The Financial Journal (Global): Factor Investing and Asset Allocation: A Business Cycle Perspective

Factor Investing and Asset Allocation: A Business Cycle Perspective
Vasant Naik | Mukundan Devarajan | Andrew Nowobilski | Sébastien Page, CFA | Niels Pedersen
https://www.cfainstitute.org/learning/products/publications/rf/Pages/rf.v2016.n4.1.aspx

Foreword

Factor investing—building portfolios with exposure to macroeconomic or statistical factors that explain the return differences between securities
The authors’ specific emphasis is on translating macroeconomic forecasts into alpha-generating portfolios or positions, in the spirit of arbitrage pricing theory. They focus on forming optimal portfolios of multiple asset classes. Most of the literature on factors deals with single asset classes, such as equities. But the ultimate problem of the asset owner is to allocate among asset classes. The monograph analyzes this problem via the risk factor approach. Moreover, the habits of mind revealed in this monograph are useful for understanding, developing, and implementing any kind of quantitative or factor-based approach to investing, not just one that relies on macroeconomic forecasts.

Preface

The authors emphasize the use of a valuation-driven approach for estimating expected returns on asset classes. This approach blends an evolving assessment of the prospects for global growth and inflation with the valuation of risk factors.
While it is convenient to think of risk in the factor space, it is important to recognize that opportunities for alpha generation present themselves in asset classes
and securities.
Marrying these two concepts—recognizing asset specific valuation characteristics and mapping them back onto the factor space for risk management—is the key to applying many of the concepts of this monograph successfully in an investment organization.

1. Introduction

Any investment process comprises a top-down component and a bottom-up component.

The “top-down” part focuses on the macro risk environment and its impact on key risk factors that drive most asset returns and determines the optimal portfolio exposure to these risk factors.
The bottom-up (or security selection) component, on the other hand, assesses the relative value of individual securities and firms.
A good investment process should excel along both these dimensions. Our focus in this monograph is on the top-down process.

The final result of the top-down process is a set of exposures to key risk factors that represents the best trade-off between risk taken and risk premia expected to be earned.

How to define the key risk factors driving returns in global financial markets
How to measure and estimate risk and risk premia
How to use these estimates to construct an optimal portfolio of risk exposures
Most importantly, what we can learn about these issues from historical evidence

Risk and return in financial markets are strongly influenced by global macroeconomic fundamentals that determine the expected path of real growth, inflation, and monetary and fiscal policy, and the variability around these expectations.
Another dimension that we emphasize in our discussion is the importance of market valuations in investment decision making.
In this monograph, we discuss how valuations are useful, albeit imprecise, signals of risk premia.

(value opportunities) = (changes in risk premia) + (changes in expectations)
While the former (changes in risk premia) creates a genuine investment opportunity and the need to consider rebalancing one’s portfolio, the latter (changes in expectations) does not.

Chapter 2: Key Risk Factors in Bond and Equity Markets

The investment universe available to a global investor is vast. The first task in optimal portfolio construction is to reduce the number of decision variables in this problem. There is a manageable number of risk factors that describe the core of the global investable universe.
One can summarize a large part of the risk in even the most complex portfolios in terms of their exposures to a parsimonious set of systematic risk factors. Doing so allows
investors to replace optimal asset allocation with optimal risk factor allocation—a much more tractable exercise. We can then focus on the optimal allocation of a risk budget to key risk factors rather than directly on a large menu of assets in financial markets.
In this chapter, we discuss the key risk factors for publicly traded bonds and equities. In Chapter 7, we show that alternative assets are exposed to the same set of risk factors that describe the behavior of bonds and equities.

2.1. Key Risk Factors for Bonds

Bonds are conceptually simpler than equities because their promised cash flows are known more or less with certainty and most of their risk can therefore be attributed to variations in discount rates (interest rates).
(discount rate) = (time-to-maturity dimension) + (credit risk dimension)

The variations in both these components are governed by a small set of risk factors.

We first focus on the default-free discount rates and then consider the risk factor characterization of credit spreads.

2.1.1. Characterizing the Variations in the Default-Free Yield Curve

For most countries, government bonds issued in their own currency effectively have no default risk in nominal terms. A typical riskless yield curve contains yields for a maturity spectrum that ranges from short maturities of three to six months to maturities as long as 30 (or even 50) years. But we do not need to consider the risk of variations in yield of every maturity. There are two reasons for this: First, riskless rates of all maturities are impacted by the common macroeconomic forces of monetary policy, real growth, and expected inflation. Second, arbitrage ensures that bonds that are close substitutes trade at similar yields; hence, yields for bonds with similar maturities
are highly correlated.
We can use principal components analysis (PCA) to demonstrate that variations in riskless yields of various maturities can be explained by a small
number of factors (that is, there is a low-dimensional representation of their dynamics). As documented in Litterman and Scheinkman (1991), most of the returns on any default-free government security or portfolio of such securities can be explained by just three factors: the level factor, the slope factor, and the curvature factor. The level factor represents parallel shifts in the yield curve. Bond yields of all maturities are affected by (or “load on”) this factor roughly equally. The slope factor captures the changes in the steepness or slope of the yield curve, and it accounts for the fact that yields of different maturities do not always move in parallel. Finally, the curvature factor captures the movements of the “belly” of the yield curve relative to long and short maturities.
PCA derives the principal components purely statistically, as linear combinations of the underlying data. Also, the linear combination that describes the principal components
can change over time. For making risk allocation decisions, therefore, it is often preferable to use a small number of exactly identified (named) and economically interpretable factors instead of “disembodied” principal components, whose interpretation can vary, as descriptors of yield curve movements. In the choice of named factors, it is useful to recognize that the yield curve is determined by two quantities: (1) the short-, medium-, and long-term expectations of the path of policy rates and (2) the term structure of risk premia for bearing interest rate risk. The path of policy rates in turn is governed by expectations of real growth and inflation in the economy. Therefore, we find
it intuitive to use changes in the following four factors as key drivers of the yield curve:

The yield of short-maturity (say, 1-year) government bonds (or “1-year yield”): a proxy for changes in short-term interest rates, which are typically set by central banks in the conduct of monetary policy
The 1-year yield one year forward less the 1-year yield: This rate depends on the market expectation of the 1-year rate in one year’s time and the risk premium for the risk of variations in short-term interest rates.
The 5-year yield five years forward less the 1-year yield: The advantage of using the 5-year × 5-year yield is that we incorporate information about growth and inflation expectations beyond the current monetary policy cycle.
The slope or difference between the 30-year yield and the 10-year yield: This variable helps to capture the behavior of the long end of the yield curve, which likely is determined by clientele effects in addition to macroeconomic factors.

regression for the change in the riskless par yield for maturity of n years (movements in the entire yield curve):

Δy(n) = α + β1 Δ(1-year yield) + β2 Δ(1-year x 1-year yield - 1-year yield) + β3 Δ(5-year x 5-year yield - 1-year yield) + β4 Δ(30-year - 10-year yield) + ε

The correlations (and volatilities) of yield curve factors, such as short rates and slopes of the yield curve at different points, are influenced strongly by the stage of the business cycle and the expected stance of monetary policy. Consequently, in our analysis in later chapters, we use explicitly defined factors for allocation of the risk budget to interest rate risk.

2.1.2. Characterizing the Variations in Credit Spreads.

Only bonds issued by sovereign governments (issuing in their own currency) come close to being default-risk free. All other bonds embed credit risk in addition to interest rate risk.
Risk of bonds = interest rate risk + “spread” risk (Spread risk is the risk of fluctuations in a bond’s price due to non-interest-rate factors, the most important of which are related to default or credit risk.)
excess returns over risk-free bonds: the portion of returns that is due to variations in the credit spread, not to changes in the default-free yield curve.
Credit Model 1

(Excess return (ith test portfolio)) / (Spread duration (i) x Spread (i)) = α(i) + β(i) x (Excess return (market portfolio)) / (Spread duration (market) x Spread (market)) + ε
Spread duration: sensitivity of the bond price to changes in the credit spread of the bond
Over short time horizons, excess returns on a portfolio of credit-risky bonds are approximately equal to the negative of spread durationtimes the change in the spread of the portfolio. Hence, excess returns per year of spread duration are approximately equal to the negative of the change in the spread of the portfolio.
The volatility of credit excess returns is higher when spreads are high (reflecting the fact that higher fundamental risk leads to higher spreads and higher volatility of spread changes).
Our specification ensures that the distribution of dependent and independent variables (excess return over spread duration times spread) has nearly constant volatility over time (that is, the distribution is homoskedastic over time) and an application of the standard regression methodology is valid.
Beta coefficient equals the correlation between the dependent and independent variables multiplied by the ratio of their standard deviations. The market betas
for these portfolios are close to 1 under the specification using excess returns normalized by spread duration times spread.

Curve effect: the differential between (normalized) excess returns of a portfolio of short-maturity bonds and those of a portfolio of long-maturity bonds.
Broad sector effect: outperformance of financial sector credits over nonfinancial credits

2.2. Key Risk Factors for Equities

Broad market equity
Sector
Region

2.3. Currencies as Risk Factors in Global Portfolios

All foreign assets come with currency exposures that affect total volatility unless currency risk has been fully hedged.
High-interest-rate currencies, such as the Australian dollar and various emerging market currencies, tend to correlate positively with stock market exposures, while low-interest-rate currencies, such as the Japanese yen, exhibit the opposite behavior.
Emerging markets’ debt, equities, and currencies all tend to rally or decline together.
The first two principal components explain 66% and 12%, respectively, of the variation in returns across currencies over the past 17 years as of December 31, 2015.
The first principal component appears to be a broad US dollar factor; the dominant role played by the US dollar in bilateral exchange rate markets due to its status as a reserve currency.
The second principal component has opposite signs on the exposures of low-interest-rate currencies, such as the Japanese yen and the Swiss franc, and high-interest-rate currencies, such as the Australian dollar. “currency carry trade” factor
The third principal component: a broad commodity index.
The three factors explain a similar proportion (roughly 40%) of the variance of emerging market currencies on average, as we show in Exhibit 2.20.
as of December 31, 2015

2.4. A Short List of Risk Factors for a Top-Down Asset Allocation Exercise

See Exhibit 2.21. A Short List of Risk Factors for a Global Asset Allocation Exercise.

3. Risk Factor Volatilities and Correlations and Their Macroeconomic Determinants

To solve the portfolio construction problem, we need, as inputs, estimates of the volatility of various risk factors, of the correlations between them, and of the risk premia associated with them.
We describe the empirical properties of these volatilities and correlations and their macroeconomic drivers. We begin with the familiar "Shiller critique" that asset markets are more volatitle than fundamentals. This effect may give rise to value opportunities where prices have deviated from fundamentals. We also note that volatility and correlations tend to be countercyclical, becoming extreme in times of economic contractions and crises.
This phenomenon also causes a level-dependence in the volatility of fixed-income markets, especially of credit spreads. It is important to account for all these features in portfolio risk management over a tactical horizon as well as over a longer horizon, such as the business cycle.
Financial market volatility far exceeds growth, earnings growth, and inflation--potentially reflecting overreaction in asset prices.
Volatility does not remain constant over time. Of particular interest to us is the dependence of asset return volatility on the stage of the business cycle. Financial market volatility is strongly countercyclical; that is, it is higher in economic downturns. In fixed-income markets (particularly credit markets), there is robust evidence that the volatility of credit returns is dependent on the level of credit spreads, with higher volatility when the spread is large. Finally, there is evidence of short-term reversals and medium-term persistence in volatility. Large shocks to volatility reverse partially in the short term, but a part of their effect is long lasting.
Correlations, not just volatilities, are time varying and sensitive to the economic cycle. In particular, correlations become more extreme in recessions. “Risk-on” assets (such as equities and credit, which generally perform well when the economy is expanding) tend to become more correlated during recessions, while correlations between
risk-on and “risk-off ” assets (such as government bonds, which perform well during economic downturns) become more negative.
Correlation between the returns on default-free bonds and equities. This correlation has switched from being mildly positive in the postwar period to being negative over the last 20 years.

3.1. Empirical Properties of the Volatility of Key Risk Factors

The fact that equity return volatility estimates are significantly higher than the level, which is fundamentally justifiable, is consistent with the notion that a large part of the variation in equity returns is due to the time variation of risk premia.
The significance of this excess volatility for investors is that it implies that
risk premia might change frequently and sometimes excessively.
valuations in asset markets tend to exhibit the property of slow mean reversion over medium horizons.
the task of a macro-aware investment process is to judge when and where risk premia may have become extreme and position portfolios to take advantage of attractive
valuations (and avoid unattractive ones).
Counter-cyclical Movements in Volatility. The fact that volatility in financial markets seems excessive relative to the volatility of fundamentals does not mean that market volatility shows no dependence on the macroeconomy.
a key consideration in a forward-looking assessment of risk is the macroeconomic outlook. Volatility is significantly higher in recessions than in expansions across the board. Over the past 60 years, equity return volatility has been 20% per year in recessions, compared with 13% per year in economic expansions. Credit markets are particularly sensitive to recessions.
In the more recent sample, since 1986, the volatility of Treasury yields has been much less variable across expansions and recessions
when the economy is expanding, financial market volatility is low; when the economy decelerates and falls into recession, volatility is relatively higher.
Level Dependence of Volatility in Fixed-Income Markets. The volatility of credit excess returns exhibits systematic covariation with the level of spreads. As a result, the volatility of changes in spreads of lower-rated bonds (higher spreads) tends to be higher than that for higher-rated bonds, and spreads are more volatile during economic downturns
and periods of financial stress.
a good measure of the spread risk exposure of a credit-risky bond is its duration times spread.
Level dependence of interest rate volatility. This level dependence of volatility is, in fact, present in interest rates as well.
Mean Reversion of Volatility in Financial Markets. Given the relationship of volatility to the business cycle, as well as the dependence of volatility on yield levels and spreads in fixed-income markets, it should not be surprising that asset market volatility also exhibits the property of mean reversion. That is, when volatility has been higher or lower than average in the recent past, it tends to revert to a normal level over time. Considering, however, that the state of the macroeconomy tends to exhibit some persistence, we would expect that financial market volatility would mean-revert somewhat slowly.

3.2. Correlation of Risk Factors

“risk-on” factors tend to react favorably to positive shocks to growth and risk appetite
“risk-off” factors react positively when the news about the economy is negative and risk aversion is rising
As expected, the risk factor that is most highly correlated with equities is the credit spread component of default-risky bonds.
Credit spreads have consistently tended to widen when equities have sold off.
Over the full sample, Treasury returns and equity returns have been modestly positively correlated. Importantly, however, over the last 20 years, this correlation has shifted to a markedly negative correlation of about -0.2 (for both short-term and intermediate rates).(*) Hence, more recently, we see that duration exposure tends to effectively diversify risk-on assets, which perform well when equities perform poorly. The curve steepener (a long position in 10-year Treasuries and a short position in 20-year Treasuries) also behaves as a mildly risk-off asset; that is, the curve tends to steepen during sell-offs in equity and credit markets.
(*) This change partly reflects a shift in macroeconomic risk from inflation, which was the dominant risk in the 1970s and early 1980s, to volatility in real growth from 1990 to the present day, a period when inflation expectations have been fairly stable.
The US dollar tends to strengthen against the euro (and, in fact, against most currencies) in crisis episodes when US assets are perceived as a “safe haven.” Such a scenario increases demand for liquid US assets, such as Treasuries, and puts upward pressure on the US dollar. The positive correlation with equities would also be a feature of the returns of emerging market currencies versus the US dollar, considering that the demand for emerging market assets wanes markedly in periods of stress. The Japanese yen is a notable exception in this respect, behaving as a risk-off asset versus equities and credit.

3.3. The Correlation between Default-Free Bonds and Equities

Investments in bonds play a pivotal role in multi-asset portfolios, both because short-horizon returns of bonds (i.e., returns over monthly or quarterly periods) tend to be negatively correlated with those of equities and because bonds tend to outperform in periods of economic weakness, when equities underperform. For these reasons, the correlation between stocks and bonds is arguably the most important correlation input to the asset allocation decision.
The bond–equity correlation has changed substantially over time and has responded to secular as well as cyclical changes in macroeconomic conditions.
The persistence of the recent “regime,” with a negative correlation between bond returns and equity returns, stands out as exceptional in the long historical time series.
The recessions in which both assets underperformed were characterized by large inflation shocks (and by the extraordinary monetary policy response to such shocks during the twin recessions of 1979–1982).
A Macro Factor Model for the Stock–Bond Correlation. One important question for investors is whether the correlation is going to stay negative in the future, providing strong diversification benefits between bonds and equities, or whether this correlation will become less negative or perhaps positive going forward.
An econometric model that relates the stock–bond correlation to the volatilities and correlations of three key macroeconomic factors: inflation, unemployment, and growth.
Short-term and longer-run correlations may differ. In the short run, stocks and bonds tend to respond in opposite directions to fluctuations in investor risk appetite. During flight-to-safety episodes, when the expected risk premium on risky assets increases, we typically observe a negative correlation.
However, at longer frequencies, shorter-term fluctuations in risk premia may be less important and the correlation can be dominated by more persistent shocks to inflation, which could result in a more positive correlation.
The short-run correlation is negative and close to –0.3 because the “flight to quality” effect dominates the effects of interest rates and inflation. However, as the investment horizon increases, the correlation becomes less negative because shocks to inflation drive bond yields and earnings yields in the same direction (higher inflation means higher yields and lower asset prices), and these shocks are relatively persistent.
A sensitivity analysis of the impact of inflation volatility and real business cycle volatility (growth and unemployment) on our model-implied correlations weaker central bank credibility can go along with more positive correlation between bonds and equities.
The real business cycle volatility gets higher in growth and unemployment, the correlation becomes more negative.
If inflation volatility does not change significantly, the correlation should remain negative, even at the 2-year horizon. On the other hand, if inflation volatility increases, our model clearly shows some “tail risk” in the hedging effect of bonds on equity risk. If inflation volatility increases significantly, the stock–bond correlation rises to +20% for 2-year returns.
The outlook for macroeconomic risk is a critical consideration in evaluating the attractiveness of duration (=bonds) as a hedge for equity and credit spread exposures.

3.4. The Covariation between Equities and Credit Spreads

The beta of investment-grade credit spread returns to equity returns has varied significantly over time.
The sensitivity of credit spreads to equities rises significantly in periods of stress.
The dramatic increase in the beta of credit spreads to equities is expected, based on the dependence of credit excess return volatility on the level of credit spreads. In periods of stress, spread levels widen, leading to an increase in the volatility and equity beta of credit excess returns. The increase in the equity sensitivity of credit can also be justified theoretically. Merton’s (1974) valuation model for the capital structure of the firm posits that equity should behave like a call option on the assets of the firm, whereas risky debt should behave like a riskless bond plus a short position in a put option on the firm’s assets.
The relationship between credit and equity returns becomes stronger in the Merton model when firm and equity valuations fall.
Credit risk exposure represents a nonlinear exposure to the value of the firm. As long as the firm is well capitalized and the leverage applied to the firm’s equity is low, there is an equity cushion that protects debtholders from default and loss of principal.
The phenomenon of higher equity sensitivity observed in time series is also seen in the cross section of credit sectors.
The equity sensitivity of credit rises as ratings deteriorate.
For lower-quality credit, the option to default is closer to being in the money—leading to a tighter relationship with equity.
Equity Sensitivity of Credit in Down Markets. The sensitivity to equities is larger on the downside than it is on the upside.
Even after adjusting for duration and spread levels, the spread sensitivity of credit excess returns tends to be larger in down markets than in up markets.
Short-maturity credit spreads tend to exhibit greater downside sensitivity to equities than the broad credit universe.
The return distribution of credit exhibits negative skewness, particularly in highly rated and low-duration credit buckets.

3.5. Lessons for Asset Allocation

The properties of risk factor volatilities—such as their countercyclicality, their level dependence, their mean reversion over short horizons, and their persistence over the medium term—are robustly present in the data. An asset allocation exercise must account for these properties.
Correlations that are critical for portfolio construction, such as the stock–bond correlation, have varied substantially over time.
The relative importance of real growth risk and inflation risk are key determinants of the stock-bond correlation.The more predominant the real growth risk, the more negative
the stock–bond correlation, while the greater relative importance of inflation risk induces a more positive correlation.
Credit spread exposure can be an important contributor to the equity beta of a portfolio, particularly in periods of stress, when the volatility of spreads increases dramatically.

4. Risk Premia in Financial Markets

The second key input into optimal asset allocation is the risk premia per unit of risk (i.e., Sharpe Ratio) for various risk factors.
Risk premia on the business cycle

Procyclical (equity, credit) or Countercyclical (excess returns on interest rate duration - excess returns of long-term bonds over the short-term interest rate - riskless bonds outperform substantially during recessions)
The stage of the business cycle

Equities, government bonds, and default-risky bonds outperform in early expansions.
As expansions mature, monetary conditions tighten and slowing earnings growth increases the incidence of bondholder-unfriendly corporate actions. As a result, the performance of government bonds and credit assets suffers markedly.

Factor risk premia are not constant over time but vary over the business cycle.
Portfolio formation should attempt to take advantage of this cyclical variation by orienting the portfolio mix towards risky assets in the late stages of a recession and the early stages of an expansion and by reducing risk as expansions begin to mature. This, however, is easier said than done. Only a minority of asset managers are likely to be able to consistently profit from timing the business cycle.

4.1. Factor Risk Premia: Theoretical Underpinnings vs. Historical Experience

One of the fundamental principles of asset pricing is that risk-averse investors demand a risk premium for bearing economy-wide risk that cannot be diversified away. Procyclical risk factors, such as the broad equity market factor, which do badly in bad times, should earn a positive risk premium. Factors that do well in bad times should have lower—possibly even negative—expected returns (over the riskless rate), since investors ought to be willing to pay to get exposure to them. The magnitude of these excess returns should also be related to the degree of risk aversion among investors (i.e., their aversion to losses in bad economic conditions) and the degree of covariation of returns with aggregate wealth.
Reconciling the Countercyclicality of Treasury Returns with Positive Risk Premia on Bonds.

4.2. Risk Premia in Interest Rates, Equities, and Credit Spreads: Beyond the Market Factor

Yield curve three risk factors: a level, or duration, factor and two curve factors (“steepener” positions in the front end and the long end of the curve)
Considering that long-dated bonds have significantly higher convexity than short-dated ones, the average returns of the steepener trade can also be attributed to the volatility risk premium embedded in the pricing of the yield curve.
Equity Factors: Size, Value, and Momentum.

A market factor which tracks the broad market.
A value factor (the so-called HML factor, defined as returns on a portfolio of stocks that is long high-book-to-market, or value, firms and short low-book-to-market firms)
A size factor (returns on a portfolio that is long small firms and short big firms, in market value terms, or SMB).
A momentum factor (returns on a portfolio that is long recent outperformers and short underperformers).
The HML, SMB, and momentum factors all had positive average excess returns over the history. The outperformance of these factors has been the subject of intense debate—and has been attributed to features that can arise in models with rational forward-looking investors and also to those arising in the presence of investors who are subject to behavioral biases.

The HML factor overweights value stocks with high book-to-market (B/M) ratios which are thought by some to be riskier. Thus, the outperformance of the HML factor may simply represent compensation for bearing this risk. This explanation is supported only weakly by empirical evidence, considering that the beta of this factor to equity markets is negative and that this factor covaries weakly with the business cycle. Behavioral explanations, on the other hand, appeal to the possibility of the overreaction of stock prices to good or bad news—which can be exploited by a value-oriented strategy such as HML.
The performance of the SMB factor since 1955 has been a lot weaker than that of both the HML factor and the equity market factor. Interestingly, the performance of the SMB factor in the first half of the sample far exceeds its performance in the past 30 years. This more recent weakness in performance has led to increased skepticism about SMB as a priced factor.
The momentum factor, have been compelling. While this factor also shows a marked weakening in performance in the past 30 years (versus the first half of the sample since 1955), it has continued to offer positive returns on average. The outperformance of the momentum factor is difficult to explain in a framework with rational forward-looking investors, and many behavioral explanations of the phenomenon rely on barriers to quick information dissemination and underreaction by investors.
Credit Spreads. Sharpe ratios of credit (over duration-matched Treasuries) by rating and maturity buckets: Sharpe ratios of low-rated investment-grade and high-rated high-yield rating buckets are larger than those of the other buckets. More specifically, since 1988, Ba rated credit appears to be the sweet spot - “fallen angel” premium. Due to restrictions in investment mandates (effectively leading to investor segmentation), investors are often forced to sell credits when they are downgraded below investment grade—which leads to selling pressure on downgraded names. This exceptional spread widening often reverses, leading to a systematic outperformance of Ba rated bonds.
Several factors contribute to the outperformance of the front end of the credit curve. First, front-end credit buckets have a larger negative skew in their returns distribution than the long end—which is due to the front end’s worse performance per unit of risk in periods of economic weakness. So the incremental performance can be thought of as compensation for bearing greater recession risk. Other explanations include the possibility of investor segmentation. For example, liability-driven investors tend to bid up—and perhaps overpay for—long corporate bonds in order to match their liabilities, which are discounted at corporate bond yields in many cases. Furthermore, the outperformance of short-maturity credit is not unlike the outperformance of short-maturity government bonds. This result can be related to the high versus low beta effect that comes from leverage-averse investors bidding up the prices of high-beta assets.
The credit quality factor is systematically long Baa rated credit and short Aa rated credit, and the credit slope factor is long the 1–3 years bucket and short the 10+ years bucket. Both factors are defined to be spread duration neutral; they are therefore immune to parallel movements in the spread curve across quality and maturity buckets, respectively.
The performance of these factors compared with constant exposure of one year of spread duration in the investment-grade credit index. As expected, all factors (credit market, credit quality, and credit slope) have positive average excess returns. Both the quality and slope factors retain a positive beta to the credit market and tend to have more negatively skewed returns distributions than the market. Both of these aspects can justify the average outperformance of these factors.

4.3. Cyclical Variations in Risk Premia

Equities and credit spreads (over Treasuries) covary positively with the business cycle. US Treasuries have the reverse property. There are, however, some nuances relating to the performance of these risk factors in the first and second halves of expansions. US Treasuries have outperformed cash in the early stages of expansions—coincident with an easy monetary policy regime and slack in the economy. It is only when expansions mature, and arguably as monetary policy is tightened, that US Treasuries underperform.
Performance of Equities and Credit in Expansions. Equities outperform throughout expansions, but they exhibit a substantial weakening in performance in the late stages of expansions. The outperformance of credit-risky bonds in expansions is almost entirely restricted to the expansion’s early stages.
As expansions mature, corporate profit growth tends to slow significantly (in comparison to the early stages of expansions), posing the risk of a decline in equity prices. In anticipation of this effect, corporate management teams have incentives to take shareholder-friendly actions at the expense of bondholders. They begin to compensate shareholders by increasing cash yields, in the form of either increased dividends or share buybacks, hoping for an expansion in price/EBITDA multiples. The increase in cash yields to shareholders is often financed by depleting cash from the balance sheet or by raising additional debt, which leads to an expansion of net debt and leverage—and eventually to a deterioration in credit quality. This chain of events contributes to the underperformance of credit markets in mature expansions.
Cyclical Performance of Other Risk Factors in Rate, Equity, and Credit Markets. In Exhibit 4.11, we present the Sharpe ratios of duration-neutral 5- to 10-year and 10- to 20-year steepener positions in US Treasuries, conditional on the stage of the business cycle. Despite the fact that the beta of these factors to returns of US 10-year Treasuries was low— only about 0.1—their performance over the cycle is quite similar to that of 10-year Treasuries: Outperformance is largely restricted to recessions, and the positions underperform in late-stage expansions. This result is consistent with the observation that the yield curve stays steep through the early stages of expansions and flattens noticeably only as monetary policy tightening gets under way in late expansions.
Risk factors in equity markets also exhibit variations in performance over the business cycle. Both the SMB (small minus big) factor and the HML (high B/M minus low B/M) factor have similar Sharpe ratios in recessions and expansions; as previously mentioned, this apparent lack of cyclicality represents a challenge to efforts to theoretically justify the risk premia earned by these factors. The momentum factor, on the other hand, underperforms in recessions relative to expansions. However, its underperformance is concentrated in the late stages of recessions as stock valuations come past their trough. There is a break in the trend of stock prices in these periods, and momentum-based investing, which relies on trend continuation for its success, underperforms. In early recessions, the momentum factor retains its outperformance.
As we partition the historical record into calendar halves of business cycles, we find that the SMB and HML factors outperform more strongly in early-stage expansions than in late-stage expansions. The relative weakening of performance of the HML factor in late expansions is at least weakly consistent with the notion that its performance covaries positively with the economic cycle. One possible reason for this behavior is that firms with high book-to-market ratios tend to have significant amounts of assets in place, which reduces their flexibility in responding to economic downturns. The SMB and HML factors perform differently in early and late stages of recessions. The outperformance of HML in early-stage recessions and underperformance in late-stage recessions points to overshooting in the valuation of low-book-to-market stocks; by contrast, the SMB factor underperforms in early recessions, along with the overall market.
The performance of the credit quality and slope factors. Strikingly, both factors perform similarly to the credit market factor over different stages of the business cycle. The quality and slope factors both underperform in recessions and in late expansions versus early expansions. The cyclicality of the slope factor is consistent with the notion that relative outperformance of the front end of the credit curve is in part attributable to higher jump-to-default risk, which is heightened in periods of economic weakness.

4.4. Lessons for Asset Allocation

There are reasonably well-identified systematic risk factors in financial markets that earn a risk premium, and these risk premia are time varying. The stage of the business cycle matters for assessing risk premia. A top-down asset allocation exercise would benefit from an assessment of the current stage of the economic cycle and how it might change over the decision horizon. However, the exercise of predicting the business cycle is necessarily a hard one, and not all portfolio managers may be skilled at it. Just the fact that risk premia follow predictable patterns over the business cycle does not imply that all investors are able to exploit these predictable variations successfully.
The analysis of both unconditional and conditional risk premia presented above provides guidelines regarding the Sharpe ratio inputs investors can use for asset allocation. While one should take heed of the old adage that “past performance is no guarantee of future results,” investors should also be cautious of accepting the argument that “this time is different.” The evidence that there are a number of risk factors other than the market factor that earn a risk premium implies that the optimal portfolio should carefully balance exposures to all these risk factors. The task of constructing such an optimal portfolio involves careful judgment about whether a particular factor is a genuine systematic factor that will reliably earn a risk premium in the future or whether it is just a short-lived anomaly or, worse still, an artifact of data mining.

5. Valuations and Risk Premia

We focus on the link between market valuation and risk premia.Since asset prices fall as expected risk premia increase and vice versa (holding all else constant), large variations in valuation metrics such as dividend yield and earnings yield (and their equivalents in the bond markets) should be indicative of significant changes in risk premia.
The challenge is to distinguish between value opportunities and value traps. (A value trap is an asset that appears cheap but is not.) Simple valuation metrics can help in this task. However, a mechanical implementation of such metrics may not work.

Risk premia vary systematically with the business cycle. However, which stage of the cycle the economy is in is not known in real time with certainty. Asset prices continuously reflect changing assessments by investors about the current state of the economy and its future evolution. In this process, time-varying risk premia become embedded in asset prices. Prices themselves contain information about risk premia: Holding other things constant, prices should be lower if risk premia are high and vice versa. Asset allocators may infer some of this information about risk premia from prices and use it in allocation decisions. However, other things are hardly ever constant. Asset prices may be lower because expectations of cash flows have fallen and not because risk premia have increased. A portfolio rebalancing would be in order only if price declines are coming from an increase in the expected return premium per unit of risk.
The task of separating changes in cash flow expectations from changes in risk premia is, however, subject to substantial imprecision. Expectations of cash flows, risk aversion, risk, and the price of risk are all unobservable quantities that cannot be accurately inferred from market prices alone. Even so, an active investment style that does not consider valuations is hard to defend as reasonable. In this chapter, we discuss some standard valuation metrics that are used in equity and bond markets. We review the evidence that valuation metrics such as price-to-earnings ratios (P/Es) and bond yields have some predictive power in forecasting returns. We show that while simple valuation metrics are generally informative of risk premia (both in the cross section and over time), they must be employed with care. Measures such as the cyclically adjusted P/E (CAPE) ratio can drift for long periods of time away from their long-run averages, posing significant challenges to their use in value-driven investment strategies.

5.1. Excess Volatility and the Promise of Value

Fundamental asset-pricing equations link valuation metrics to expected returns. For example, a higher dividend yield in equities must correspond to either higher expected returns or lower expected dividend growth. In a simplified setting, consider first the Gordon and Shapiro (1956) growth model, as shown below:
D / P = (r + λ) − g,
where D is the current dividend, r is the long risk-free real rate, λ is the equity risk premium, and g is the real long-term expected dividend growth rate.
Gordon model: P = D / (r + λ - g)
The dividend yield is higher when expected returns are higher and when expected growth is lower. As shown in Campbell and Shiller (1988) and as we demonstrated in Chapter 3, the observed variation in the dividend yield is too large to be accounted for by volatility in dividend growth alone. Thus, a large part of the variation in valuation indicators such as the dividend yield ought to come from variations in the equity risk premium.
There are many possible explanations for the excessive volatility of asset prices. One hypothesis is that households’ demand for liquidity accentuates fundamental risks, especially during recessions.
Excess volatility in asset prices can present opportunities for active, value-driven investors if that volatility is accompanied by evidence of mean reversion in valuations. Indeed, implicit in the hypothesis that institutional investors demand a concession for providing liquidity to households in recessions is the view that institutional investors are value oriented—that is, that they purchase equities in recessions with the expectation that valuations will revert to more normal levels.
Real yields are likely to be stationary but inflation expectations need not mean-revert, given the evolving nature of monetary policy over this history.
Whatever its causes, the excess volatility of financial markets has important implications for optimal portfolio construction. As prices cannot diverge from fundamentals without limit, the excess volatility of financial markets also suggests return forecastability—giving rise to opportunities for value-oriented investors.

5.2. Valuation Metrics in Equity Markets

Valuation metrics in equity markets typically involve a comparison of the market capitalization of a firm (or group of firms) to fundamental measures of the firm’s earnings or asset base. Some key valuation metrics often used by investors include trailing dividend yield (e.g., past 12-month dividend per share divided by the current price per share), earnings yield, cash flow yield, and the ratio of the book value of the equity of a firm (or group of firms) to its market value (book-to-price ratio). The basic justification for using such valuation ratios (which all have the market value of firms in their denominator) is simply that all such measures should increase if risk premia increase, holding other things constant. As a result, if one were to overweight assets with higher earnings (dividend) yields and underweight those with lower yields, one might be able to capture some of the risk premium that is embedded in these yield differentials.
The use of current earnings or current dividends is subject to the short-coming that these measures may be too sensitive to business cycle movements. We therefore often use the cyclically adjusted earnings yield (CAEY), defined as the ratio of long-run average earnings to the market capitalization of the firm, as a good indicator of the risk premium. Long-run average earnings—in this case, 10-year average earnings—better represent long-term profitability. Instead of using yields, of course, one can equivalently use price-to-earnings ratios or price-to-dividend ratios, which are simply the inverse of the yields measures mentioned above.
Indicators of valuation can be used in two ways: to position portfolios to be overweight undervalued stocks (or groups of stocks) and underweight overvalued stocks, or to be overweight the market when valuations in the aggregate are cheap and underweight when valuations are expensive. These “cross-sectional” and “time-series” strategies, while related, can have different empirical properties, especially because while the former does not typically have exposures to the market factor, the latter does.
Valuations in the Cross Section. We begin by analyzing a simple cross-sectional valuation strategy implemented across country indices. Consider the performance of a hypothetical investment strategy that chooses which country indices to overweight or underweight based on their earnings yield, dividend yield, and aggregate book-to-market ratios.
Even such simple strategies would have historically realized a positive Sharpe ratio. This experiment suggests that simple valuation metrics could be capturing information about relative risk premia in aggregate equity indices and may be helpful in decisions about which stocks or countries/regions to underweight and overweight in a global equity portfolio. A similar conclusion is obtained by looking at the evidence on the behavior of a cross section of individual stocks classified according to their book-to-market ratios.
Time-Series Strategies: Using Valuations for Timing the Market. We next analyze market-timing strategies based on valuation signals.
The cyclically adjusted earnings yield—a remarkably simple measure of valuation—appears to be a reasonable starting point for estimating the current long-term risk premium on equities.
Uncovering risk premia from valuation metrics is a difficult task and, in a sense, the central challenge of active investing. Valuations are not determined by risk premia alone: Expectations of growth in cash flows and the expected path of riskless interest rates also influence valuations. The economic environment can change in a secular manner, altering long-term expectations of growth and the path of interest rates, so the fact that earnings yields or dividend yields are too low relative to history (or a valuation metric based on price-to-earnings ratios or price-to-dividend ratios is too high) may not signal an abnormally low risk premium. Secular movements in the economy can make the time series of valuation metrics such as CAEY extremely persistent, so that the historical record is effectively shorter than it seems. This scenario leads to considerable sampling uncertainty in estimates of the distribution of the valuation metric. The evidence we have documented here demonstrates that a judicious mix of analytical and subjective inputs is required for successful value investing.

5.3. Valuation Metrics for Interest Rates and Credit Spreads

In this section, we analyze valuation signals that anchor interest rates and credit spreads to underlying fundamentals, in a manner similar to using the cyclically adjusted earnings yield to evaluate the attractiveness of the equity market. Encouragingly, we find that the evidence for mean reversion in valuations is somewhat stronger in these markets than in the equity market.
Estimating the Risk Premium in Government Bonds. The determinant of the duration risk premium is how much yields are expected to change relative to what is priced into the yield curve. We can think of the nominal yield to maturity on any riskless zero-coupon bond with maturity τ years, y(t,τ), as being given by

y(t,τ) = Et (Average nominal policy rate over [t,t + τ])
+ Interest rate risk premium + Convexity adjustment.

Typically, the adjustment for convexity explains a small component of variations in bond yields over time. So the key determinant of the interest rate risk premium is one’s view on the expected average policy rate over the time to maturity of the bond.
Carry and roll down of government bonds: Expected returns under a random walk. The assumption of a random walk in interest rates helps compute a first-cut estimate of expected excess returns on government bonds. This estimate of expected returns, known as “carry,” is a commonly used concept in fixed-income markets.
Exhibit 5.9 shows the average shape of the US Treasury yield curve from December 1985 to December 2015. On average, the yield curve is upward sloping in various subsamples. There are two interpretations of an upward- sloping yield curve. The first is that the yield curve simply reflects expectations for higher interest rates in the future. The second is that since interest rate movements are uncertain, a part of the slope of the yield curve is attributable to a premium for bearing interest rate risk (after adjusting for convexity at the long end). More specifically, if riskless yields followed a random walk, the expected excess return on a bond over the horizon to its maturity should be approximately equal to the spread between the yield of the bond and the short rate (the “carry” of the bond). Over shorter horizons—such as, say, one year—one also ought to include an estimate of “roll down” (the price return from “rolling down” the yield curve as time to maturity shrinks).
The literature finds that, although yields do not quite follow a random walk, yield changes are sufficiently unpredictable to make the slope and carry signals fairly robust quantitative predictors for average excess returns.
A simple investment strategy of buying duration in high-carry countries and selling it in low-carry markets has generated positive excess returns in our sample. This observation is consistent with the idea that the carry (or, more generally, the slope of the curve) itself contains information about the interest rate premium in various yield curves.
Beyond carry: Macroeconomic conditions and interest rates. Although carry provides a reasonable starting point for estimating expected returns, it is helpful to anchor valuations to macroeconomic fundamentals as well. There is a natural link between interest rates and economic growth and inflation. Since the expected path of policy rates is determined by medium-term expectations of real growth and inflation, it is reasonable to postulate that the difference between nominal yields and measures of medium-term expected inflation and real growth information about the risk premium embedded in the yield curve.
To exploit these linkages, we formulate a simple valuation metric for 5-year × 5-year Treasury yield based on the macroeconomic outlook. We focus on this part of the curve because, by construction, it looks beyond the horizon over which rate expectations are linked tightly to central bank policy. Beyond five years, rate expectations ought to be more related to medium-horizon forecasts of macroeconomic variables and long-term anchors of policy rates.
In particular, we specify the following valuation metric:
Value measure = 5-year × 5-year forward yield less expected inflation less expected growth,
where expected in ation and expected real growth are meant to be measured over a medium-term horizon. In our implementation of this metric, expected in ation is estimated as the 10-year trailing average of the year-over-year growth in consumer price index (CPI) while expected real growth is esti- mated as the 10-year trailing average of year-over-year real GDP growth. The 10-year window is designed to reduce the in uence of shorter-term cyclical fluctuations, in a manner similar to the averaging of real earnings in the CAEY measure for equities. Exhibit 5.11 displays the history of this metric from 1960 to 2015.
This simple metric has been a reasonable signal of excess returns of 5-year forwards on the US Treasury 5-year yield. Exhibit 5.12 displays aver- age forward-looking returns conditional on the beginning-of-period quintile of the valuation signal. Both 10-year and 3-year returns are increasing in signal “cheapness” (high yields relative to inflation and growth).
Estimating the Credit Risk Premium. The primary differences between corporate bonds and government bonds are the risk of default, relative illiquidity, and embedded options (e.g., callability). A useful starting point for assessing value in corporate credit is therefore simply the credit spread over government bonds. By adjusting the credit spread for the expected loss from default, we have a simple yet fairly reliable estimate of the expected hold-to-maturity excess returns of corporate bonds over default-free securities.

5.4. Testing Valuation Signals in Treasury and Credit Markets

The investment-grade credit spreads suggest that value-based timing strategies can exhibit positively skewed returns and thereby improve the distributional properties of the overall portfolio. This property of value-oriented strategies can be particularly valuable in improving the properties of returns to credit spread exposures, which tend to be negatively skewed.

5.5. Valuation-Based Investing: Key Takeaways

The basic logic of valuation-based investing is simple: Prices vary inversely with the risk premium, holding everything else constant. Therefore, over-weighting assets whose prices, relative to some measure of their cash flows, are low and underweighting assets whose situation is reversed should capture some of the risk premium differential that may exist between these assets. We have presented evidence across several markets suggesting that valuation-based investing does seem to generate excess returns on average. To the extent that price fluctuations are exaggerated by short-term surges in demand for liquidity, value investing provides liquidity when it is in high demand. Investors with a relatively long-term investment horizon ought to be in a position to engage in such liquidity provision and be compensated for it. Also, to the extent that the logic of value investing is based on earning a risk premium rather than taking advantage of a short-lived market inefficiency, such an investing style is probably sustainable in the long term. It can therefore be argued that the core of a sound investment process should be built around value investing. However, one has to find ways to deal with a few challenges that value investing poses—some of which should be apparent from our discussions in this chapter.
First, prices do not depend on risk premia alone. Variations in prices or yields could also be the result of changes in expectations of cash flows of underlying assets. It is therefore incumbent on value investors to take a nuanced view of expectations over both cyclical and secular horizons. This approach might require investors to blend qualitative and quantitative considerations.
Secondly, no single valuation metric should be used to the exclusion of others.
Lastly, value-driven investment strategies can see sustained periods of underperformance, given that asset prices tend to mean-revert fairly slowly. It is often useful to complement valuation-based signals with momentum-driven ones, since the latter conveniently capture high-frequency information embedded in indicators such as fund flows.

6. Putting It All Together: Optimal Portfolio Construction

We show how to combine the inputs of volatilities, correlations, and risk premia to construct an optimal portfolio of exposures to key global risk factors. We use the simple one-period mean–variance model for this purpose. We show that despite its simplicity, this approach can be fruitfully used to construct realistic portfolios. The key message is elementary but too often forgotten in practice: The optimal portfolio consists of a balance of procyclical and countercyclical exposures.
avoids doubling up on macro bets (overconfidence in opinions about market outcomes, the predictability of which is typically much lower than is commonly thought) and tries to exploit relative value between correlated macro risk factors
tail risk (e.g., equity returns and credit spreads)

The objective of active asset allocation should be to construct a portfolio that delivers the highest risk-adjusted returns. In the previous chapters, we have provided a framework for characterizing the expected return and risk of different risk factors. The next step is to synthesize the resulting views into a portfolio.
In a generalized setting, portfolio choice can be viewed as a utility maximization problem in a multi-period for the investor.
First, we assume that the investor’s risk preferences and liability constraints are embedded in the choice of a benchmark or policy portfolio. For example, an investor who has a highly binding liability constraint would choose a more fixed-income-oriented benchmark, while a working-age individual with a large amount of human capital in his or her “portfolio” would choose an equity-heavy policy portfolio. The optimization problem is effectively one that maximizes the expected returns of the overlay versus the benchmark, subject to the constraint that the tracking error volatility of the portfolio (i.e., the standard deviation of the portfolio’s excess return over the return of the benchmark) is less than a given limit.
Second, we focus on a one-period optimization problem rather than a multi-period one. This assumption reduces the mathematical complexity of the problem while still offering a realistic reflection of the portfolio construction exercise undertaken by institutional investors, whose performance is often measured over annual horizons.
Third, we assume that the trade-off between expected returns and risk can be quantified in terms of the mean and variance of returns on various risk factors.
Although portfolio managers in practice do not mechanically construct their portfolios according to any particular optimization program, we show that a simple MVO setup can deliver rich insights into the trade-offs between risk and return.

6.1. Key Trade-Offs in Portfolio Construction: A Simple Example

To illustrate the main trade-offs in a typical portfolio construction problem, we begin with a simple example. We consider the problem of an investor who seeks an optimal portfolio over a tactical horizon (say, 12 months) and measures performance relative to a benchmark. The investor’s objective is to choose an overlay to maximize expected excess return over the benchmark subject to the constraint that the tracking error volatility of the portfolio (i.e., the standard deviation of the portfolio’s excess return over the return of the benchmark) is less than a given limit.
Important considerations in portfolio construction.

First, some assets, such as government bonds, hedge portfolio returns in recessions. An optimally constructed portfolio would hold positive exposures to government bonds even if their Sharpe ratios were relatively poor. This case demonstrates the central role that fixed-income investments play in diversified portfolios, especially when investors seek to limit losses in “left-tail” events.
Second, risk exposures to “relative value” positions appear frequently in unconstrained optimizations. These relative value views tend to emerge from differences between the Sharpe ratios of correlated assets. These views ought to be stress-tested because the optimization program would likely capitalize on small presumed differences in expected returns across correlated risk factors. The possible antidotes to this problem are to examine the return assumptions carefully and to constrain the size of relative value positions, if necessary.

6.2. Portfolio Construction in a Macro-Oriented Setting: Main Ingredients

We now turn to the more realistic tactical asset allocation problem of a global multi-asset investor. A business cycle–oriented approach has natural applications in this context. The key inputs to this asset allocation problem—forecasts of expected returns of factors—are intimately related to views on the state of the macroeconomy. We demonstrate that the use of a macro-oriented approach generates a nuanced set of forward-looking inputs and therefore a more soundly constructed portfolio.
Estimating Expected Returns and Sharpe Ratios.
Defining a Tractable Value Metric. The first step in estimating ex ante Sharpe ratios is to select an analytically tractable valuation metric. For the US equity market factor, we take this metric to be cyclically adjusted earnings yield (CAEY), which is the inverse of the cyclically adjusted price-to-earnings multiple. As discussed before, this valuation metric compares prices to a trailing 10-year average of earnings, thereby smoothing out cyclical variations. In general, we prefer earnings yield to its inverse, since it is more robust at the extremes of earnings: As earnings fall towards zero, the price-to-earnings multiple increases non-linearly, while the earnings yield declines more gradually.
As we discussed in Chapter 5, equity valuation metrics such as the CAEY mean-revert over 3- to 5-year horizons. The mean to which they revert, however, has not been constant over time.
Some of the decline in earnings yields since the early 1980s can be attributed to the secular decline in real interest rates. We therefore de fine the equity risk premium (ERP) as

ERP ≡ CAEY × Payout ratio + E (Real earnings growth ) − E (Real interest rate).

The first term is an estimate of “cyclically adjusted” dividend yields to shareholders, which should include cash returned via both dividends and share buybacks. The second term incorporates the expectation of growth into these cash flows. The last term adjusts for the effect of interest rates in pricing the present value of cash flows.
In the history of our estimate of the ERP since 1950, despite having adjusted for the effect of interest rates, we find that the estimate of ERP has declined systematically over the past 65 years. The decline has often been attributed to one-off effects, such as advances in technology that have improved liquidity and the efficacy of arbitrage capital in US equity markets—a view that would prompt us to believe that average ERP going forward will be closer to recent experience. On the other hand, the post-1980s period (until the financial crisis of 2008) was also one of extraordinary calm in the macroeconomy and financial markets, which might not recur. This view would prompt us to believe that the average ERP going forward will be higher than it has been since the mid-1980s. Given these competing arguments, we use the unconditional average since 1950 of 3.1% as the mean to which ERP is likely to revert over the next three to five years.
Systematic Variations in the Valuation Metric over the Business Cycle. On average, ERP was higher than its trend by 0.8% in recessions and lower than its trend by 0.2% in expansions, coinciding with the underperformance of equities in recessions and their outperformance, on average, in expansions.
Using deviations from trend as our guide to cyclical variations in ERP, rather than its level, allows us to conveniently account for our forward-looking views on the long-term average of equity valuations. We alter the distribution of these deviations slightly to be consistent with the view that ERP averages going forward are likely to be the same as the average over the past 65 years (i.e., the secular decline in ERP will not continue in the future and the average deviation from trend would be zero rather than negative). The variations of the metric around this “re-centered” mean are preserved in our calculations to be consistent with historical experience.
Estimating Expected Return on Equities Given a View on the State of the Economy. The volatility of risk factor returns also varies systematically over the business cycle.
It is important to take a view on the business cycle in arriving at expected return forecasts.
Expected Returns of Other Risk Factors. Expected returns of other key risk factors are estimated along similar lines. e.g., x-axis: Sharpe Ratio with Recession Probability = 15%, y-axis: Sharpe Ratio with Recession Probability = 50%

6.3. Practical Considerations in Portfolio Construction: Imposing Constraints

Three key practical considerations that ought to be embedded as constraints in the portfolio construction problem.
Incorporating Tail Awareness into Portfolio Choice. Limit losses in a “left-tail” event.
A trade-off directly between expected returns and forecasts of losses in periods of stress, often defined as conditional expectations of returnsin the bottom 5%–10% of the distribution (CVaR, or “conditional value at risk”). While this “Mean vs. CVaR” approach could be appropriate for a portfolio of particularly convex assets, such as out-of-the-money options, in our view, it is not ideal for a top-down asset allocation exercise whose opportunity set is dominated by modestly skewed risk factors. The degree of estimation error is significantly larger when one focuses on a small part of the returns distribution of risk factors rather than its entirety. Our preferred formulation of the problem is as a trade-off between mean and variance, while including tail awareness through an additional constraint that expected losses in the “left tail” (bottom 5%–10% of the distribution) be no larger than a predetermined limit.
A constraint that limits expected portfolio returns in the left tail would therefore curtail exposures to credit spreads, which as higher left-tail risk unless the incremental benefit to expected portfolio returns were enough to compensate for the greater risk.
Constraints on Relative Value Positions.
It is often useful to stress test the assumptions behind large differences in forecasts of expected returns—and assess whether these assumptions are robust to varying sets of parameter inputs and qualitative views. The easiest way to control relative value positions is to constrain the risk allocated to each relative value position. In the presence of such a constraint, the solution to the problem would by definition be consistent with the investor’s ex ante conviction in the position—that is, the investor’s assessment of his
or her skill in determining the attractiveness of a given relative value position.
Institutional Constraints on Portfolio Allocation. Constraints on portfolio allocation can also relate to investment mandates. The most common is a constraint that prohibits portfolio managers from selling securities short. Similarly, investment mandates might disallow holding securities if they fall below a certain credit quality (often defined in terms of ratings). Institutional mandates might also disallow excessive leverage in portfolios. Leverage is often employed by portfolio managers to gain exposures to certain risk factors (such as a duration-neutral steepener trade). However, such exposures can be fraught with a significant degree of risk in scenarios of stress, often because of the inability to refinance positions—a vulnerability that warrants limits on the amount of leverage allowed. In order for portfolio managers to abide by such rules, constraints might have to be imposed on the size of allowable positions.

6.4. Optimal Portfolio Construction: A Case Study

6.5. Conclusion

The basic idea of MVO is simple: Find the allocation that maximizes expected return for a given portfolio volatility. However, despite having been available as a portfolio construction methodology for several decades now, and despite there being a multitude of so-called commercial “optimization tools” that employ the mean–variance approach, MVO has perhaps not gained as much currency among practitioners as one might expect.
The main criticism of MVO in a practical setting is that implementing it from first principles requires investors to take views on a large number of parameters, particularly those relating to the expected returns of risk factors. Typically, investors have a small number of high-conviction views, which is not sufficient to run a full-blown optimization exercise. A related criticism is that mean–variance solutions can lead to large long–short positions in risk factors with high correlations, even if their Sharpe ratio differences were small and potentially caused by estimation error.
Several alternatives have been forwarded to address these concerns, the most notable being the “reverse optimization” method of Black and Litterman (1990). The Black–Litterman (BL) method takes a market portfolio as the starting point for the portfolio construction exercise. Further, it provides a convenient way, via Bayesian analysis, to blend an investor’s small number of views with this starting point in order to arrive at an optimal portfolio. The BL setup is ideally suited for building portfolios of stocks in which the number of views that active investors might have tends to be quite small.
However, the use of a “neutral point” as an exogenous input to the portfolio construction exercise is a convenient feature to incorporate into multi-assetcontexts as well. Our discussion in this chapter has focused only on the determination of active tilts versus a neutral point, which could be a benchmark or a policy portfolio.
To deal with the issue of optimal solutions requiring large long–short positions, it is often useful to constrain relative value positions. In addition, it is essential to have an ongoing interaction between setting up the optimization problem and assessing the confidence in the inputs to it. For example, an investment process that emphasizes relative valuations of risk factors across countries or sectors should allocate a greater amount of the risk budget to relative value positions than one that is more top-down in nature.
The institutional context may necessitate modifications to the simplistic MVO approach. An asset manager who is mandated to manage portfolios against a fixed-income benchmark is typically expected to construct portfolios that outperform in regimes of economic weakness. Active overlays on such benchmarks ought to incorporate the additional constraint that they will not underperform significantly in weak economic conditions.
Despite these challenges, a formal portfolio construction exercise is an important tool to help portfolio managers navigate the complex trade-offs inherent in any large set of investment opportunities. With the judicious use of tail awareness, position constraints, and parsimony in problem formulation, even simple methodologies like MVO can yield rich insights into the optimal risk–return trade-off in realistic situations.

7. Moving beyond Stocks and Bonds: Alternative Investments

Alternative assets (those beyond publicly traded stocks, bonds, and currencies) in the optimal portfolio include real estate, farm-land, timber, infrastructure assets, and assets managed by specialist managers, such as hedge funds, private equity managers, and venture capitalists
return smoothing due to infrequent return calculations - realistic estimate of volatilities, correlations, and Sharpe ratios

So far, we have concentrated on the problem of portfolio construction with traditional assets, such as equities, government bonds, and corporate bonds. In a real-life asset allocation exercise, the investor also has an opportunity to invest in a variety of so-called alternative investments. These investments can be classified broadly into three groups:

Private equity and venture capital
Real assets: real estate, infrastructure, farmland, timberland, and natural resources
Hedge funds and exotic beta strategies (momentum, carry, value, volatility, etc.)

In this chapter, we discuss the considerations that need to be accounted for when including alternatives in an optimal portfolio.
We review the risk properties and diversification benefits of alternatives vis-à-vis publicly traded equities and bonds. We show that the lack of mark-to-market data may lure investors into the misconception that alternative asset classes and strategies represent something of a “free lunch.” This misconception arises because return indices for privately held assets often are artificially smoothed, which biases both volatility and correlation estimates downward (particularly in down markets) and, correspondingly, biases measures of risk-adjusted performance, such as the Sharpe ratio, upward.
To address this problem, the statistical methods used to estimate correlations and volatilities must be adjusted to control for reporting biases in the illiquid return series. We show how to estimate risk factor exposures when the available asset return series may be smoothed due to the difficulty of obtaining market-based valuations. This adjustment provides a way of obtaining a more realistic view of the risks in alternative and illiquid investments. We find that alternative investments are exposed to many of the same risk factors as those that drive stock and bond returns. Risk models that fail to capture the systematic risk factor exposures of these investments may consequently overestimate their diversification benefits, resulting in the potential for overinvestment in alternatives or higher-than-expected downside risk in crisis episodes.
The bottom line of our analysis in this chapter is that alternative investments are riskier than their reported index returns would generally suggest. Similarly, their correlations with other asset classes are higher once we control for reporting biases. These features of alternatives should be taken into account in selecting portfolios that can allocate risk to these assets.

7.1. Risk Factor Exposures of Alternative Investments

The main challenge in including alternatives in an optimal portfolio with traditional assets is to model the exposure of alternatives to the same (or similar) risk factors to which traditional assets are exposed. An assessment of which factors to include requires the use of econometric methods as well as judgment. A “kitchen sink” regression approach, which starts from a large set of risk factors, however sophisticated it may be, will tend to isolate factors that improve the fit in sample but can produce exposures without a clear economic interpretation. For this reason, our approach to assigning risk factor exposures to alternative asset classes consists of two steps:

First, we rely on economic intuition to narrow down the set of factors that should be relevant for a particular alternative asset class or strategy. This process relies on basic valuation principles and knowledge of the underlying investments.
Second, we use econometric techniques to estimate exposures to each of the factors based on historical returns. To adjust for the smoothing effect, our model assumes that observed index returns represent a “moving average” of the current and past “true” investment returns. Dimson (1979) and Scholes and Williams (1977) present some of the theoretical foundations for this approach.

As the first step in our empirical analysis, we discuss and identify the most important set of risk factors for each asset class. If we accept that investors value alternative assets as discounted cash flow streams, we should expect their volatility to be driven mostly by the same factors that drive expected growth and discount rates for stocks and bonds. For assets with stable and less cyclical cash flow dynamics, valuation changes should be dominated by changes in interest rates—just as interest rates drive most of the volatility for bonds—while valuations for more speculative and highly cyclical investments should be driven by changes in the risk premia that investors require for risky assets and should consequently exhibit more equity-like characteristics.
Based on this logic, we posit that private equity, venture capital, and real assets are exposed to the following risk factors: the three Fama–French equity factors (i.e., equity market beta, small size, and value) and, additionally, credit spreads, real interest rates, and a liquidity factor.
Equity beta represents most of the mark-to-market risk across alternatives because equity market returns reflect changes in the way that investors value and discount risky cash flow streams at a broad level. As for corporate earnings, cash flows for private assets are linked to general economic growth. Company profitability and earnings growth can be expected to be high during expansions and low during recessions, irrespective of whether a specific company is traded privately or publicly. The same logic applies to real estate and infrastructure investments, whose cash flows—and therefore market values—vary with the level of economic activity. For example, a recession may reduce demand for office and retail space, which in turn negatively affects the occupancy rates and net operating income of commercial real estate properties. Hence, in general, changes in prospective equity market earnings should also be positively correlated with changes in projected cash flows from private investments.
Other equity factor betas help better capture asset class–specific risk exposures. Our models incorporate the size and value factors to account for exposures that may be independent of broad equity beta. Venture capital investments typically have strong growth (negative value) exposures, whereas other private equity strategies that aim at acquiring undervalued firms through levered buyouts can be characterized as having a distinct value tilt.
Credit spread duration captures bond-like cash flow risk and financing effects. While equity returns capture some of the common variation in discount rates across alternative asset classes, credit spreads may play a distinct role in shaping the returns for some alternatives, such as real estate and infrastructure.
Due to the nature of their bond-like cash flows, the pricing of these real assets may fluctuate more directly with bond spreads than with equity valuations. In other words, credit spreads are a key component of the discount rate applied by investors to the cash flow streams of real asset investments because these assets are viewed in part as substitutes for bonds. In addition, most private equity, real estate, and infrastructure portfolios may be exposed to financing or refinancing risks. Due to this exposure, anticipated returns can be particularly vulnerable to changes in the cost and availability of debt financing, both of which change with credit spreads.
Real interest rate duration represents the inflation-hedging characteristics of certain alternative asset classes. Real estate investments, for instance, provide real cash flows that are broadly insensitive to the level of inflation and nominal cash flows that track inflation over the medium to long term. Rent payments can, for example, be modeled as cash flows that are similar to coupon payments on an inflation-indexed bond, since rent changes tend to reflect the general level of inflation. Similarly, managers of infrastructure investments (such as toll roads and electricity producers) often have opportunities to at least partially adjust prices in response to inflation. Therefore, real estate and infrastructure investments could be particularly exposed to changes in real interest rates and less sensitive to changes in nominal rates. (In certain cases, where inflation pass-through is limited, it is appropriate to also consider assigning some nominal duration in the risk factor model.)
Liquidity beta represents an important, yet often overlooked, component of the investment risk of most alternative asset classes. Indeed, decisions to allocate to private and illiquid asset classes are often made without serious consideration of their exposure to liquidity risk. To capture the potential exposure of illiquid assets to fluctuations in liquidity, we include Pastor and Stambaugh’s (2003) liquidity factor in our models for real estate, private equity, and infrastructure. The Pastor–Stambaugh factor captures excess returns on stocks that have large exposures to changes in aggregate liquidity.
Pastor and Stambaugh construct their liquidity measure for each stock by estimating the return reversal effect associated with a given order flow (volume). They rely on the idea that lower-liquidity stocks will experience greater return reversals following high volume. These stock-level liquidity estimates are aggregated to form a marketwide liquidity measure at each point in time.
The return to the liquidity risk factor in a given period is defined by the returns of a long–short portfolio of stocks that have been sorted according to their sensitivity to changes in market liquidity (“liquidity betas”). This methodology is similar to the methodology used to derive the Fama–French (1992) factors.
Recent academic research by Franzoni, Nowak, and Phalippou (2012) confirms that realized private equity returns are affected by their significant exposure to the Pastor–Stambaugh liquidity factor. The authors describe the economic channel that links private equity to public market liquidity, explaining how changes in illiquidity affect returns through the availability and costs of financing for private equity deals:
Due to their high leverage, private equity investments are sensitive to the capital constraints faced by the providers of debt to private equity, who are primarily banks and hedge funds. Therefore, periods of low market liquidity are likely to coincide with periods when private equity managers may find it difficult to finance their investments, which in turn translate into lower returns for this asset class.
The effects of funding liquidity and market liquidity are not confined to private real assets. Liquidity conditions should affect the viability of all levered investments and should drive correlation across assets, especially during stress periods. A common liquidity beta across alternative assets may help capture this effect.
It should be noted, however, that liquidity conditions generally fluctuate with aggregate market volatility and that changes in liquidity premia are also embedded in credit spreads; hence, the liquidity betas that we estimate must be interpreted as exposures to “incremental systemic liquidity,” net of the liquidity effect embedded in other factors.
Risk Factors for Hedge Funds. One might consider a more extensive list of risk factors to capture the risks of hedge funds and include specialized “alternative beta”–type risk factors, such as FX carry, volatility, and momentum (trend following) in the analysis. We have found, however, that hedge fund style index returns are well explained by exposures to a conventional set of risk factors, and for that reason, we keep the set of risk factors parsimonious. The motivation for including hedge fund allocations in multi-asset portfolios is generally to diversify exposure to equity risk. It is therefore especially important to estimate the relationship between hedge fund returns and the equity factor and to evaluate how robust the relationship is likely to be in stressed markets. Most hedge fund styles tend to have significant exposures to equity risk (direct or indirect) that may lie dormant until a crisis occurs.

7.2. Econometric Estimation of Factor Exposures

Returns to a given asset can be expressed as a linear combination of risk factor returns. To derive the econometric specification, we assume that the observed “smoothed” returns for each of the illiquid assets can be viewed as a weighted average of the recent history of actual but unobserved returns. Thus, the observed return series on alternatives can be viewed as a so-called “moving average” process of past realized returns. If these realized (but unobserved) returns have a factor representation, then we can establish a relationship between moving averages of risk factor returns and the observed (smoothed) returns on alternatives.

7.3. Computing Risk Estimates

Factor-based volatility. To estimate volatility from risk factors for a given asset class, we use the standard portfolio risk formula, but we replace weights, volatilities, and correlations with risk factor exposures, risk factor volatilities, and risk factor correlations.
Non-factor-based volatility (idiosyncratic risk). We add idiosyncratic volatility such that total volatility matches the unsmoothed index volatility. Idiosyncratic volatility can come from security selection, factor timing, and a variety of other nonsystematic, non-factor-based risk exposures. Idiosyncratic volatility is assumed to have zero correlation with factor-based volatility.

7.4. The Bottom Line on Risk Factor Models for Alternatives

In many cases, public and private investment vehicles provide exposure to the same underlying assets and represent claims to similar or highly correlated cash flows. Consequently, public and private investments in the same underlying asset or economic activity should be distinct only from the point of view of liquidity, tax structure, dividend distribution profile, and to some extent, leverage.
Our risk factor framework and our econometric modeling approach reveal that alternative assets indeed have significant exposure to the same risk factors that drive volatility in publicly traded stocks and bonds. Returns on alternative assets depend on changes in interest rates, as well as the way that investors value risky cash flows, as reflected in equity market valuations and credit spreads. Lastly, liquidity and other specialized factors also play a role. In addition to higher volatility, expected drawdowns, and tail risk exposures, the risk factor–based approach generally generates higher correlations between alternative investments and their public market counterparts, especially when their equity beta is high.
The characteristics of privately held assets can appeal to different investors and segments of the market and thereby possibly drive a wedge between the valuations in private and public markets, and the expected returns of privately held assets may differ at different phases of the business cycle and/or funding cycle.

Summary

Our goal is to show how we apply existing research and its extensions to solve real-life portfolio allocation problems. We intend to convey the insights we have obtained from this amalgam of research and practice in numerous portfolio construction exercises. In our applications, we have found that a broad choice of models works best. Also, we prefer parsimonious and simple models over complex ones. We believe that simplicity and modularity lend substantial robustness to investment analysis.

The Financial Journal (Global)

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Wednesday, April 4, 2018

Factor Investing and Asset Allocation: A Business Cycle Perspective

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Deep Learning (Regression, Multiple Features/Explanatory Variables, Supervised Learning): Impelementation and Showing Biases and Weights

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