This article is available for download as a PDF here.

Summary

The low current market outlook for stocks and bonds paints a gloomy picture for retirees under common retirement forecasting assumptions.
However, assumptions such as net investment returns and retirement spending can have a large impact on forecasted retirement success, even for small changes in parameters.
By boosting returns through a combination of broader asset class and strategy diversification, considering lower fee options for passive exposures, and nailing down how retirement spending will evolve over time, we can arrive at retirement success projections that are both more reflective of a retiree’s actual situation and more in line with historical experience.

A few weeks back, we wrote about the potential impact that high core asset valuations – and the associated muted forward return expectations – may have on retirement[1].

In the post, we presented the following visualization:

Historical Wealth Paths for a 4% Withdrawal Rate and 60/40 Stock/Bond Allocation

Source: Shiller Data Library. Calculations by Newfound Research. Credit to Reddit user zaladin for the graph format. Analysis assumes the reinvestment of dividends. Returns are hypothetical index returns and are gross of all fees and expenses. Results may differ slightly from similar studies due to the data sources and calculation methodologies used for stock and bond returns.

The horizontal (x-axis) represents the year when retirement starts. The vertical (y-axis) represents a given year in history. The coloring of each cell represents the savings balance at a given point in time. The meaning of each color as follows:

Green: Current account value greater than or equal to initial account value (e.g. an investor starting retirement with $1,000,000 has a current account balance that is at least $1,000,000).
Yellow: Current account value is between 75% and 100% of initial account value
Orange: Current account value is between 50% and 75% of the initial account value.
Red: Current account value is between 25% and 50% of the initial account value.
Dark Red: Current account value is between 0% and 25% of initial account value.
Black: Current account value is zero; the investor has run out of money.

We then recreated the visualization, but with one key modification: we adjusted the historical stock and bond returns downward so that the long-term averages are in line with realistic future return expectations[2] given current valuation levels. We did this by subtracting the difference between the actual average log return and the forward-looking long return from each year’s return. With this technique, we capture the effect of subdued average returns while retaining realistic behavior for shorter-term returns.

Historical Wealth Paths for a 4% Withdrawal Rate and 60/40 Stock/Bond Allocation with Current Return Expectations

One downside of the above visualizations is that they only consider one withdrawal rate / portfolio composition combination. If we want the see results for withdrawal rates ranging from 1% to 10% in 1% increments and portfolio combinations ranging from 0/100 stocks/bonds to 100/0 stocks/bonds in 20% increments, we would need sixty graphs!

To distill things a bit more, we looked at the historical “success” of various investment and withdrawal strategies. We evaluated success on three metrics:

Absolute Success Rate (“ASR”): The historical probability that an individual or couple will not run out of money before their retirement horizon ends.
Comfortable Success Rate (“CSR”): The historical probability that an individual or couple will have at least the same amount of money, in real terms, at the end of their retirement horizon compared to what they started with.
Ulcer Index (“UI”): The average pain of the wealth path over the retirement horizon where pain is measured as the severity and duration of wealth drawdowns relative to starting wealth[3].

As a quick refresher, below we present the ASR for various withdrawal rate / risk profile combinations over a 30-year retirement horizon first using historical returns and then using historical returns adjusted to reflect current valuation levels.

Absolute Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition – 30 Yr. Horizon

Source: Shiller Data Library. Calculations by Newfound Research. Analysis assumes the reinvestment of dividends. Returns are hypothetical index returns and are gross of all fees and expenses. Results may differ slightly from similar studies due to the data sources and calculation methodologies used for stock and bond returns.

Absolute Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations – 30 Yr. Horizon

Overall, our analysis suggested that retirement withdrawal rates that were once safe may now deliver success rates that are no better – or even worse – than a coin flip.

Over the coming weeks, we want to delve a bit deeper into this topic. Specifically, we are going to explore some key properties of distribution portfolios – portfolios from which investors take regular withdrawals to finance retirement spending – as well as some strategies that investors may consider in order to improve retirement outcomes.

This week we are going to focus on the high degree of sensitivity that retirement planning outcomes can have to initial assumptions. In upcoming weeks, we will explore other retirement investment topics, including:

The sequence of returns and risk management.
The impact of behavioral finance and investor emotions.
Finding the right portfolio risk profile through retirement.

The Butterfly Effect in Retirement Portfolios

Quoting from a great piece on distribution portfolio theory by James Sandidge[4]:

“The butterfly effect refers to the ability of small changes early in a process that lead to significant impact later. It gets its name from the idea that a butterfly flapping its wings in Brazil could trigger a chain of events that would culminate in the formation of a tornado in Texas[5]. The butterfly effect applies to distribution portfolios where even small changes early in retirement can have significant impact long-term.”

One example of the butterfly effect in the context of retirement planning is the impact of small changes in long-term average returns. These differences could arise from investment outperformance or underperformance, fees, expenses, or taxes.

In the example below, we consider 60/40 stock/bond investor with a 30-year investment horizon and a 4% target withdrawal rate, adjusted each year for inflation. We consider three scenarios:

Pessimistic Scenario: Average annual portfolio returns are 100bps below our long-term assumption (e.g. we picked bad managers, allocated assets poorly, paid high fees, etc.).
Base Case Scenario: Average annual portfolio returns are equal to our long-term assumption.
Optimistic Scenario: Average annual portfolio returns are 100bs above our long-term assumption (e.g. we picked good managers, nailed our asset allocation, paid lower than expected fees, etc.).

We see that varying our return assumption by just +/-100bps can swing our probability of fully funding retirement – without decreasing withdrawals below plan – from 48% to 74%. Similarly, the probability of ending retirement with our original nest egg fully intact ranges from 11% in the pessimistic scenario to 47% in the optimistic scenario.

In the optimistic scenario, the median ending wealth after 30 years is $800k for an initial investment of $1mm. Not outstanding but certainly nothing to complain about. In the pessimistic scenario, however, our median ending wealth is zero, meaning the most likely outcome is running out of money!

The Butterfly Effect and Changes to Average Long-Term Return Assumption:
30-Yr. Horizon, 60/40 Allocation, 4% Withdrawals

Below, we present one example that is particularly telling: an investor that retired in 1973[6]. We see that a 100bps difference in returns in either direction can literally be the difference between running out of funds (gray), sweating every dollar and cent (orange), or a relatively comfortable retirement (blue).

Camouflaged Butterflies: Assumptions in Spending Rate Changes

An example of a secondary input that sometimes may be glossed over, but nonetheless can have a large impact on outcomes is the assumption regarding how quickly withdrawals will increase relative to inflation. Again, we consider three scenarios:

Withdrawals increase at a rate that is 1% slower than inflation (i.e. spending will rise by 2% year-over-year when inflation is 3% – spending falls in real terms).
Withdrawals increase at the same rate of inflation (spending stays constant in real terms).
Withdrawals increase at a rate that is 1% faster than inflation (i.e. spending will rise by 4% year-over-year when inflation is 3% – spending rises in real terms). This is probably an unrealistic scenario, for reasons that we will discuss later, but it still helps illustrate the sensitivity of planning analysis to its inputs.

The Butterfly Effect and Changes to the Spending Growth Assumption:
30-Yr. Horizon, 60/40 Allocation, 4% Withdrawals

Overall, the results are very similar in magnitude to what we saw when we adjusted the return assumption.

Implications of the Butterfly Effect

The examples above provide clear evidence that retirement success is significantly impacted by both primary and secondary assumptions. But what does this mean for investors? We think there are two main implications.

Getting the details right is crucial.

First, it’s important to get the details right when planning for retirement. To highlight this, let’s return to the topic of spending. Many financial calculators assume that spending increases one-for-one with inflation through retirement. Put differently, this assumes that spending is constant after adjusting for inflation.

Data from the Employee Benefit Research Institute (“EBRI”) suggests that this is generally an erroneous assumption. Instead, spending tends to decline as retirees age. Specifically, EBRI found that on average spending declines 20% from age 50-64 to 65-79, 22% from age 65-79 to 80-89, and 12% from age 80-89 to 90+.

(Note: This is obviously a gross oversimplification of actual spending behavior. At the end of this commentary, we discuss a few interesting research pieces on this topic. They make clear the importance of customizing spending assumptions to each client’s situation and preferences.)

Source: “Adaptive Distribution Theory” by James B. Sandidge

Implementing more realistic spending assumptions makes a material difference in our Absolute Success Rate (“ASR”), Comfortable Success Rate (“CSR”), and Ulcer Index stats.

Below, we recreate our ASR, CSR, and Ulcer Index tables assuming that real spending declines by 1% per year. We also compare these measures across three scenarios for a 4% withdrawal rate:

Historical return assumptions and constant real spending
Current return assumptions and constant real spending
Current return assumptions and 1% per year decline in real spending

We see that our adjusted spending assumption helps to close the gap between the historical and forward-looking return scenarios. This is especially true when we look at the ASR.

For example, a 60/40 portfolio and 4% constant real withdrawal rate produced an ASR of 99% across all historical market scenarios. The success rate dropped all the way to 58% when we adjusted the historical stock and bond returns downward for our future expectations. Changing to the declining spending path increases the success rate from 58% to 75%.

Absolute Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations and Real Spending Declining by 1% Per Year – 30 Yr. Horizon

Comfortable Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations and Real Spending Declining by 1% Per Year – 30 Yr. Horizon

Ulcer Index for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations and Real Spending Declining by 1% Per Year – 30 Yr. Horizon

Incremental increases (decreases) in portfolio returns (spending) matter, a lot.

Reducing spending is a very personal topic, so we will focus on some potential ways to grind out some incremental portfolio gains. (Note: another important topic when constructing withdrawal portfolios is to manage sequence of return risk. We will address this topic in a future post).

First, it’s important to be strategic, not static. To us, this means having a thoughtful, forward-looking outlook when setting a strategic asset allocation. A big part of this is fighting the temptation of home-country bias.

Source: https://personal.vanguard.com/pdf/icrrhb.pdf

This tendency to prefer home-country assets not only leaves quite a bit of diversification on the table, but also puts U.S. investors on the wrong side current equity market valuations.

Source: https://personal.vanguard.com/pdf/icrrhb.pdf

Based upon a blended set of capital market assumptions sourced from J.P. Morgan, Blackrock, and BNY Mellon, we see that it’s possible to increase long-term expected returns by between 30bps and 50bps, depending on desired risk profile, by moving beyond U.S. stocks and bonds[7]. Last week we discussed the “weird portfolios” that may be best positioned for the future.

Source: J.P. Morgan, Blackrock, BNY Mellon, Newfound Research. Return forecasts are forward-looking statements based upon the reasonable beliefs of Newfound Research and are not a guarantee of future performance. Forecasts are not representative of any Newfound Research strategy or fund. Forward-looking statements speak only as of the date they are made and Newfound assumes no duty to and does not undertake to update forward-looking statements. Forward looking statements are subject to numerous assumptions, risks, and uncertainties, which change over time. Actual results may differ materially from those anticipated in forward-looking statements. Returns are presented gross of taxes and fees.

Second, we recommend using a hybrid active/passive approach for core exposures given the increasing availability of evidence-based, favor-driven investment strategies. Now this sounds great in theory, but with over 300 factors now identified across the global equity markets and the proliferation of “smart beta” ETFs, it is reasonable to wonder how in the world one can have a view of which factors will actually work going forward. To dig into this a bit deeper, let’s look at one of our favorite examples of factor-based investing.

This portfolio, suggested by Vanguard, buys companies whose tickers start with the letters S, M, A, R, or T. This is not a real portfolio that anyone should invest in; yet it has identified an anomalous outperformance pattern. On a backtested basis, the S.M.A.R.T. Beta portfolio nearly doubled the annualized return of the S&P 500.

In order to determine the validity of this so-called factor, we need to understand:

What is the theory that explains why the factor works (provides excess return)? Without a theory for why something works, we cannot possibly form an intelligent view as to whether or not it will world in the future.
How has the factor performed on an out-of-sample basis? This is math speak for the following types of questions: How as the factor performed after its discovery? How does the factor work with slightly alternative implementations? Does the factor perform well in other assets classes and geographies?

In the case of the S.M.A.R.T. Beta factor, these questions allow us to quickly dismiss it. There is obviously no good reason – at least no good reason we can think of off the top of our heads – for why the first letter in a stock’s ticker should drive returns[8]. While we have not tested S.M.A.R.T. Beta across asset classes and geographies, we know that this was simply a tongue-in-cheek example presented by Vanguard trying to get the point across that it’s easy to find something that works in the past, but much harder to find something that works in the future. We suspect that if we did test the strategy in other countries, as an example, that it would probably outperform in some cases and underperform in others. This lack of robustness would be a clear sign that our level of confidence in this factor going forward should be very low.

So, what factors do meet these criteria (in our view)? Only four that are applicable to stocks:

Value: Buy cheap stocks and sell expensive ones
Momentum: Buy outperforming securities and sell underperforming ones
Defensive: But lower risk/higher quality securities and sell higher risk/lower quality ones
Size/Liquidity: Buy smaller/less liquid companies and sell larger/more liquid ones[9]

Data Source: AQR, Calculations by Newfound Research. Value is the HML Devil factor. Momentum is the UMD factor. Defensive is a blend of the BAB and QMJ factors. Size is the SMB factor. Equal Weight is an equally weighted blend of all four factors, rebalanced monthly. Returns include the reinvestment of dividends and are gross of all fees and expenses. Past performance does not guarantee future results.

Going back to 1957, an equally-weighted blend of the four factors mentioned above would have generated in excess of 500bps of excess annualized return before fees and expenses. Even if we discount future performance by 50% for reduced strategy efficacy and fees, the equal weight factor portfolio could add nearly 160bps for a 60/40 investor[10].

Third, we recommend looking beyond fixed income for risk management. Broadly speaking, we divide asset classes and strategies into two categories: return generators and risk mitigators.

Over the last 30+ years, investors have been very fortunate that their primary risk mitigator – fixed income – happened to experience an historic bull market.

Unfortunately, our situation today is much different than the early 1980s. Current yields are very low by historical standards, implying that fixed income is likely to be a drag to portfolio performance especially after accounting for inflation. However, that does not mean that bonds should not still play a key role in all but the most aggressive portfolios. It simply means that the premium for using bonds as a form of portfolio insurance is high relative to historical standards. As a result, we advocate looking for complementary risk management tools.

One option here would be to employ a multi-strategy, unconstrained sleeve like we constructed in a recent commentary[11]. When constructed with the right objectives in mind, these types of portfolios can act as an effective buffer to equity market volatility without the cost of large fixed income positions in a low interest rate environment. Let’s take the Absolute Return strategy that we discussed in that piece. It was constructed by optimizing for an equal risk contribution across the following seven asset classes and strategies:

U.S. Treasuries: 25%
Low volatility equities: 8%
Trend-based tactical asset allocation: 9%
Value-based tactical asset allocation: 12%
Unconstrained fixed income: 25%
Risk Parity: 9%
Managed Futures: 12%

Now let’s consider our typical 60/40 investor. Historically, a 25% allocation to this unconstrained sleeve with 18.8% (3/4 of the 25%) taken from fixed income and 6.3% (1/4 of the 25%) taken from equities would have left the investor in the same place as the original 60/40 from a risk perspective. This holds true whether we measure risk as volatility or maximum drawdown.

When we regress the absolute return strategy on world equities and U.S. Treasuries, we get the following results (data for this analysis covers the period from January 1993 to June 2016):

A loading to global equities of 0.25
A loading to U.S. Treasuries of 0.49
Annualized alpha of approximately 2%
Annualized residual volatility of 2.2%.
An R-squared of around 0.77

From the relatively high R-squared, we can conclude that a decent way to think of the absolute return portfolio is as a combination of three positions: 1) a 25% allocation to world stocks, 2) a 49% allocation to U.S. Treasuries, and 3) a 100% allocation to an unconstrained long/short portfolio with historical performance characterized by a 2% excess return and 2.2% volatility.

Using this construct, we can get at least a very rough idea of what to expect going forward by plugging in our capital market assumptions for world equities and U.S. Treasuries and making a reasonable assumption for what the long/short portfolio can deliver going forward on a net-of-fee basis. Let’s assume as we did for the factor discussion that the long/short portfolio only captures around 50% of its historical performance after fees. This would still imply an expected forward-looking return of 4.1% compared to an average expected return of 2.5% for U.S. core bonds[15]. For the 60/40 investor, this could mean close to 25bps of incremental return.

Finally, we should seek to reduce fees, all else being equal. Four things that we think are worth mentioning here.

We need to consider fees holistically. This means looking beyond expense ratios and considering factors like execution costs (e.g. bid/ask spread), commissions, and ticket charges.
The “all else being equal” part is really important. We want to be fee-conscious, not fee centric. Just like you probably don’t always buy the cheapest home, clothes, and electronics, we don’t believe in defaulting to the lowest cost investment option in all cases. We want to find value in the investments we choose. If market-cap weighted equity exposure costs 5bps and we can get multi-factor exposure for 25bps, we will not eliminate the factor product from consideration just due higher fees if we believe it can offer more than 20bps in incremental value. Fortunately, the proliferation of passive investment vehicles effectively being offered for free has helped put downward pressure on products throughout the industry.
We have to remember that while there are many, many merits to a passive, market-cap weighted approach, the rise of this type of investing has largely coincided with upward trends in equity and bond valuations. In other words, the return pie has been very big and therefore the name of the game has been capturing as much of the pie as possible, usually by minimizing fees and staying disciplined (after all, a passive approach to investing, like any other approach, only works long-term if we can stick with it, and behavioral science and experience suggests there are real difficulties doing so especially when markets get volatile). Today, we are in a fundamentally different situation. The pie is nearly as small as it’s ever been. For many investors, even capturing 100% of the pie may not be enough. Instead, many must search out ways to expand the pie in order to meet their goals.
From a behavioral perspective, there is nothing wrong with channeling our inner Harry Markowitz and going with a hybrid active[13]/passive approach within the same portfolio. Markowitz, who helped revolutionize portfolio construction theory with his landmark paper “Portfolio Selection,” famously explained that when building his own portfolio he knew he should have “…computed the historical covariances of the asset classes and drawn an efficient frontier.” Instead, he said, “I visualized my grief if the stock market went way up and I wasn’t in it – or if it went way down and I was completely in it. So, I split my contributions 50/50 between stocks and bonds.” We are strong advocates for passive, just not for 100% concentration in passive.

Let’s say as an example that by using these techniques, we are able to improve returns by 150bps annually. What would the impact be on ASR, CSR, and Ulcer Index using our same framework? For this analysis, we retain our assumption from earlier that real spending declines by 1% per year.

Absolute Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations Plus 150bps and Real Spending Declining by 1% Per Year – 30 Yr. Horizon

Comfortable Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations Plus 150bps and Real Spending Declining by 1% Per Year – 30 Yr. Horizon

Ulcer Index for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations Plus 150bps and Real Spending Declining by 1% Per Year – 30 Yr. Horizon

Conclusion: The Sum of All Assumptions in Retirement

Retirement projections are based on many different assumptions including asset class returns, time horizon, allocation strategies, inflation, and how withdrawals evolve over time. Small changes in many of these assumptions can have a large impact on retirement success rates (the Retirement Butterfly Effect).

High valuations of core assets in the U.S. suggest that retirement withdrawal rates that were once safe may now deliver success rates that are no better – or even worse – than a coin flip. However, by focusing our efforts on refining the assumptions that go into retirement planning, we can arrive at results that do not spell doom and gloom for retirees.

While getting all the details right is ideal, there are specific areas that matter the most.

For returns, increasing net returns is what matters, which means there are many knobs to adjust. Incorporating factor based strategies and broader diversification are good initial starting points. Expanding the usage of international equity and unconstrained strategy exposure can be simple modifications to traditional U.S. equity and bond heavy portfolios that may give a boost to forward-looking returns.

Fees, expenses, and taxes can be other areas to examine as long as we keep in mind that it is best to be fee/expense/tax-conscious, not fee/expense/tax-centric. Slight fee or tax inefficiencies can cause a “guaranteed” loss of return, but these effects must be weighed against the potential upside.

For many exposures (e.g. passive and long-only core stock and bond exposure), minimizing cost is certainly appropriate. However, do not let cost considerations preclude the consideration of strategies or asset classes that can bring unique return generating or risk mitigating characteristics to the portfolio.

These are all ideas that help form the foundation for our QuBe Model Portfolios.

With spending, the assumption that retirees will track inflation with their withdrawals throughout a 30 year retirement is not applicable across the board. Nailing down spending is tough, but improved assumptions can have a big impact on retirement forecasts. A thorough conversation on housing, health care, travel, insurance, and general consumption is critical.

As with any model that produces a forecast, there will always be errors in retirement projections. When asset class returns are strong, as they have been in previous decades, we can comfortably brush many assumptions under the rug. However, with muted future returns, achieving financial goals requires a better understanding of model sensitivities and more diligent research into how to equip portfolios to thrive in such an environment.

Appendix: Retiree Spending Behavior

Estimating the True Cost of Retirement[14]

David Blanchett, Head of Retirement Research for Morningstar Investment Management, argues that the common assumptions of a generic replacement rate[15], constant real spending, and a fixed retirement horizon do not accurately capture the highly personalized nature of a retiree’s spending behavior.

Key takeaways include:

From a category perspective, the main changes through retirement are a decline in relative spending on insurance and pensions and an increase in health care spending.
Source: Blanchett’s Estimating the True Cost of Retirement
Forecasts on spending by category can be used to determine a customized spending inflation rate for a given household. For example, Blanchett plots general inflation vs. medical inflation. Using this relationship, we can predict that 2% general inflation would lead to medical cost inflation of approximately 4%. One theme of many research papers on the topic of retirement spending is that health care planning should be accounted for in a separate line item. Not only does the future of the health care system have the potential to look much different from the past, but the actual financial impact of health care costs can differ greatly depending on each individual’s insurance situation. Blanchett also finds that health care spending does not differ materially across income levels.
Source: Blanchett’s Estimating the True Cost of Retirement
Blanchett finds that spending does decline through retirement and on average follows a “U” pattern whereby spending declines accelerate before age 75 and decelerate afterwards.
Source: Blanchett’s Estimating the True Cost of Retirement
Blanchett decomposed the population of his dataset into four groups based on spending and net worth. $30,000 was the threshold for separating spenders into high and low groups. $400,000 was the threshold for dividing the population by net worth. He found that households with “matched” spending and net worth (i.e. low spending and low net worth or high spending and high net worth) exhibited the “U” pattern that we saw with the full dataset. However, households with mismatched spending/net worth behaved differently. High net worth and low spending households saw spending increase through retirement, although the rate of this increase was faster earlier in retirement. Conversely, households with high spending and low net worth reduced their spending more aggressively than the other groups.
Source: Blanchett’s Estimating the True Cost of Retirement

How Does Household Expenditure Change with Age for Older Americans? [16]

The EBRI studied linked above also documents spending reductions through retirement. It presents very interesting data on the distribution of health care spending by age group. We see that the distribution widens out significantly over time with the largest increases occurring in the right tail (90^th and 95^th percentile of spending).

Source: EBRI

Spending in Retirement [17]

In this piece, J.P. Morgan analyzed retirement spending using a unique dataset of 613,000 households that utilize the Chase platform (debit cards, credit cards, mortgage payments, etc.) for the majority of their spending. The authors found the same general trend of declining spending as in the EBRI and Morningstar pieces.

Spending declines were largest in the transportation, apparel & services, and mortgage categories. The overall and category-specific patterns were generally consistent across wealth levels. The researchers were able to classify households into five categories: foodies, homebodies, globetrotters, health care spenders, and snowflakes. This categorization is relevant because each group can expect to see their spending needs evolve differently over time. Some key takeaways for each group are:

Foodies
1. Most common group
2. Generally frugal
3. Low housing expenses due to mortgages being paid off and low property tax bills
4. Tend to spend less as they get older and so an assumption of faster declines in real spending may be appropriate
Homebodies
1. High share of spending on mortgages, real estate taxes, and ongoing maintenance
2. May be prudent to assume that expenses track inflation
3. For planning purposes, it’s important to consider future plans related to housing
Globetrotters
1. Highest overall spending
2. More common among households with higher net worth
3. May be prudent to assume that expenses track inflation
Health care spenders
1. Medicare-related expenses were the largest share of spending for these households
2. These expenses may grow faster than inflation.
3. For further reading, see:
  1. Health care costs in retirement [18]
  2. Guide to Retirement [19]
Snowflakes
1. These households are more unique and do not fit into one of the other four categories.

[1] https://blog.thinknewfound.com/2017/08/impact-high-equity-valuations-safe-retirement-withdrawal-rates/

[2] Specifically, we use the “Yield & Growth” capital market assumptions from Research Affiliates. These capital market assumptions assume that there is no valuation mean reversion (i.e. valuations stay the same going forward). The adjusted average nominal returns for U.S. equities and 10-year U.S. Treasuries are 5.3% and 3.1%, respectively, compared to the historical values of 9.0% and 5.3%.

[3] Normally, the Ulcer Index would be measured using true drawdown from peak, however, we believe that using starting wealth as the reference point may lead to a more accurate gauge of pain.

[4] References to ideas similar to the butterfly effect date back as far as the 1800s. In academia, the idea is prevalent in the field of chaos theory.

[5] https://www.imca.org/sites/default/files/current-issues/JIC/JIC172_AdaptiveDistributionTheory.pdf

[6] We continue to adjust returns to account for current valuations. Therefore, this example takes the actual returns for U.S. stocks and bonds from 1973 to 2003 and then adjusts them downward based on the Research Affiliates’ long-term return assumptions.

[7] Potential increases in expected return, based upon the capital market assumptions of the three institutions listed, are actually larger than what we present here. This results from two aspects of the QuBe investment process. First, we utilize a simulation-based approach that incorporates downside shocks to the correlation matrix and that accounts for parameter estimate uncertainty. Second, we consider two behaviorally-based optimizations, one that attempts to smooth the absolute path of returns and another that attempts to smooth the path of returns relative to a common benchmark, which is tilted toward U.S. equities. Both of these techniques reduce the expected returns generated when we combine the resulting weights with the stated capital market assumptions.

[8] There actually has been research published suggesting evidence that stock tickers can be useful in picking stocks. For example, “Would a stock by any other ticker smell as sweet?” by Alex Head, Gary Smith, and Julia Wilson find evidence that stocks with “clever” tickers (e.g. Southwest’s choice of LUV to reflect its brand) outperform the broader market. Their results were robust to the Fama-French 3-factor model. As a rationale for these results, the authors posited that clever tickers might signal manager ability or that the memorable tickers feed into the behavioral biases of investors.

[9] The size premium is probably the most hotly debated of the four today. Recent research suggests that that size prospers once we control for quality (i.e. we want to buy small, high quality companies not just small companies).

[10] As we’ve written about in the past, factor portfolios do not have to generate excess returns to justify an allocation in equity portfolios. Even with zero to slightly negative premiums, moderate allocations to these strategies would have historically led to increased risk-adjusted returns due to the diversification that they provide to market-cap weighted portfolios.

[11] https://blog.thinknewfound.com/2017/07/building-unconstrained-sleeve/

[12] Again using data from J.P. Morgan, Blackrock, and BNY Mellon.

[13] When we say active, we usually (but not always) mean systematic strategies that are factor-based and implemented using a quantitative and rules-based investment process.

[14] Blanchett, David. 2013. Estimating the True Cost of Retirement. Working paper, Morningstar Investment Management. https://corporate.morningstar.com/ib/documents/MethodologyDocuments/ResearchPapers/Blanchett_True-Cost-of-Retirement.pdf

[15] Quoting from Blanchett, “The replacement rate is the percentage of household earnings need to maintain a similar standard of living during retirement.

[16] Banerjee, Sudipto. 2014. How Does Household Expenditure Change with Age for Older Americans? Employee Benefits Research Institute. Notes 35, no. 9 (September). https://www.ebri.org/pdf/notespdf/Notes.Sept14.EldExp-Only.pdf

[17] Roy, Katherine and Sharon Carson. 2015. Spending in Retirement. J.P. Morgan. https://am.jpmorgan.com/gi/getdoc/1383244966137.

[18] Carson, Sharon and Laurance McGrath. 2016. Health care costs in retirement. J.P. Morgan. https://am.jpmorgan.com/blob-gim/1383331734803/83456/RI_Healthcare%20costs_2016_r4.pdf?segment=AMERICAS_US_ADV&locale=en_US

[19] Roy, Katherine, Sharon Carson, and Lena Rizkallah. 2016. Guide to Retirement. J.P. Morgan. https://am.jpmorgan.com/blob-gim/1383280097558/83456/JP-GTR.pdf

Impact of High Equity Valuations on Safe Retirement Withdrawal Rates

By Justin Sibears

On August 14, 2017

In Sequence Risk, Weekly Commentary

This post is available as a PDF here.

Summary

While valuation-based market timing is notoriously difficult, present and future retirees should prepare for muted U.S. stock and bond returns relative to historical experience.
High valuations suggest that retirement withdrawal rates that were once safe may now deliver success rates that are no better – or even worse – than a coin flip.
This outlook is by no means a call for despair, but rather highlights the increasing need for taking control of one’s destiny by controlling both investment and non-investment factors that can improve the odds of successfully meeting one’s retirement goals.

We are always on the lookout for interesting data visualizations related to the financial markets. Recently, two such charts have come across our computer screens.

The Drumbeat of High Equity Valuations Grows Louder

The first chart is from a recent article from Goldman Sachs Asset Management (“GSAM”)[1]. It reinforces the importance of developing realistic forward-looking expectations for asset class returns. This is a topic that we have droned on and on about over the last couple of years and one that we feel is especially important today, when the valuation backdrop for many core asset classes are stretched by historical standards.

The clear takeaway, at least in GSAM’s eyes, is found in the blue text in the upper right: “In 99% of the time at current valuation levels, equity returns have been single digit or negative.”

Now, there are a few complicating factors with the chart and this conclusion:

There is some hindsight bias embedded in the chart. In December 1999, when the S&P 500 reached an all-time high Shiller CAPE of 44.2, there was no way of knowing with certainty that valuations weren’t going even higher. After all, for an example of higher than tech bubble valuations, we need look no further than Japan.
The median rolling 10-year return for the S&P 500 over this period was 8.5%, so be careful in drawing the following conclusion: Equity returns have been “bad” 99% of the time when we’ve been at or near current valuation levels. A better conclusion to draw would be something like: Equity returns have tended to be average to below average when we’ve been at or near current valuation levels. When S&P 500 valuations were between the 75^th and 100^th percentile, subsequent 10-year returns were below the median of 8.5% approximately 80% of the time. The odds of a negative 10-year return, even at these valuation levels, is a pretty modest one in eight.
Mean reversion in valuations can take a very, very long time. For those looking to sell high and buy low (or vice-versa), the path to success can be terribly frustrating, requiring Buffett-like discipline to capture the eventual rewards. For example, Shiller’s CAPE rose above the 75^th percentile in January 1992. From this already high point, equities rallied another 300%+ before valuations peaked in late 1999. CAPE would not fall below the January 1992 value of 19.8 until October 2008.
There is a strong argument that valuations are driven by behavioral factors. For example, Jeremy Grantham discussed such a behavioral model in GMO’s most recent quarterly letter. He argues that the two factors most important in explaining high valuations are high profit margins and low inflation volatility. Viewed in this way, mean reversion would require one or both of these conditions to reverse course.

Visualizing Retirement Success and Failure

The second visualization comes from a recent post on Reddit; a news aggregation, web content rating, and discussion website; by a user going by the name zaladin. The graph shows the retirement wealth paths for various combinations of withdrawal rates and stock/bond splits.

However, before we start we want to point out that this is a highly simplified example. We only consider U.S. stocks and bonds, we don’t consider taxes or fees, etc.

In reality, the following factors can play a significant role in developing a retirement strategy: Alpha (investment performance vs. broadly diversified market portfolios), fees, taxes, desire to leave an inheritance to heirs, longevity/time horizon, diversification/risk management, spending flexibility, risk tolerance, valuation environment, etc.

Returning to our simplistic world, we’ve recreated the graph for a 4% inflation-adjusted withdrawal rate and a 60/40 stock/bond split below. In order to present data going back more than a century, we stick to U.S. equities for our stock exposure and 10-Year U.S. Treasuries for our bond exposure.

Green: Current account value greater than or equal to initial account value (e.g. an investor starting retirement with $1,000,000 has a current account balance that is at least $1,000,000).
Yellow: Current account value is between 75% and 100% of initial account value
Orange: Current account value is between 50% and 75% of the initial account value.
Red: Current account value is between 25% and 50% of the initial account value.
Dark Red: Current account value is between 0% and 25% of initial account value.
Black: Current account value is zero; the investor has run out of money.

The diagonal gray lines represent 20, 30, 40, and 50 years, respectively, after retirement.

Historical Wealth Paths for a 4% Withdrawal Rate and 60/40 Stock/Bond Allocation

Source: Shiller Data Library. Calculations by Newfound Research. Credit to Reddit user zaladin for the graph format. Analysis uses real returns and assumes the reinvestment of dividends. Returns are hypothetical index returns and are gross of all fees and expenses. Results may differ slightly from similar studies due to the data sources and calculation methodologies used for stock and bond returns.

One downside of the above visualization is that it only considers one withdrawal rate / portfolio composition combination. If we want the see results for withdrawal rates ranging from 1% to 10% in 1% increments and portfolio combinations ranging from 0/100 stocks/bonds to 100/0 stocks/bonds in 20% increments, we would need sixty graphs!

To distill things a bit more, we will look at the historical “success” of various investment and withdrawal strategies. We will evaluate success on three metrics:

Absolute Success Rate (“ASR”): The historical probability that an individual or couple will not run out of money before their retirement horizon ends.
Comfortable Success Rate (“CSR”): The historical probability that an individual or couple will have at least the same amount of money, in real terms, at the end of their retirement horizon compared to what they started with.
Ulcer Index (“UI”): The average pain of the wealth path over the retirement horizon where pain is measured as the severity and duration of wealth drawdowns relative to starting wealth. [Note: Normally, the Ulcer Index would be measured using true drawdown from peak, however, we believe that using starting wealth as the reference point may lead to a more accurate gauge of pain.]

We will evaluate these three metrics over a 30-year retirement horizon. Please feel free to reach out if you’d like to see the analysis for different horizon length.

Absolute Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition – 30 Yr. Horizon

Source: Shiller Data Library. Calculations by Newfound Research. Analysis uses real returns and assumes the reinvestment of dividends. Returns are hypothetical index returns and are gross of all fees and expenses. Results may differ slightly from similar studies due to the data sources and calculation methodologies used for stock and bond returns.

We see that withdrawal rates of 3% or less succeeded 95%+ of the time based on “ASR” regardless of asset allocation. A 4% withdrawal likewise succeeded with 90%+ historical probability as long as some equity exposure was incorporated into the portfolio. No stock/bond mix was able to support a withdrawal rate of 5% or more while succeeding at least nine times out of ten.

Comfortable Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition – 30 Yr. Horizon

The results with “CSR” as our success measure largely mirror the “ASR” results. The only main differences are:

A 100% bond portfolio with a 3% withdrawal rate only leaves the investor with 100% of more of their initial wealth at the end of retirement in about two-thirds of scenarios. For an investor to achieve 90%+ CSR success with a 3% withdrawal rate, some equity is required.
Succeeding 90%+ of the time with a 4% withdrawal rate requires holding more stocks than bonds.

Ulcer Index for Various Combinations of Withdrawal Rate and Portfolio Composition – 30 Yr. Horizon

The Ulcer Index is a measure that summarizes the severity and duration of wealth drawdowns. We like this metric as it provides us some idea of how emotionally stressful a given market path is for investors. In our view, high investing stress not only is unenjoyable, but also raises the likelihood of making poor, emotionally-charged decisions.

Interpreting an individual Ulcer index alone can be difficult, but the relative values provide context. For example, for a 4% withdrawal rate, even though the portfolios with some equity had 90%+ ASRs, the 60/40 portfolio had the least stress, on average – even less than the slightly more successful (from a CSR standpoint) 80/20 portfolio.

So, what do these equity valuation and retirement visualizations have to do with one another?

For many investors, market returns are only the means to an end. Ultimately, investors are looking to achieve their financial goals. We certainly know that muted long-term returns in core stocks and bonds are not a good thing. But it can be hard to immediately understand what the true impact of such an outcome would be.

To see the effect of muted returns more clearly, we are going to recreate the retirement visualizations from earlier, but with one key modification: we adjust historical stock and bond returns downward so that the long-term averages are in line with realistic future return expectations given current valuation levels. We do this by subtracting the difference between the actual average log return and the forward-looking log return from each year’s return. By doing this, we reflect subdued average returns while retaining the peaks and valleys that we would expect in actual rolling 30-year periods.

Specifically, we use the “Yield & Growth” capital market assumptions from Research Affiliates. These capital market assumptions assume that there is no valuation mean reversion (i.e. valuations stay the same going forward). The adjusted average nominal returns for U.S. equities and 10-year U.S. Treasuries are 5.3% and 3.1%, respectively, compared to the historical values of 9.0% and 5.3%.

Historical Wealth Paths for a 4% Withdrawal Rate and 60/40 Stock/Bond Allocation with Current Return Expectations

With updated return assumptions, we see a dramatically different picture with a lot less green and a lot more of the dreaded black (i.e. fully exhausting one’s savings). The results are similar across withdrawal rates and asset allocations.

We see that only withdrawal rates of 2% or less would have achieved 90%+ success over thirty years regardless of asset allocation. High success rates can still be attained with a 3% withdrawal rate assuming investors are willing to bear the risk of moderate to aggressive equity allocations. Unfortunately, a 4% withdrawal rate no longer offers the safety that actual experience has suggested.

Absolute Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations – 30 Yr. Horizon

In our example, passing on starting wealth to heirs at the end of retirement looks difficult except at withdrawal rates of less than 3%. The same can be said for investors looking for a stress-free journey as Ulcer Index values are much higher in this scenario for 3%+ withdrawal rates than what we saw using historical returns.

Comfortable Success Rate for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations – 30 Yr. Horizon

Ulcer Index for Various Combinations of Withdrawal Rate and Portfolio Composition with Average Stock and Bond Returns Equal to Current Expectations – 30 Yr. Horizon

Conclusion: Taking Control of Retirement

High valuations of core assets in the U.S. suggest that retirement withdrawal rates that were once safe may now deliver success rates that are no better – or even worse – than a coin flip. Unfortunately, we cannot control the returns of U.S. stocks or bonds (or any asset class returns for that matter).

But we can take control of the factors that we can influence.

For a current or future retiree, this means controlling to the extent possible factors like taxes, saving, and spending. From an investment perspective, it means:

Being strategic, not static: Have a thoughtful, forward-looking outlook when developing a strategic asset allocation. This means having a willingness to diversify U.S. stocks and bonds with the ever-expanding palette of complementary asset classes and strategies.
Directly address the role of behavioral finance by recognizing that an investor must have the willingness to stick with a plan in order to succeed (e.g. the journey is just as important as the destination).
Utilize a hybrid active/passive approach for core exposures given the increasing availability of evidence-based, factor-driven investment strategies.
Be fee-conscious, not fee-centric. For many exposures (e.g. passive and long-only core stock and bond exposure), minimizing cost is certainly appropriate. However, do not let cost considerations preclude the consideration of strategies or asset classes that can bring unique return generating or risk mitigating characteristics to the portfolio.
Look beyond fixed income for risk management given low interest rates.
Recognize that the whole can be more than the sum of its parts by embracing not only asset class diversification, but also strategy/process diversification.

These are all ideas that help form the foundation for our QuBe Model Portfolios.

Retirement success and muted future returns are not mutually exclusive. However, achieving financial goals in such an environment requires careful planning for factors that may have been safely ignored given the generous market tailwinds of prior decades.

[1] Goldman Sachs Asset Management, “The Synchronized Expansion.” https://www.gsam.com/content/gsam/us/en/advisors/market-insights/market-strategy/market-know-how/2017/Q32017.html#section-background_ebd2_background_moduletitle_874b

Building an Unconstrained Sleeve

By Corey Hoffstein

On July 31, 2017

In Portfolio Construction, Risk Management, Weekly Commentary

We’re often asked about how to build an unconstrained sleeve in a portfolio.

Our view is that your mileage will largely vary by where you are trying to go. With that in mind, we focus on three objectives:

Sleeves that seek to hedge equity losses.
Sleeves that seek significant equity upside capture while reducing downside.
Sleeves that seek an absolute return profile.

We explore how these sleeves can be built using common strategies such as tactical equity, minimum volatility equity, managed futures, risk parity, global contrarian, alternative income, and traditional U.S. Treasuries.

You can find the full presentation below.

(If the above slideshow is not working, you can view an online version here or download a PDF version here.)

Combining Tactical Views with Black-Litterman and Entropy Pooling

By Corey Hoffstein

On July 17, 2017

In Portfolio Construction, Weekly Commentary

This post is available as a PDF download here.

Summary

In last week’s commentary, we outline a number of problems faced by tactical asset allocators in actually implementing their views.
This week, we explore popular methods for translating a combination of strategic views and tactical views into a single, comprehensive set of views that can be used as the foundation of portfolio construction.
We explore Black-Litterman, which can be used to implement views on returns as well as the more recently introduced Entropy Pooling methodology of Meucci, which allows for more flexible views.
For practitioners looking to implement tactical views into a number of portfolios in a coherent manner, the creation of posterior capital market assumptions via these methods may be an attractive process.

Note: Last week’s commentary was fairly qualitative – and hopefully applicable for practitioners and non-practitioners alike. This week’s is going to be a bit wonkier and is primarily aimed at those looking to express tactical views in an asset allocation framework. We’ll try to keep the equations to a minimum, but if the question, “how do I create a posterior joint return distribution from a prior and a rank view of expected asset class returns?” has never crossed your mind, this might be a good week to skip.

In last week’s commentary, we touched upon some of the important details that can make the actual implementation and management of tactical asset allocation a difficult proposition.[1] Specifically, we noted that:

Establishing consistent measures across assets is hard (e.g. “what is fair value for a bond index and how does it compare to equities?”);
There often are fewer bets being made, so position sizing is critical;
Cross-asset dynamics create changing risk profiles for bets placed.
Tactical decisions often explicitly forego diversification, increasing the hurdle rate.

We’ll even add a fifth, sixth, and seventh:

Many attractive style premia (e.g. momentum, value, carry, and trend) trades require leverage or shorting. Many other tactical views (e.g. change in yield curve curvature or change in credit spreads) can require leverage and shorting to neutralize latent risk factors and allocate risk properly.
Combining (potentially conflicting) tactical views is not always straight forward.
Incorporating tactical views into a preexisting policy portfolio – which may include long-term strategic views or constraints – is not obvious.

This week, we want to address how points #2-7 can be addressed with a single comprehensive framework.[2]

What is Tactical Asset Allocation?

As we hinted in last week’s commentary, we’re currently smack dab in the middle of writing a book on systematic tactical asset allocation.

When we sat down to write, we thought we’d start at an obvious beginning: defining “what is tactical asset allocation?”

Or, at least, that was the plan.

As soon as we sat down to write, we got a case of serious writer’s block. Which, candidly, gave us deep pause. After all, if we struggled to even write down a succinct definition for what tactical asset allocation is, how in the world are we qualified to write a book about it?

Fortunately, we were eventually able to put digital ink to digital paper. While our editor would not let us get away with a two sentence chapter, our thesis can be more or less boiled down to:

Strategic asset allocation is the policy you would choose if you thought risk premia were constant; tactical asset allocation is the changes you would make if you believe risk premia are time-varying.[3]

We bring this up because it provides us a mental framework for thinking about how to address problems #2 – 7.

Specifically, given prior market views (e.g. expected returns and covariances) that serve as the foundation to our strategic asset allocation, can our tactical views be used to create a posterior view that can then serve as the basis of our portfolio construction process?

Enter Black-Litterman

Fortunately, we’re not the first to consider this question. We missed that boat by about 27 years or so.

In 1990, Fischer Black and Robert Litterman developed the Black-Litterman model while working at Goldman Sachs. The model provides asset allocators with a framework to embed opinions and views about asset class returns into a prior set of return assumptions to arrive at a bespoke asset allocation.

Part of what makes the Black-Litterman model unique is that it does not ask the allocator to necessarily come up with a prior set of expected returns. Rather, it relies on equilibrium returns – or the “market clearing returns” – that serve as a neutral starting point. To find these returns, a reverse optimization method is utilized.

Here, R is our set of equilibrium returns, c is a risk aversion coefficient, S is the covariance matrix of assets, and w is the market-capitalization weights of those assets.

The notion is that in the absence of explicit views, investors should hold the market-capitalization weighted portfolio (or the “market portfolio”). Hence, the return views implied by the market-capitalization weights should be our starting point.

Going about actually calculating the global market portfolio weights is no small feat. Plenty of ink has been spilled on the topic.[4] For the sake of brevity, we’re going to conveniently ignore this step and just assume we have a starting set of expected returns.

The idea behind Black-Litterman is to then use a Bayesian approach to combine our subjective views with these prior equilibrium views to create a posterior set of capital market assumptions.

Specifically, Black-Litterman gives us the flexibility to define:

Absolute asset class return views (e.g. “I expect U.S. equities to return 4%”)
Relative asset class return views (e.g. “I expect international equities to outperform U.S. equities by 2%”)
The confidence in our views

Implementing Black-Litterman

We implement the Black-Litterman approach by constructing a number of special matrices.

P: Our “pick matrix.” Each row tells us which asset classes we are expressing a view on. We can think of each row as a portfolio.
Q: Our “view vector.” Each row tells us what our return view is for the corresponding row in the pick matrix.
O: Our “error matrix.” A diagonal matrix that represents the uncertainty in each of our views.

Given these matrices, our posterior set of expected returns is:

If you don’t know matrix math, this might be a bit daunting.

At the highest level, our results will be a weighted average of our prior expected returns (R) and our views (Q). How do compute the weights? Let’s walk through it.

t is a scalar. Generally, small. We’ll come back to this in a moment.
S is the prior covariance matrix. Now, the covariance matrix represents the scale of our return distribution: i.e. how far away from the expectation that we believe our realized returns could fall. What we need, however, is some measure of uncertainty of our actual expected returns. g. If our extracted equilibrium expected returns for stocks is 5%, how certain are we it isn’t actually supposed to be 4.9% or 5.1%? This is where t comes back. We use a small t (generally between 0.01 and 0.05) to scale S to create our uncertainty estimate around the expected return. (tS)^-1, therefore, is our certainty, or confidence, in our prior equilibrium returns.
If O is the uncertainty in our view on that portfolio, O^-1 can be thought of as our certainty, or confidence, in each view.
Each row of P is the portfolio corresponding to our view. P’O^-1P, therefore, can be thought of as the transformation that turns view uncertainty into asset class return certainty.
Using our prior intuition of (tS)^-1, (tS)^-1R can be thought of as certainty-scaled prior expected returns.
Q represents our views (a vector of returns). O^-1Q, therefore, can be thought of as certainty-scaled P’O^-1Q takes each certainty-scaled view and translates it into cumulative asset-class views, scaled for the certainty of each view.

With this interpretation, the second term – (tS)^-1R + P’O^-1Q – is a weighted average of our prior expected returns and our views. The problem is that we need the sum of the weights to be equal to 1. To achieve this, we need to normalize.

That’s where the first term comes in. (tS)^-1 + P’O^-1P is the sum of our weights. Multiplying the second term by ((tS)^-1 + P’O^-1P)^-1 is effectively like dividing by the sum of weights, which normalizes our values.

Similar math has been derived for the posterior covariance matrix as well, but for the sake of brevity, we’re going to skip it. A Step- by-Step Guide to Black-Litterman by Thomas Idzorek is an excellent resource for those looking for a deeper dive.

Black-Litterman as a Solution to Tactical Asset Allocation Problems

So how does Black-Litterman help us address problems #2-7 with tactical asset allocation?

Let’s consider a very simple example. Let’s assume we want to build a long-only bond portfolio blending short-, intermediate-, and long-term bonds.

For convenience, we’re going to make a number of assumptions:

Constant durations of 2, 5, and 10 for each of the bond portfolios.
Use current yield-to-worst of SHY, IEI, and IEF ETFs as forward expected returns. Use prior 60 months of returns to construct the covariance matrix.

This gives us a prior expected return of:

	E[R]
SHY	1.38%
IEI	1.85%
IEF	2.26%

And a prior covariance matrix,

	SHY	IEI	IEF
SHY	0.00005	0.000177	0.000297
IEI	0.000177	0.000799	0.001448
IEF	0.000297	0.001448	0.002795

In this example, we want to express a view that the curvature of the yield curve is going to change. We define the curvature as:

Increasing curvature implies the 5-year rate will go up and/or the 2-year and 10-year rates will go down. Decreasing curvature implies the opposite.

To implement this trade with bonds, however, we want to neutralize duration exposure to limit our exposure to changes in yield curve level and slope. The portfolio we will use to implement our curvature views is the following:

We also need to note that bond returns have an inverse relationship with rate change. Thus, to implement an increasing curvature trade, we would want to short the 5-year bond and go long the 2- and 10-year bonds.

Let’s now assume we have a view that the curvature of the yield curve is going to increase by 50bps over the next year. We take no specific view as to how this curvature increase will unfold (i.e. the 5-year rate rising by 50bps, the 5-year rate rising by 25bps and each of the 2-year and 10-year rates falling by 25bps, etc.). This implies that the curvature bond portfolio return has an expected return of negative 5%.

Implementing this trade in the Black-Litterman framework, and assuming a 50% certainty of our trade, we end up with a posterior distribution of:

	E[R]
SHY	1.34%
IEI	1.68%
IEF	1.97%

And a posterior ovariance matrix,

	SHY	IEI	IEF
SHY	0.000049	0.000182	0.000304
IEI	0.000182	0.000819	0.001483
IEF	0.000304	0.001483	0.002864

We can see that while the expected return for SHY did not change much, the expected return for IEF dropped by 0.29%.

The use of this model, then, is that we can explicitly use views about trades we might not be able to make (due to leverage or shorting constraints) to alter our capital market assumptions, and then use our capital market assumptions to build our portfolio.

For global tactical style premia – like value, momentum, carry, and trend – we need to explicitly implement the trades. With Black-Litterman, we can implement them as views, create a posterior return distribution, and use that distribution to create a portfolio that still satisfies our policy constraints.

The Limitations of Black-Litterman

Black-Litterman is a hugely powerful tool. It does, however, have a number of limitations. Most glaringly,

Returns are assumed to be normally distributed.
Expressed views can only be on returns.

To highlight the latter limitation, consider a momentum portfolio that ranks asset classes based on prior returns. The expectation with such a strategy is that each asset class will outperform the asset class ranked below it. A rank view, however, is inexpressible in a Black-Litterman framework.

Enter Flexible Views with Entropy Pooling

While a massive step forward for those looking to incorporate a variety of views, the Black-Litterman approach remains limited.

In a paper titled Fully Flexible Views: Theory and Practice[5], Attilio Meucci introduced the idea of leveraging entropy pooling to incorporate almost any view a practitioner could imagine. Some examples include,

A prior that need not be normally distributed – or even be returns at all.
Non-linear functions and factors.
Views on the return distribution, expected returns, median returns, return ranks, volatilities, correlations, and even tail behavior.

Sounds great! How does it work?

The basic concept is to use the prior distribution to create a large number of simulations. By definition, each of these simulations occurs with equal probability.

The probability of each scenario is then adjusted such that all views are satisfied. As there may be a number of such solutions, the optimal solution is the one that minimizes the relative entropy between the new distribution and the prior distribution.

How is this helpful? Consider the rank problem we discussed in the last section. To implement this with Meucci’s entropy pooling, we merely need to adjust the probabilities until the following view is satisfied:

Again, our views need not be returns based. For example, we could say that we believe the volatility of asset A will be higher than asset B. We would then just adjust the probabilities of the simulations until that is the case.

Of course, the accuracy of our solution will depend on whether we have enough simulations to accurately capture the distribution. A naïve numerical implementation that seeks to optimize over the probabilities would be intractable. Fortunately, Meucci shows that the problem can be re-written such that the number of variables is equal to the number of views.[6]

A Simple Entropy Pooling Example

To see entropy-pooling in play, let’s consider a simple example. We’re going to use J.P. Morgan’s 2017 capital market assumptions as our inputs.

In this toy example, we’re going to have the following view: we expect high yield bonds to outperform US small-caps, US small-caps to outperform intermediate-term US Treasuries, intermediate-term US Treasuries will outperform REITs, and REITs will outperform gold. Exactly how much we expect them to outperform by is unknown. So, this is a rank view.

We will also assume that we are 100% confident in our view.

The prior, and resulting posterior expected returns are plotted below.

We can see that our rank views were respected in the posterior. That said, since the optimizer seeks a posterior that is as “close” as possible to the prior, we find that the expected returns of intermediate-term US Treasuries, REITs, and gold are all equal at 3%.

Nevertheless, we can see how our views altered the structure of other expected returns. For example, our view on US small-caps significantly altered the expected returns of other equity exposures. Furthermore, for high yield to outperform US small-caps, asset class expectations were lowered across the board.

Conclusion

Tactical views in multi-asset portfolios can be difficult to implement for a variety of reasons. In this commentary, we show how methods like Black-Litterman and Entropy Pooling can be utilized by asset allocators to express a variety of views and incorporate these views in a cohesive manner.

Once the views have been translated back into capital market assumptions, these assumptions can be leveraged to construct a variety of portfolios based upon policy constraints. In this manner, the same tactical views can be embedded consistently across a variety of portfolios while still acknowledging the unique objectives of each portfolio constructed.

[1] https://blog.thinknewfound.com/2017/07/four-important-details-tactical-asset-allocation/

[2] For clarity, we’re using “addressed” here in the loose sense of the word. As in, “this is one potential solution to the problem.” As is frequently the case, the solution comes with its own set of assumptions and embedded problems. As always, there is no holy grail.

[3] By risk premia, we mean things like the Equity Risk Premium, the Bond Risk Premium (i.e. the Term Premium), the Credit Risk Premium, the Liquidity Risk Premium, et cetera. Active Premia – like relative value – confuse this notion a bit, so we’re going to conveniently ignore them for this discussion.

[4] For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2352932

[5] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1213325

[6] Those looking to implement can find Meucci’s MatLab code (https://www.mathworks.com/matlabcentral/fileexchange/21307-fully-flexible-views-and-stress-testing) and public R code (https://r-forge.r-project.org/scm/viewvc.php/pkg/Meucci/R/EntropyProg.R?view=markup&root=returnanalytics) available. We have a Python version we can likely open-source if there is enough interest.

Growth Optimal Portfolios

By Corey Hoffstein

On July 5, 2017

In Portfolio Construction, Weekly Commentary

This post is available as a PDF download here.

Summary

Traditional portfolio management focuses explicitly on the trade-off between risk and return.
Anecdotally, investors often care more about the growth of their wealth. Due to compounding effects, wealth is a convex function of realized returns.
Within, we explore geometric mean maximization, an alternative to the traditional Sharpe ratio maximization that seeks to maximize the long-term growth rate of a portfolio.
Due to compounding effects, volatility plays a critical role in the growth of wealth. Seemingly lower return portfolios may actually lead to higher expected terminal wealth if volatility is low enough.
Maximizing for long-term growth rates may be incompatible with short-term investor needs. More explicit accounting for horizon risk may be prudent.

In 1956, J.L. Kelly published “A New Interpretation of Information Rate,” a seminal paper in betting theory that built off the work of Claude Shannon. Within, Kelly derived an optimal betting strategy (called the Kelly criterion) for maximizing the long-term growth rate of a gambler’s wealth over a sequence of bets. Key in this breakthrough was the acknowledgement of cumulative effects: the gambler would be reinvesting gains and losses, such that too large a bet would lead to ruin before any probabilistic advantage was ever realized.

Around the same time, Markowitz was laying the foundations of Modern Portfolio Theory, which relied upon mean and variance for the selection of portfolios. Later work by Sharpe and others would identify the notion of the tangency portfolio: the portfolio that maximizes excess return per unit of risk.

Without leverage, however, investors cannot “eat” risk-adjusted returns. Nor do they, anecdotally, really seem to care about it. We, for example, have never heard of anyone opening their statement to look at their Sharpe ratio.

More academically, part of the problem with Markowitz’s work, as identified by Henry Latane in 1959, was that it did not provide an objective measure for selecting a portfolio along the efficient frontier. Latane argued that for an investor looking to maximize terminal wealth (assuming a sequence of uncertain and compounding choices), one optimal strategy was to select the portfolio that maximized geometric mean return.

The Math Behind Growth-Optimal Portfolios

We start with the idea that the geometric mean return, g, of a portfolio – the value we want to maximize – will be equal to the annualized compound return:

With some slight manipulation, we find:

For[1],

We can use a Taylor expansion to approximate the log returns around their mean:

Dropping higher order terms and taking the expected value of both sides, we get:

Which can be expressed using the geometric mean return as:

Where sigma is the volatility of the linear returns.

Multi-Period Investing: Volatility is a Drag

At the end of the last section, we found that the geometric mean return is a function of the arithmetic mean return and variance, with variance reducing the growth rate. This relationship may already be familiar to some under the notion of volatility drag.[2]

Volatility drag is the idea that the arithmetic mean return is greater than the geometric mean return – with the difference being due to volatility. Consider this simple, albeit extreme, example: on the first day, you make 100%; on the second day you lose 50%.

The arithmetic mean of these two returns is 25%, yet after both periods, your true compound return is 0%.

For less extreme examples, a larger number of periods is required. Nevertheless, the effect remains: “volatility” causes a divergence between the arithmetic and geometric mean.

From a pure definition perspective, this is true for returns. It is, perhaps, somewhat misleading when it comes to thinking about wealth.

Note that in finance, we often assume that wealth is log-normally distributed (implying that the log returns are normally distributed). This is important, as wealth should only vary between [0, ∞) while returns can technically vary between (-∞, ∞).

If we hold this assumption, we can say that the compounded return over T periods (assuming constant expected returns and volatilities) – is[3]:

Where is the random return shock at time t.

Using this framework, for large T, the median compounded return is:

What about the mean compounded return? We can re-write our above framework as:

Note that the random variable is log-normal, the two terms are independent of one another, and that

Thus,

The important takeaway here is that volatility does not affect our expected level of wealth. It does, however, drive the mean and median further apart.

The intuition here is that while returns are generally assumed to be symmetric, wealth is highly skewed: we can only lose 100% of our money but can theoretically make an infinite amount. Therefore, the mean is pushed upwards by the return shocks.

Over the long run, however, the annualized compound return does not approach the mean: rather, it approaches the median. Consider that the annualized compounded return can be written:

Taking the limit as T goes to infinity, the second term approaches 1, leaving only:

Which is the annualized median compounded return. Hence, over the long run, over one single realized return path, the investor’s growth rate should approach the median, not the mean, meaning that volatility plays a crucial role in long-term wealth levels.

The Many Benefits of Growth-Optimal Portfolios

The works of Markowitz et al. and Latane have subtle differences.

Sharpe Ratio Maximization (“SRM”) is a single-period framework; Geometric Mean Maximization (“GMM”) is a multi-period framework.
SRM maximizes the expected utility of terminal wealth; GMM maximizes the expected level of terminal wealth.

Over time, a number of attributes regarding GMM have been proved.

Breiman (1961) – GMM minimizes the expected time to reach a pre-assigned monetary target V asymptotically as V tends to infinity.
Hakansson (1971) – GMM is myopic; the current composition depends only on the distribution of returns over the next rebalancing period.
Hakansson and Miller (1975) – GMM investors never risk ruin.
Algoet and Cover (1988) – Assumptions requiring the independence of returns between periods can be relaxed.
Ethier (2004) – GMM maximizes the median of an investor’s fortune.
Dempster et al. (2008) – GMM can create value even in the case where every tradeable asset becomes almost surely worthless.

With all these provable benefits, it would seem that for any investor with a sufficiently long investment horizon, the GMM strategy is superior. Even Markowitz was an early supporter, dedicating an entire chapter of his book Portfolio Selection: Efficient Diversification of Investments, to it.

Why, then, has GMM largely been ignored in favor of SRM?

A Theoretical Debate

The most significant early challenger to GMM was Paul Samuelson who argued that maximizing geometric mean return was not necessarily consistent with maximizing an investor’s expected utility. This is an important distinction, as financial theory generally requires decision making be based on expected utility maximization. If care is not taken, the maximization of other objective functions can lead to irrational decision making: a violation of basic finance principles.

Practical Issues with GMM

Just because the GMM provably dominates the value of any other portfolio over a long-horizon does not mean that it is “better” for investors over all horizons.

We use quotation marks around better because the definition is largely subjective – though economists would have us believe we can be packaged nicely into utility functions. Regardless,

Estrada (2010) shows that GMM portfolios are empirically less diversified and more volatile than SRM portfolios.
Rubinstein (1991) shows that it may take 208 years to be 95% confident that a Kelly strategy beats an all-cash strategy, and 4700 years to be 95% sure that it beats an all-stock strategy.

A horizon of 208 years, and especially 4700 years, has little applicability to nearly all investors. For finite horizons, however, maximizing the long-term geometric growth rate may not be equivalent to maximizing the expected geometric return.

Consider a simple case with an asset that returns either 100% or -50% for a given year. Below we plot the expected geometric growth rate of our portfolio, depending on how many years we hold the asset.

We can see that for finite periods, the expected geometric return is not zero, but rather asymptotically approaches zero as the number of years increases.

Finite Period Growth-Optimal Portfolios

Since most investors do not have 4700 hundred years to wait, a more explicit acknowledgement of holding period may be useful. There are a variety of approximations available to describe the distribution of geometric returns with a finite period (with complexity trading off with accuracy); one such approximation is:

Rujeerapaiboon, Kuhn, Wiesemann (2014)[4] propose a “robust” solution for fixed-mix portfolios (i.e. those that rebalance back to a fixed set of weights at the end of each period) and finite horizons. Specifically, they seek to maximize the worst-case geometric growth rate (where “worst case” is defined by some probability threshold), under all probability distributions (consistent with an investor’s prior information).

If we simplify a bit and assume a single distribution for asset returns, then for a variety of worst-case probability thresholds, we can solve for the maximum growth rate.

As we would expect, the more certain we need to be of our returns, the lower our growth rate will be. Thus, our uncertainty parameter, , can serve, in a way, as a risk-aversion parameter.

As an example, we can employ J.P. Morgan’s current capital market assumptions, our simulation-based optimizer, the above estimates for E[g] and V[g], and vary the probability threshold to find “robust” growth-optimal portfolios. We will assume a 5-year holding period.

Source: Capital market assumptions from J.P. Morgan. Optimization performed by Newfound Research using a simulation-based process to account for parameter uncertainty. Certain asset classes listed in J.P. Morgan’s capital market assumptions were not considered because they were either (i) redundant due to other asset classes that were included or (ii) difficult to access outside of private or non-liquid investment vehicles.

To make interpretation easier, we have color coded the categories, with equities in blue, fixed income in green, credit in orange, and alternatives in yellow.

We can see that even with our uncertainty constraints relaxed to 20% (i.e. our growth rate will only beat the worst-case growth rate 80% of the time), the portfolio remains fairly diversified, with large exposures to credit, alternatives, and even long-dated Treasuries largely used to offset equity risk from emerging markets.

While this is partly due to the generally bearish view most firms have on traditional equities, this also highlights the important role that volatility plays in dampening geometric return expectations.

Low Volatility: A Geometric Mean Anomaly?

By now, most investors are aware of the low volatility anomaly, whereby strategies that focus on low volatility or low beta securities persistently outperform expectations given by models like CAPM.

To date, there have been three behavioral arguments:

Asset managers prefer to buy higher risk stocks in effort to beat the benchmark on an absolute basis;
Investors are constrained (either legally or preferentially) from using leverage, and therefore buy higher risk stocks;
Investors have a deep-seeded preference for lottery-type payoffs, and so buy riskier stocks.

In all three cases, investors overbid higher risk stocks and leave low-risk stocks underbid.

In Low Volatility Equity Investing: Anomaly or Algebraic Artifact, Dan diBartolomeo offers another possibility.[5] He notes that while the CAPM says there is a linear relationship between systematic risk (beta) and reward, the CAPM is a single-period model. In a multi-period model, there would be convex relationship between geometric return and systematic risk.

Assuming the CAPM holds, diBartolomeo seeks to solve for the optimal beta that maximizes the geometric growth rate of a portfolio. In doing so, he addresses several differences between theory and reality:

The traditional market portfolio consists of all risky assets, not just stocks. Therefore, an all stock portfolio likely has a very high relative beta.
The true market portfolio would contain a number of illiquid assets. In adjusting volatility for this illiquidity – which in some cases can triple risk values – the optimal beta would likely go down.
In adjusting for skew and kurtosis exhibited by financial time series, the optimal beta would likely go down.
In general, investors tend to be more risk averse than they are growth optimal, which may further cause a lower optimal beta level.
Beta and market volatility are estimated, not known. This causes an increase in measured asset class volatility and further reduces the optimal beta value.

With these adjustments, the compound growth rate of low volatility securities may not be an anomaly at all: rather, perception of outperformance may be simply due to a poor interpretation of the CAPM.

This is both good and bad news. The bad news is that if the performance of low volatility is entirely rational, it’s hard for a manager to demand compensation for it. The good news is that if this is the case, and there is no anomaly, then the performance cannot be arbitraged away.

Conclusion: Volatility Matters for Wealth Accumulation

While traditional portfolio theory leads to an explicit trade-off of risk and return, the realized multi-period wealth of an investor will have a non-linear response – i.e. compounding – to the single-period realizations.

For investors who care about the maximization of terminal wealth, a reduction of volatility, even at the expense of a lower expected return, can lead to a higher level of wealth accumulation.

This can be non-intuitive. After all, how can a lower expected return lead to a higher level of wealth? To invoke Nassim Taleb, in non-linear systems, volatility matters more than expected return. Since wealth is a convex function of return, a single bad, outlier return can be disastrous. A 100% gain is great, but a 100% loss puts you out of business.

With compounding, slow and steady may truly win the race.

It is worth noting, however, that the portfolio that maximizes long-run return may not necessarily best meet an investor’s needs (e.g. liabilities). In many cases, short-run stability may be preferred at the expense of both long-run average returns and long-term wealth.

[1] Note that we are using here to represent the mean of the linear returns. In Geometric Brownian Motion, is the mean of the log returns.

[2] For those well-versed in pure mathematics, this is an example of the AM-GM inequality.

[3] For a more general derivation with time-varying expected returns and volatilities, please see http://investmentmath.com/finance/2014/03/04/volatility-drag.html.

[4] https://doi.org/10.1287/mnsc.2015.2228

[5] http://www.northinfo.com/documents/559.pdf

Flirting with Models

The Research Library of Newfound Research

The Butterfly Effect in Retirement Planning

Summary

The Butterfly Effect in Retirement Portfolios

Camouflaged Butterflies: Assumptions in Spending Rate Changes

Implications of the Butterfly Effect

Conclusion: The Sum of All Assumptions in Retirement

Appendix: Retiree Spending Behavior

Impact of High Equity Valuations on Safe Retirement Withdrawal Rates

Summary

The Drumbeat of High Equity Valuations Grows Louder

Visualizing Retirement Success and Failure

So, what do these equity valuation and retirement visualizations have to do with one another?

Conclusion: Taking Control of Retirement

Building an Unconstrained Sleeve

Combining Tactical Views with Black-Litterman and Entropy Pooling

Summary

What is Tactical Asset Allocation?

Enter Black-Litterman

Implementing Black-Litterman

Black-Litterman as a Solution to Tactical Asset Allocation Problems

The Limitations of Black-Litterman

Enter Flexible Views with Entropy Pooling

A Simple Entropy Pooling Example

Conclusion

Growth Optimal Portfolios

Summary

The Math Behind Growth-Optimal Portfolios

Multi-Period Investing: Volatility is a Drag

The Many Benefits of Growth-Optimal Portfolios

A Theoretical Debate

Practical Issues with GMM

Finite Period Growth-Optimal Portfolios

Low Volatility: A Geometric Mean Anomaly?

Conclusion: Volatility Matters for Wealth Accumulation

The Butterfly Effect in Retirement Planning

Summary

The Butterfly Effect in Retirement Portfolios

Camouflaged Butterflies: Assumptions in Spending Rate Changes

Implications of the Butterfly Effect

Conclusion: The Sum of All Assumptions in Retirement

Appendix: Retiree Spending Behavior

Impact of High Equity Valuations on Safe Retirement Withdrawal Rates

Summary

The Drumbeat of High Equity Valuations Grows Louder

Visualizing Retirement Success and Failure

So, what do these equity valuation and retirement visualizations have to do with one another?

Conclusion: Taking Control of Retirement

Building an Unconstrained Sleeve

Combining Tactical Views with Black-Litterman and Entropy Pooling

Summary­­

What is Tactical Asset Allocation?

Enter Black-Litterman

Implementing Black-Litterman

Black-Litterman as a Solution to Tactical Asset Allocation Problems

The Limitations of Black-Litterman

Enter Flexible Views with Entropy Pooling

A Simple Entropy Pooling Example

Conclusion

Growth Optimal Portfolios

Summary­­

The Math Behind Growth-Optimal Portfolios

Multi-Period Investing: Volatility is a Drag

The Many Benefits of Growth-Optimal Portfolios

A Theoretical Debate

Practical Issues with GMM

Finite Period Growth-Optimal Portfolios

Low Volatility: A Geometric Mean Anomaly?

Conclusion: Volatility Matters for Wealth Accumulation

Summary

Summary