value of money, mean–variance portfolio math, geometric means, the capital asset pricing model (CAPM), and similar basics. The more-advanced tools we present here were developed in more detail elsewhere, making this article something of a literature review. Similar to Jennings et al. [2011], we offer a curated selection of the large literature of private wealth management. With our narrower focus on methods and tools, we hope to convey the underlying intuition and relevance in plain English.

In the following, we present tools for synthesizing the difference between geometric and arithmetic average returns, accurately calculating real returns, quantifying the benefits of diversification, and managing clients’ expectations—both in terms of how correlated assets’ returns will differ and the appropriate portfolio spending rate, given expected return and risk tolerance levels. For each, we discuss the relevance of the tool, give an example, and demonstrate how to perform the analysis in Microsoft Excel.

We augment the article and the spreadsheet discussion with some custom Excel functions. These custom Excel functions allow the private wealth manager to more easily implement the mathematical tools; we provide details in the Appendix.

**WHAT RETURN SHOULD I USE TO FORECAST?**

Selecting a rate of return is critical to estimating future portfolio values. Private wealth managers and investors alike have long used historical market return averages to make informed decisions about the future. Some use historical values directly, while others use historical levels and relationships to inform forward-looking capital market assumptions.

In making forecasts, investors face a dilemma in choosing between historical arithmetic averages and geometric averages for making projections. Recall that the arithmetic average characterizes a typical individual year, whereas the geometric average characterizes the compound growth of an investment over multiple years. They can differ substantially. How best to generate an unbiased forecast of ending portfolio values?

**Significance**

Choosing the correct average is a nontrivial decision. The difference between the arithmetic and geometric averages is typically larger than 1% for core equity investments. For the riskiest asset classes, such as private equity, small-capitalization stocks, and emerging market equities, it is higher still. For diversified portfolios, the difference between the arithmetic and geometric averages can be 0.5% or more at typical risk levels.

These differences are economically significant because private wealth management investment involves long decisions. At a 30-year horizon, forecasting with the two different averages produces a terminal wealth difference of about 25% for a typical diversified portfolio. For the riskiest asset classes, the terminal wealth varies by a factor of two—a 100% difference (see Jacquier, Kane, and Marcus [2003], for a similar multiple).

An arithmetic average will always be higher than a geometric average, but as return variation increases, so does the gap between the two types of averages. As a result, it becomes increasingly important to choose the appropriate rate as the portfolio mix or asset classes considered becomes riskier (because return–variation risk widens the gap between the two rates).

A problem emerges when using an arithmetic average to make forecasts of cumulative returns. Compounding using an arithmetic average provides a cumulative arithmetic return that has positive, or right, skew. The result is not representative of what an investor is likely to actually experience and overstates the typical investment gains. Investors have less than a 50/50 chance of achieving forecasted wealth that is calculated using arithmetic averages. (Statistically, this is *bias*.) Even if return estimates are unbiased, terminal wealth estimates will be biased. Symmetrical errors in the expected return produce asymmetrical errors in wealth. In fact, the odds of attaining the forecast wealth, calculated using arithmetic averages, decreases substantially as the investment horizon increases. *Private wealth managers should not use the arithmetic average*.

In light of this bias, some have advocated for the use of geometric averages. For example, Hughson, Stutzer, and Yung [2006] illustrate that median wealth estimates generated from geometric averages, while lower, are more useful for investors. Likewise, Jacquier, Kane, and Marcus [2003, p. 46] note that “many in the practitioner community seem to prefer geometric averages.” However, with typical forecast horizons, the geometric average gives terminal wealth estimates that are biased too low (Blume [1974], p. 636). *Private wealth managers should not use the geometric average*.

A better return path lies somewhere between the two averages. Blume [1974] first proposed this happy medium after finding that terminal wealth is closely approximated by using a weighted average of arithmetic and geometric means. Equation (1) represents Blume’s weighted-average approximation for the return to generate an unbiased terminal wealth forecast:

1where *H* is the length of the historical estimate sample and *F* is the forecast investment horizon. That is, the unbiased forecast, *R*(*F*), is a weighted average of the arithmetic and geometric averages (µ_{a} and µ_{g}, respectively). The notation *R*(*F*) highlights that the best forecast of returns is a *function* of the *F* forecast investment horizon. *
Equation (1) provides private wealth managers with a better forecast than the arithmetic or geometric mean*. We can see from Equation (1) that as the forecast horizon approaches the historical sample length, *F* → *H*, the weighted average *R*(*F*) will approach the geometric average. Conversely, as the difference between *H* and *F* increases, the arithmetic average becomes relatively more important.^{1}

Blume [1974] assumed a normal return distribution, but some research posits that lognormal distributions are more appropriate. Jacquier, Kane, and Marcus [2003], for example, reach a similar conclusion to Blume with a lognormal return distribution. One advantage of assuming a lognormal relationship is that it allows one to directly express the relationship between arithmetic and geometric averages. (Lognormality also has the attractive property that it precludes losses greater than 100%.) Equation (2) represents the Jacquier, Kane, and Marcus [2003] weighted-average formula for the return to generate an unbiased terminal wealth forecast:

2where µ_{g} and σ^{2} are the geometric return average and its variance. The first two terms, in the curly braces, equal the arithmetic average of a return series. The third term of the equation, in square brackets, reduces the growth rate estimate for an unbiased terminal wealth forecast. For long forecast horizons, *F*/*H* approaches one, and the third term cancels the second term; *R*(*F*) therefore approaches the geometric return estimate. For short forecast horizons with *F*/*H* approaching zero, *R*(*F*) converges on the arithmetic average. Conversely, we again see that the geometric average carries more weight as the investment horizon increases. Notably, the forecast horizon *F* is not restricted to be less than the historical horizon *H*, as in Equation (1). The Equation (2) technique is important, having influenced Hughson, Stutzer, and Yung [2006], Kan and Zhou [2009], and Grinblatt and Linnainmaa [2011], among others. *
Equation (2) provides private wealth managers with a better long-term forecast than the arithmetic or geometric mean*.

**Example**

Assume that an investor reallocates a tax-deferred account at the age of 35 and has a 30-year investment horizon. She plans to invest $250,000.

To evaluate this scenario, we apply Equation (2), using 50 years of returns for the S&P 500 Index. From 1965–2014, the geometric average is 9.84%. The standard deviation of this series of log returns is 16.55%.^{2} Because our forecast period is 30 years, we set *F* = 30. The return sample is 50 years in length, so *H* = 50. By Equation (2), our weighted-average estimate is now 10.38%:

Based on this forecast, the new investor’s projected wealth 30 years hence is $6.1 million at the 11.27% arithmetic rate and $4.8 million at the unbiased 10.38% rate, both from the single $250,000 investment. Thus, the arithmetic approach overstates the prospective ending portfolio value by about 25%. In contrast, using an unbiased weighted-average growth rate provides a portfolio value that is smaller but more likely to occur. Using arithmetic averages is a good start but a flawed one. The resulting cumulative arithmetic returns are overly optimistic. They overstate the wealth investors will typically experience since they positively skew the distribution of potential outcomes. As investment horizons expand and errors compound, the projected cumulative returns become increasingly unrealistic.

Likewise, the geometric average is unduly pessimistic. Using the geometric average of 9.84% gives projected investor wealth of $4.2 million. This is lower than the unbiased value by about 14%. While the geometric average is more conservative and closer to Equation (2) than the arithmetic average, it produces an economically meaningful difference in ending portfolio wealth.^{3}

A cumulative return that uses a weighted average of arithmetic and geometric returns found in Equation (2) is a better approach that is more likely to actually occur for the investor. Equation (2) enables more-realistic planning by the private wealth manager.

**Excel Implementation**

We re-create the Jacquier, Kane, and Marcus [2003] weighted-average growth estimate from Equation (2) using a simple Excel function available in the downloadable file detailed in the Appendix:

Four inputs are required to use this function: the geometric return, the lognormal standard deviation, the length of the estimation period, and the length of the forecast period.^{4}

**WHAT IS THE ACCURATE WAY TO ADDRESS INFLATION?**

Portfolio managers understand the ability of inflation to destroy purchasing power. The cumulative effect of rising price levels can be deceptive. The effects can be quite large, particularly over long investment horizons. Accordingly, many managers account for inflation using an approximation that relates real inflation-adjusted rates of return and nominal rates of return. The typical real-rate approximation used is:

3This approximation is clearly superior to merely ignoring inflation. However, this approximation results in real rates that are systematically too high.

**Significance**

A more precise inflation-adjusted rate is found in Fisher [1930]:

4Under many nominal rate and inflation rate assumptions, the difference between the two approaches, *i _{nominal}
* ×

*i*, may appear negligible. However, akin to the previous section’s arithmetic

_{inflation}*versus*geometric discussion, the small deviation compounds to become meaningfully important over the longer investment horizons relevant to private wealth management. Equation (4) is more accurate. Further, because it is lower, Equation (4) will be more conservative.

**Example**

We apply Equation (3) and Equation (4) to highlight the different results. Let us assume a nominal rate of 10% and an inflation rate of 3%. An approximate real rate of return is 7%, per Equation (3) and most investors’ intuition. Using the precise formula, we see that the approximation is overly optimistic—the accurate rate, per Equation (4), is 6.796%.

The precise calculation is more than 20 basis points lower than the Fisher approximation. Again, a first impression may be that this is a minor disparity, yet it becomes noteworthy as investment horizons expand.

Failing to account precisely for inflation results has potentially large cumulative effects. Let us assume that a 25-year-old investor inherits $500,000 and begins an investment program to retire at age 65, for a 40-year pre-retirement investment horizon. If the $500,000 inheritance is invested for 40 years at the approximate 7.00% real rate, the ending wealth at age 65 is $7.5 million in today’s dollars. In contrast, the precise estimate—at the correct 6.796% real rate—is $6.9 million. The approximation is 8% higher.

If this lump sum is then converted into a 30-year annuity, the approximate real rate indicates a monthly annuity of $49,524, while the precise real rate shows a monthly annuity of $44,953. The approximation is 10% higher.

It is good that financial planners and private wealth managers systematically incorporate the effects of inflation, but the precise Equation (4) adjustment for inflation will make overestimating investment gains less likely. A difference of a few basis points becomes an economically significant difference at long wealth-management time horizons.

**Excel Implementation**

We implement the exact Fisher Equation (4) real return calculation using a simple Excel function available in the downloadable file detailed in the Appendix:

Only two inputs are required to use this Excel function: the nominal interest rate and an inflation forecast.

**WHAT IS THE RETURN I GAIN FROM DIVERSIFICATION?**

Modern managers continually seek additional sources of diversification for multiasset portfolios. Whereas 1970s portfolios might include only stocks and bonds, today’s portfolios include additional asset classes such as real estate, hedge funds, emerging market bonds, and inflation-linked bonds. Sometimes, however, this is “DINO”—diversification in name only.

Leibowitz and Bova [2005] show that U.S. equity market exposure is the chief driver of portfolio risk for most portfolios.^{5} Their “allocation betas” capture the bulk of the risk in diversified portfolios. The widespread embrace of multiasset diversification and the “endowment model” have only marginally changed the overall risk profile of typical diversified investment pools.

Their claim is that the U.S. equity market represents the key driver of portfolio risk. This occurs because most asset classes include significant embedded exposure to the U.S. stock market. Leibowitz and Bova show this with their formula for allocation beta:

5where ρ*
_{j,us}
* is the correlation of asset

*j*with U.S. stocks, σ

_{j}is the risk of asset

*j*, and σ

_{us}is the risk of U.S. stocks. This formula follows naturally from the CAPM when applied to asset classes.

After accounting for their allocation beta, an “allocation alpha” remains:

6where
is the return on asset *j*,
is the return on U.S. stocks, and *r*
_{f} is the risk-free rate. This allocation alpha reflects the true diversification benefit beyond that achievable by simply changing the portfolio beta with core assets. Allocation alpha is the return benefit of diversification.^{6}

**Significance**

The Leibowitz and Bova [2005] allocation alpha gives the expected gain from diversification. It is useful and important to know the return benefit of adding a new asset class. A positive allocation alpha implies benefit to the portfolio, but a negative allocation alpha would be detrimental to the portfolio. Moreover, the scale of the allocation alpha relative to the return generated by Equation (5) calibrates the value of diversification. The Leibowitz and Bova [2005] technique is important, having influenced Carmichael and Coen [2008]; Brown, Garlappi, and Tiu [2010]; Anson [2012]; and Jennings and Payne [2016]; among others. Indeed, Peter Bernstein [2007], in *Capital Ideas Evolving*, highlighted the Leibowitz and Bova [2005] model as one of six key practitioner advances since his seminal *Capital Ideas* book. *
Equation (6) helps private wealth managers better calibrate returns due to diversification*.

**Example**

As an example, consider international equities. Exhibit 1 implements Equation (6) and shows that international stocks have an allocation beta of 0.92 (ß_{j} = 0.8 × 0.23/0.20 = 0.92) under plausible capital market assumptions. This 0.92 beta is the proportional co-movement of international stocks with U.S. equities; international stocks have exposure to 92% of the systematic risk of U.S. stocks.

Exhibit 1 further shows that international stocks have an allocation alpha of 1.36% (α_{j} = 0.07 - 0.92(0.06 - 0.015) - 0.015), again under plausible assumptions. This means that international stocks earn approximately 1 and
% more than the return explained by their co-movement with U.S. stocks. So, international stocks are expected to earn 7.00% from three sources—1.50% from the risk-free rate, 4.14% from co-movement with the U.S. equity market (0.92 allocation beta exposure to the 4.50% equity risk premium), and 1.36% from independent diversification return.

With the risk–return assumptions in Exhibit 1, international stocks offer a positive return to diversification—the Leibowitz and Bova [2005] allocation alpha is positive. Moreover, international stocks have less systematic risk than U.S. equities—the Leibowitz and Bova [2005] allocation beta is less than 1. With these particular capital market assumptions, international stocks are a worthy diversifier.

**Excel Implementation**

We implement the Leibowitz and Bova [2005] calculations of allocation beta and alpha, respectively, with two simple Excel functions available in the downloadable file detailed in the Appendix:

Only three inputs are required to use this Excel function: the risk of the asset class being evaluated and its correlation with U.S. equities, along with the risk of U.S. equity.

The second function is:

Six inputs are required to use this Excel function: the return and risk of the asset class being evaluated, along with its correlation with U.S equities, and the return and risk of U.S. equities, along with the risk-free rate.

**HOW MUCH SHOULD TWO ASSETS’ RETURNS DIVERGE?**

Private wealth managers understand well the advantages of diversification. They analyze the risk–return trade-off of incorporating new asset classes and subclasses into portfolios. In this process, a new asset’s correlation with other assets often becomes a primary focus.

Private wealth managers form expectations from these correlations but are then frequently surprised when the new investments perform unexpectedly relative to other investments. The surprising fact is that the returns of highly correlated investments can diverge significantly.

**Significance**

This divergence in investment returns, known as return gaps, can help private wealth managers in their diversification efforts. A better understanding of prospective return gaps can enhance portfolio diversification and can make asset mix decisions more robust and sustainable. Private wealth managers who are able to communicate to clients that significant return gaps occur, even between highly correlated assets, are able to build resilience into their portfolios. Clients are less likely to be disappointed, and managers are better able to meet expectations.

Investors should expect returns between assets to be as different as the normal return gap, on average, even if two investments have the same expected return. Investors should also expect that the differences should be favorable half of the time and unfavorable half of the time for such assets. A natural consequence of this reality is that investors may perceive diversification as failing half of the time. Alternatively, clients who have a realistic perspective on this potential return difference will be less likely to “lose faith” in a diversifying investment. (The same can be said of the private wealth managers themselves.) Private wealth managers can better avoid myopic client reactions and the subsequent abandonment of a sound long-term diversification policy.

Solnik and Roulet [2000] use return gaps to calibrate correlation and diversification benefits instantaneously and without the need for a long time series. They present correlation in terms of the dispersion of potential outcomes for individual assets. Statman and Scheid [2006, 2008] make the case that it is more appropriate to focus on return gaps than correlations—most investors have a “faulty intuition about correlation” (Statman and Scheid [2006, p. 25]). Return gaps better characterize diversification benefits than the typical approach using correlation. Jennings, O’Malley, and Payne [2016] develop a more general approach to the normal return gap:

7where asset *i* is distributed *N*(µ_{i},
) and the two assets have correlation ρ. Intuitively, Equation (7) shows the normal return gap is increasing in the expected returns difference. This first effect is largest at high correlations and lower expected returns. Further, the correlation between assets helps determine the expected return gap. All else equal, high correlations correspond to smaller return gaps. Lastly, divergence in the variance of the two assets magnifies expected return gaps. This third effect is also largest when the two assets have low expected returns and high correlations. Return gaps are important, having influenced Chua, Kritzman, and Page [2009]; Vermorken, Medda, and Schroder [2012]; Grant and Satchell [2016]; and Jennings, O’Malley, and Payne [2016]; among others.

**Example**

Consider two classic examples of diversification—adding international stocks and incorporating equity styles. Most private wealth managers do not expect large differences in returns between U.S. and international equities or between value and growth stocks, even with currency fluctuations and Fama and French [1993] factor returns.

Exhibit 2 illustrates, however, that these investments will typically have large return gaps, even though the investments are highly correlated. In the first panel, we examine international diversification and assume a 0.90 correlation. Despite this high correlation, the expected gap in returns is 6.67%. This return gap is the same order of magnitude as the expected returns for equity investments themselves. We suspect that many private wealth managers will be surprised by this magnitude. In the second panel, we examine style diversification and assume a 0.95 correlation. Again, despite the high correlation, the expected return gap is large—4.29%, in this case. Even with a high 0.95 correlation, the expected return gap is half the magnitude of the expected returns for equity investments. Again, we suspect many private wealth managers will be surprised that the gap is this large.

The size of both gaps is more surprising when one notes that the first two panels of Exhibit 2 assume no difference in returns. When we incorporate such differences, a 1.00% difference in expected returns in the third and fourth panels, we demonstrate that the disparity in expected returns is not particularly important to predicting return gaps. The expected return gaps are only 0.04%–0.07% larger with differential return expectations. As Statman and Scheid [2008] and Jennings, O’Malley, and Payne [2016] discuss, this is because return gaps are heavily dependent on the variance in expected returns. Risk, not returns, drives the expected return gap for many asset classes. *
Equation (7) helps private wealth managers calibrate the performance-differential risks of diversification*.

**Excel Implementation**

We implement the Equation (7) calculation with a simple Excel function available in the downloadable file detailed in the Appendix:

Only five inputs are required to use this Excel function: the risk and return of each of the two asset classes being evaluated and their correlation.

**WILL MY CLIENT RUN OUT OF MONEY?**

Outliving one’s money is a perennial concern for retirement-age individuals, and private wealth managers can provide immense value by addressing this issue properly. Designing portfolios to last for a client’s life expectancy is inadequate as half the population will outlive an age-based life expectancy. Both returns and life expectancy have distributions, so addressing these *jointly* is key.

**Significance**

Exhibit 3 illustrates the error of considering them separately; Panels A and B show 10% of the life expectancies and returns separately, while Panel C shows them being addressed concurrently. The longest life expectancies and lowest returns are *not* the concern; rather, it is the worst combination of the two—those who live long and earn low returns.

Simulation allows addressing both concerns concurrently. However, not all wealth managers have the time, resources, or inclination to run million-point simulations to provide clients with a suitable answer that accommodates both distributions. Further, Monte Carlo simulation can invoke fear in the most capable of private wealth managers.

Conveniently, Milevsky and Robinson [2005] present a straightforward model that private wealth managers can use to tailor their recommendations and accommodate variability in both life expectancy and return patterns. The Milevsky and Robinson [2005] model elegantly bypasses the need for simulation, yet incorporates variability in both life expectancy and returns. Where estimation is required in their model, they note it “does not materially change the assessment of the retiree’s position” (Milevsky and Robinson [2005, p. 94]). The Milevsky and Robinson [2005] technique is important, having influenced Fraser and Jennings [2010], Waring and Siegel [2015], and Estrada [2016], among others. We believe their approach is one of the most important tools that private wealth managers should embrace.

In contrast to many studies (e.g., Bengen [1994]) that implement Monte Carlo or historical bootstrapping methods to generate acceptable spending rates, Milevsky and Robinson [2005] synthesize three probability distributions in their closed-form model. Specifically, they assume lognormally-distributed asset returns and an exponentially distributed lifetime random variable, both of which feed into the parameters of a tailored reciprocal gamma distribution shown below:

8This distribution calculates the “risk of ruin” as the probability *p*(·) that the *stochastic present value* (SPV) of all future portfolio withdrawals, *s*, outweighs the current value of the portfolio wealth, *w*. The “stochastic” part of SPV captures the distribution of returns and the distribution of longevity. This risk, *p*(*SPV _{s}
*>

*w*), of outliving one’s wealth depends on four parameters: µ and σ represent the portfolio’s anticipated mean real return and standard deviation,

*s*is the annual percentage spending rate, and λ represents the implied mortality rate. This mortality rate is calculated as λ =

*ln*(2)/

*L*, where

_{r}*L*

_{r}is the

*median remaining lifespan*in years. The outcome of this reciprocal gamma calculation in Equation (8) is the probability that a client experiences “financial ruin” and runs out of money.

Milevsky and Robinson [2005] provide illustrative tables, which we mimic below. Exhibit 4 (an excerpt of their Table 4) shows the probability of “ruin” for a 65-year-old retiree who spends between 3% and 7% of her portfolio, growing with inflation, at various levels of real portfolio risk and return. As a case in point, if the client spends an inflation-adjusted 4% from a portfolio that has a 6% return and 15% standard deviation, then the risk of outliving her money is 8.8%. This “risk of ruin” generally grows as the portfolio return/risk decreases and monotonically increases as the consumption level increases.

Exhibit 5 (a version of their Table 5) demonstrates the sensitivity of portfolio spending rates to changes in age, portfolio, and the risk of ruin. Panel A shows acceptable spend-down rates for a 10% risk of ruin, while Panel B shows the same for a more conservative 5% risk of ruin. Using the same 65-year-old case as before, we see that halving the risk of ruin from 10% to 5% means a reduction in portfolio expenditures from 4.20% to 3.22%, or a 23.3% reduction in annual retirement income.

**Example**

Consider a 65-year-old single retiree calibrating how much she can spend from her $10 million portfolio. The latest RP-2014 actuarial tables specify female life expectancy of 89 years for her. This implies an inflation-adjusted starting spending level of $583,000 if we merely use life expectancies and typical capital market assumptions.^{7} Note this amount is much higher than any of the values in Exhibit 5.

The same actuarial tables show that her 90th-percentile life expectancy is age 97. The 65-year-old female retiree calibrating spending of her $10 million portfolio could use this to determine an inflation-adjusted starting spending level of $487,000, if we account for the distribution of life expectancy. This is 17% less, yet is still higher than any portfolio in Exhibit 5.

We can also focus on 90th-percentile assurance about returns. While the baseline geometric return is 5.51%, the 10th-percentile return for a 24-year horizon is only 3.60%, or 35% lower. The 65-year-old female retiree calibrating spending of her $10 million portfolio could use this to determine an inflation-adjusted starting spending level of $482,000, if we account for the distribution of returns. Because of annuity math, this is 17% lower than the baseline scenario.

The Milevsky and Robinson [2005] tool provides 90th-percentile assurance about *both* life expectancy and returns. The 65-year-old female retiree calibrating spending of her $10 million portfolio could use this to determine an inflation-adjusted starting spending level of $322,000, if we account for the distribution of both life expectancy and returns. This is 45% less.

Thus, this tool incorporates uncertainty about longevity, portfolio returns, and portfolio risk to generate either (a) the probability an individual outlives their money—what the authors call the “risk of ruin”—for a desired spending rate or (b) the rate at which a retiree can spend down their portfolio when accepting a specified “risk of ruin” probability. *The Milevsky and Robinson [2005] techniques are vital planning tools for the private wealth manager*.^{8}

**Excel Implementation**

We implement the Milevsky and Robinson [2005] probability of ruin calculation with a simple Excel function available in the downloadable file detailed in the Appendix:

Only four inputs are required to use this Excel function: the spending rate as a percentage of the initial portfolio value, the median remaining life, the inflation-adjusted continuously compounded return, and the logarithmic volatility.^{9}

Likewise, we implement the Milevsky and Robinson [2005] sustainable spending rate, at a given probability of ruin risk level, with a simple Excel function available in the downloadable file detailed in the Appendix:

Only four inputs are required to use this Excel function, with the probability of ruin the investor is willing to accept replacing the spending rate from the previous function.^{10}

While Milevsky and Robinson [2005] made an Excel-based calculator and function available with their article,^{11} their calculations are more daunting because they rely on esoteric Excel functions with numerous internal calculations that do not clearly correspond to their article. We believe our functions have more-intuitive inputs.

A major part of this model’s elegance is its admitted simplicity. A private wealth manager can quickly calculate a client’s probability of running out of money, given a desired spend rate or an acceptable spend rate for a desired “risk of ruin,” respectively. Inputs required include either a client’s risk profile (specifically, an acceptable probability of “ruin”) or desired spend rate, *r*, median longevity in years,^{12} and assumptions about an investment portfolio’s return and risk. Doing so facilitates more digestable communication with the private wealth management client, an idea Milevsky [2016] advocates. No Monte Carlo or bootstrapping necessary!

**CONCLUSION**

We have assembled a set of practical quantitative tools that we think private wealth managers can benefit from knowing and using. Our goal was to make the case for particular underutilized quantitative tools and demonstrate the economic significance of the particular tools. Most importantly, we attempt to make the tools accessible to private wealth managers, both by conveying the intuition behind them and offering an easy, one-cell implementation in Microsoft Excel.

These tools, by design, present a coherent, albeit limited, set of choices. When managing assets—and especially clients’ expectations—using the correct return forecast becomes critical. Arithmetic averages are too high for long investment horizons; geometric averages are too low for short ones. A Blume-style tool can assist one developing a return estimate that is “just right.” Furthermore, clients must understand that this return forecast needs an adjustment for inflation. Generating a precise real return using Fisher’s calculations—versus simply estimating nominal minus inflation rates—can substantially affect a client’s investment outlook. Having the building blocks of accurate real return forecasts, asset managers can then implement some tools to manage client expectations. For instance, diversifying into different asset classes can provide portfolio diversification benefits that are independent of actively managed alpha. However, clients must also understand that in any given time period, asset class returns can differ substantially, even if the assets are highly correlated. Such return gaps are not an asset manager’s “fault,” but a statistical likelihood that high-net-worth investors and their advisors must understand. Finally, when spending down an accumulated portfolio, private wealth managers can help a client comprehend their mortality and return probabilities and degree of risk aversion. These attributes, coupled with accurate return expectations discussed in earlier sections, will provide clients with a realistic expectation about their ability to live out their years comfortably.

Our five tools are necessarily a limited subset of a range of choices. They are somewhat narrowly focused on the investment aspects of private wealth management and give shorter shrift to estate planning, insurance, and asset protection strategies, among others. We started with a longer list before winnowing it to five tools. This means we have nascent ideas for a potential sequel article; *we invite readers to contact us and nominate their suggestions for important quantitative tools that will benefit private wealth managers*.

**APPENDIX**

Our Excel software tool is available at www.williamjennings.com/. Some notes on using our Excel functions:

• It is helpful to see the input parameters of the Excel functions without having to refer to this article. To do so, press Ctrl + Shift + A after typing the custom function name and opening parenthesis. For example, you should type = JOMPToolReturnForecast (Ctrl + Shift + A to see JOMPToolReturnForecast (GeoMean,LogSD,HistoryPeriod,ForecastPeriod). After the variables appear, each variable can be highlighted in turn and replaced with the relevant input cells.

• In addition to the five main tools of this article, our spreadsheet includes four additional tools: JOMPToolLogMean(NormalMean,NormalSD)} and the related JOMPToolLogSD(NormalSD,NormalMean) convert normal means and standard deviations into lognormal means and standard deviations. We also offer the inverses JOMPToolNormalMean(LogMean,LogSD) and JOMPToolNormalSD(LogSD,LogMean) to make the opposite conversions. Each use the Kaplan [2000] elucidation of Lewis et al. [1980] for the conversion formula. These four tools may be helpful in establishing inputs for the five main tools described in this article.

• The prefix JOMP for each function is a reference to the present authors’ names.

## ENDNOTES

The opinions included are those of the authors and not necessarily those of the U.S. Air Force Academy, the U.S. Air Force, or any other federal agency.

↵

^{1}The unbiased return estimate,*R*(*F*), is a weighted average because the two parenthetical terms in Equation (1) sum to one. Note that the Equation (1) (Blume [1974]) approach works only when the forecast horizon is less than the historical sample period,*F*<*H*. However, even when a portfolio analyst is explicitly using a shorter historical sample, it is probable that her forecast assumptions are implicitly informed by knowledge of the 90-year horizon of the post-1926 Ibbotson data (H ˜ 90); that is, the implicit historical sample data,*H*, is large.↵

^{2}We used data from Aswath Damodaran’s NYU website: pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html.↵

^{3}Conservatism in expected returns requires greater savings rates, all else equal. Nominal conservatism forces a change in behavior. Similarly, the apparent conservatism of the geometric average might lead an investor to conclude—unnecessarily—that she needs to take on more risk to reach a desired retirement wealth level. Again, nominally conservative assumptions might force a change in behavior.↵

^{4}In the Appendix, we detail a supplemental tool to convert normal standard deviations into lognormal ones. Likewise, we provide a supplemental tool to convert an arithmetic mean return to a lognormal mean return, which serves as the geometric mean for input into the JOMPToolReturnForecast function.↵

^{5}The Leibowitz and Bova [2005] model is expanded upon in Leibowitz [2005], Leibowitz and Bova [2007], and Leibowitz, Bova, and Hammond [2010]; each contends that most portfolios have similar portfolio variances and similar allocation betas, and that these allocation betas give rise to allocation alphas that can enhance fund return without affecting portfolio volatility. While the Leibowitz and Bova [2005] model is univariate, it captures much of the risk of typical asset classes; Jennings and Payne [2016, footnote 10] considered a multivariate extension but concluded that the univariate approach is generalizable.↵

^{6}“These residual returns…can be viewed as ‘alpha-like’ and variously referred to as structural alphas, diversification alphas, allocation alphas, embedded alphas, or, most important, implicit alphas” [Leibowitz, quoted in Bernstein [2007], ch. 15]. They are*alphas*in the sense that they are independent residual returns unrelated to overall market movements. They are*allocation**alphas*in the sense that they do not depend on active management but are obtainable via strategic asset allocation. These allocation alphas are also labeled passive or “structural” alpha. Unlike active management alphas, they are non-zero-sum.↵

^{7}We evaluated a 50/50 stock/bond all-domestic portfolio using the widely disseminated capital market assumptions from Feser et al. [2015]; Jennings and Payne [2016] make the case that these assumptions are typical and generalizable. We make the Fisher [1930] adjustment detailed in an earlier section [the second tool].↵

^{8}The extreme consequences of miscalculating a portfolio’s drawdown rate are clear: an impoverished client at life’s end or a client who dies with substantial wealth, regretful they did not live more extravagantly when they had the financial wherewithal to do so. These extremes become real possibilities, using insufficient mortality modeling or making a small error in some of the more complex models. The simple closed-form model presented here addresses these issues. Additionally, it gives the private wealth management client input—and control—regarding the probability of ruin.*Bequests become probable.**Extension to multiple generations.**ln*(2)/*L*_{r}, becomes quite small but nonetheless still consequential. For perpetual-life endowments, the hazard rate is theoretically zero. We note, however, that even the oldest extant university is less than 1,000 years old, and most endowed charities are significantly younger. For multigenerational families, 4.5 generations (135 years) corresponds to a 0.51% hazard rate. The three generations that wealth proverbially lasts corresponds to 90 years and a hazard rate, λ of 0.77%. Monte Carlo experimentation demonstrates these hazard rates are conservative.Returning to the 65-year-old female retiree of our examples, calibrating spending of her $10 million portfolio could use a multigenerational 135-year horizon to determine an inflation-adjusted starting spending level of $201,000, if we account for the distribution of both life expectancy and returns. (Of course, to be fair, she would likely adjust her 50/50 stock/bond portfolio for such a long horizon.)

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^{9}In the Appendix, we detail an additional tool to derive this log-volatility input.↵

^{10}In both of our Milevsky and Robinson [2005] Excel functions, inputting a “lifespan” of 999 years directly handles perpetual-life endowments; i.e., we coded 999 to act as a perpetuity.↵

^{11}See the*Financial Analysts Journal*website, www.cfapubs.org.↵

^{12}While this lifespan most easily comes from standard actuarial tables, the private wealth manager can alter it subjectively based on conversations with the client, based on a “real age” assessment, or based on wealthier individuals having better health and longer life expectancies. See the actuarial tables from the Social Security Administration, http://www.ssa.gov/oact/STATS/table4c6.html, or Society of Actuaries https://www.soa.org/Research/Experience-Study/pension/research-2014-rp.aspx.

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