5.2.2.1 Measuring γ

One approach to measure the economic gain from making more intelligent financial planning decisions is to calculate the net present value of the additional income generated by the improved strategy, for example, to quantify the economic benefit that American investors can obtain from strategically timing the start of social security benefits (e.g., adjusting the initial claiming age). This approach does not require any explicit assumptions about subjective investor preferences.

5.2.2.2 How to Measure γ

For a single period, expected utility theory expects that an investor ranks alternative uncertain amounts of income by the expected utility of each. Letting I denote the random amount of income in the given period, then the expected utility of I is

(5.1)

where u(·) is an increasing concave utility function that reflects the risk tolerance of the investor.

Since values of u (·) are abstract undefined measures of “utility,” one may define the expected utility to the utility-adjusted certainty equivalent level of income:

(5.2)

which means that the investor is indifferent to the random amount of income I and the certain amount of income CE[I]. γ measures how much additional utility-adjusted income a strategy increases over and above the utility-adjusted income from a set of base-case decisions.

In the financial literature, several parametric forms of u(·) are commonly used. The most common one is the constant relative risk aversion (CRRA) utility function, which may be expressed as:

(5.3)

where θ is the risk tolerance parameter.

Owing to its analytical simplicity and its ability to represent a wide range of opinions toward risk with a single parameter, the CRRA utility function is used not only in single-period models, but also in multiperiod models.

Let I_t denote the random amount of income in period t, the expected utility of the sequence of incomes from periods 0 to T in these models is

(5.4)

where d_t is the discount factor for period t. However, the parameter θ does have a role in the calculation of utility even if there is no uncertainty – because, in a multiperiod context, θ plays the following two roles:

1. The investor's risk tolerance parameter
2. The investor's elasticity of intertemporal substitution (EOIS) preference parameter.

It was pointed out by Epstein and Zin that there is no reason that the risk tolerance parameter and the EOIS parameter are equal. The reason for setting them equal is mathematical expediency. By recursively nesting the certainty equivalence function inside of the intertemporal utility function, Epstein and Zin formulated expected utility that makes these distinct parameters:

(5.5)

where

1. V_t is the utility of the stream of income, starting at time t (measured in the same unit as income), and
2. η is the investor's elasticity of intertemporal substitution preference parameter.

On the certainty equivalent operator, the t subscript denotes that it is conditional on what is known at time t.

The utility function generalizes the recursive expected utility maximization problem formulated by Lucas (1978), whereas gamma is derived from a measure of the utility of a given set of simulated income paths. Here, one may formulate a utility function with the same EOIS and risk parameters as the Epstein–Zin utility function that may be evaluated without recursion. This may be achieved by reversing the order of the nesting of intertemporal and risk components of utility. Thus, for each simulated income path, one may calculate its utility-equivalent constant income level based on the EOIS parameter, which is denoted as II. That is, for a given simulated income path, II is the constant amount of income with the same utility as the actual income path. This is given by

(5.6)

where

1. I_t is the level of income in year t
2. q_t is the probability of surviving to at least year t
3. r is the last year for which q_t > 0

ρ is the investor's subjective discount rate (so that d_t in (5.5) is q_t(1 + ρ)^−t Note that while (5.6) contains two preference parameters (ρ and η) that describe how the investor feels about having income to consume at different points in time, it does not show how the investor feels about risk. As was discussed already, one treats the elasticity of intertemporal substitution as a parameter distinct from the risk tolerance parameter. One may introduce the risk tolerance parameter next by treating the entire path as unknown and evaluating expected utility.

One may measure expected utility using the CRRA utility function with its risk tolerance parameter θ that we introduced in (5.3).

(5.7)

where

1. M is the number of paths
2. the subscript i denotes which of M paths is being referred to

p_i is the probability of path i occurring which is set to 1/M.One may define Y as the constant value for II that yields this level of expected utility. This is the certainty equivalent of the stochastic utility-adjusted income II, and Y is given by

(5.8)

One may now define the gamma γ of a given strategy or a set of decisions as

(5.9)

As a base case, one may use the following parameter values:

Later, one may perform sensitivity analysis to explore the impact of how gamma γ is affected by the choice values for these parameter values.

5.2.4.1 The Elasticity of Intertemporal Substitution (EOIS)

To illustrate the role of the EOIS parameter, η, and its impact on gamma, consider a model in which a given amount of total wealth (financial assets plus human capital) is used to finance consumption over an infinite time horizon. Assume that the market offers a risk-free flat yield curve. If the single market interest rate equals the subjective discount factor (ρ), the optimal level of consumption is the same in every year regardless of the marginal rate of intertemporal substitution.

If the market interest rate is less than the subjective discount rate, the optimal consumption path is downward sloping:

1. If the EOIS is high (η = 0.9), the optimal level of consumption starts high and the path is steeply sloped. However, if the EOIS is low (η = 0.1), the optimal consumption path is nearly flat.

   This model of the sensitivity of gamma to the η parameter illustrates that gaging investors' willingness to reschedule income can be an important input to financial planning. While most financial planning questionnaires are designed to elicit information about an investor's investment horizon and risk tolerance, few seek to assess an investor's elasticity of intertemporal substitution.

5.3.1.1 Example of a Risk-Averse/Neutral/Loving Investor

Suppose an investor is given the choice between the following two scenarios:

1. One with a guaranteed payoff
2. One without

In the guaranteed scenario (1), the investor receives $100,000 zillion.

In the uncertain scenario (2), a coin is flipped to decide whether the person receives $200,000 (that is, 2 × $100,000) or nothing. The expected payoff for both scenarios is $100,000, meaning that an individual who was insensitive to risk would not care whether they took the guaranteed payment or the gamble.

However, individuals may have different risk attitudes.

An investor is said to be the following:

• Risk-averse (or risk-avoiding): If the investor would accept a certain payment (certainty equivalent) of less than $100,000 (e.g., $80,000), rather than taking the gamble and possibly receiving nothing.
• Risk-neutral: If the investor is indifferent between the bet and a certain $100,000 payment.
• Risk-seeking: If the investor would accept the bet even when the guaranteed payment is more than $100,000 (e.g., $120,000).

The average payoff of the gamble, called its expected value, is $100,000. The dollar amount that the individual would accept instead of the bet is called the certainty equivalent, and the difference between the expected value and the certainty equivalent is called the risk premium, that is,

(5.15)

For risk-averse individuals, the risk premium is positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium becomes negative.

5.3.1.2 Expected Utility Theory

In expected utility theory, a person (including an investor) has a utility function u(x), where x represents the value that he might receive in money or goods (in the above example x could be 0 or 100).

Time does not come into this consideration, so inflation does not appear. (The utility function u(x) is defined only up to positive linear affine transformation – in other words, a constant offset could be adjusted (added) to the value of u(x) for all x, and/or u(x) could be multiplied by a positive constant factor, without affecting the conclusions).

An agent possesses risk aversion if and only if the utility function is concave. For instance u(0) could be 0, u(100) might be 10, u(60) might be 7, and for comparison u(50) might be 6.

The expected utility of the above bet (with a 50% chance of receiving 100 and a 50% chance of receiving 0) is

(5.16)

and if the person has the utility function with u(0) = 0, u(50) = 6, and u(100) = 10, then the expected utility of the bet equals 5, which is the same as the known utility of the amount 40. Hence, the certainty equivalent is 40.

The risk premium is ($50 − $40) = $10, or in proportional terms

(5.17)

or 25%, where $50 is the expected value of the risky bet

This risk premium means that the investor would be willing to sacrifice as much as $10 in expected value in order to achieve perfect certainty about how much money will be received. That is, the person would be indifferent between the bet and a guarantee of $40, and would prefer anything over $40 to the bet.

For a more wealthy investor, the risk of losing $100 would be less significant, and for such small amounts his utility function would be likely to be almost linear, for instance if u(0) = 0 and u(100) = 10, then u(40) might be 4.0001 and u(50) might be 5.0001.

5.3.1.3 Utility Functions

The utility function for perceived gains has two key properties: an upward slope and concavity.

1. The upward slope implies that the investor feels that more is better: A larger amount received yields greater utility, and for risky bets the person would prefer a bet which is first-order stochastically dominant over an alternative bet (i.e., if the probability mass of the second bet is pushed to the right to form the first bet, then the first bet is preferred).
2. The concavity of the utility function implies that the investor is risk averse: A sure amount would always be preferred over a risky bet having the same expected value; moreover, for risky bets the investor would prefer a bet that is a mean-preserving contraction of an alternative bet (that is, if some of the probability mass of the first bet is spread out without altering the mean to form the second bet, then the first bet is preferred) (Figures 5.3a, b, and c).

A person is given the choice between two scenarios, one with a guaranteed payoff and one without. In the guaranteed scenario, the person receives $50. In the uncertain scenario, a coin is flipped to decide whether the person receives $100 or nothing. The expected payoff for both scenarios is $50, meaning that an individual who was insensitive to risk would not care whether they took the guaranteed payment or the gamble. However, individuals may have different risk attitudes.

5.4.1.1 The Rationale Behind Time-Varying Risk Aversion

While the average investor risk tolerance changes over time, there are a number of other investor and market characteristics that affect risk aversion. Two potential reasons of risk aversion are

1. historical experiences (i.e., past returns) and
2. future expectations of stock market performance.

The research has shown that there is a significant relation between risk aversion score decile and net equity mutual fund flows, CAPE Ratio, and future one-year returns. The negative relation between risk aversion and net equity flows suggests the following:

1. Investors tend to favor equities when risk aversion is the lowest, which is also when markets have the least attractive valuations based on the CAPE Ratio and lowest future one-year returns.
2. Investors appear to be less attracted to stock funds when valuations are most.

5.4.1.2 Risk Tolerance for Time-Varying Risk Aversion

Since risk aversion is time-varying, it is possible that the “run-of-the-mill” risk tolerance questionnaires (RTQs) may contribute to individual investor underperformance if the scores influence asset allocation selected by the plan providers. The more sophisticated investors may be less susceptible to risk preference bias.

One way to determine the relative investor sophistication is to look at the historical relation between net equity mutual fund flows for different types of mutual funds and risk aversion over time. The seasoned investor with constant risk preferences would rationally rebalance to equity mutual funds during a bear market. A biased investor may be less attracted to equity funds as valuations fall, increasing the portfolio share of safe assets when valuations of risky assets are most attractive. One may use three different methods to classify investors based on net equity mutual fund flows:

1. Sort by whether funds are actively or passively managed. Passive mutual funds are assumed to be those classified as “index” funds by Morningstar, Inc., while all other mutual funds are considered actively managed funds. There is evidence that investors in passive funds are more sophisticated than investors in active funds that are most often sold through the broker channel.
2. Classify mutual funds into four different fund groups: broker-sold, institutional, investor, or retirement.
3. One may further decompose the broker-sold category by the form of compensation paid to the financial advisor. Differences in the method of compensation provided through share class structure may influence whether the advisor gains from debiasing a client who is tempted to shift the portfolio to safety during an equity market decline.

The Black–Litterman (B–L) Asset Allocation Model

Overview

Combining information from two sources to create an estimate of expected returns:

1. Source 1: What does the current market tell us about the expected excess returns? Implied excess equilibrium returns?
2. Source 2: What views does the investment manager have about particular stocks, sectors, asset classes, or country?

The B–L model combines these different sources to produce estimates of excess returns.

Combining predicted and implied returns

Basic Notation

1. τ = A scalar (assume τ = 1).
2. S = Variance–Covariance matrix for all assets under consideration
3. Ω = Uncertainty surrounding your views.
4. Π = Implied excess returns
5. Q = Views on expected excess returns for some or all assets
6. P = Link matrix identifying which assets you have views about

Understanding the Formula Splitting into two sections for better understanding

Consider the second section first:

We are combining implied excess returns with our own views on excess returns: A weighted average

What are the weights?

The investor about his/her views relative to the implied excess returns.

-How confident?

Ω will be large ⇒ Ω⁻¹ will be small

Understanding the formula:

The first part of this formula:

• τ implies excess returns and our views add up to 1.
• Namely, the formula is just a weighted average!

Consider a worked example:

Consider 1 stock: AMZN (Amazon.com)

• The implied excess returns are 0.74% per month and the variance is 2.018%
• We predict excess returns of 2% per month. The uncertainty surrounding this view is reflected by a variance of 0.50%
  
• Assume τ = 1, P = 1

Substituting into the B–L formula:

Does this estimate make sense?

Since we are confident about our views, now, try a more complicated example:

Incorporating Investor Views:

1. Analysts tell you that AEP has found a way to store electricity. Based on this breakthrough, they expect AEP to outperform XOM by 1% per month.

   Stocka	Historical excess returns	CAPM excess returns	Implied excess returns
   INTCa	0.0035	0.0135	0.0092
   AEP	0.0026	0.0064	0.0032
   AMZN	0.0326	0.0111	0.0074
   MRK	0.0002	0.0057	0.0059
   XOM	0.0111	0.0052	0.0064
   a). Intel, American Electric Power Company, Inc., Amazon, Merck, Exxon Mobil

2. Given the current economic conditions, we think that INTC will outperform AMZN by 1.75% per month.
   • Relative views are common in reality.
   • Absolute views, such as AEP having returns of 2% per month, are much less common.

Views versus Implied Excess Returns:

1. View (1): AEP outperforms XOM by 1% per month
   • The difference in implied excess returns is −0.32% per month.
   • One would expect that incorporating these views would lead to an increase in one's holdings of AEP and a decrease in XOM:0.0092.
2. View (2): INTC will outperform AMZN by 1.75% per month.
   • The difference in implied excess returns is 0.18% per month.
   • We would expect that incorporating our view would increase our holdings of INTC and reduce our holdings of AMZN.

The views expressed above are relative views of assets.

Incorporating Our Views:

To link our views to implied excess returns, we need a link matrix, P.

Matrix P is constructed in the following way:

Each row represents a view, each column represents a company

The View Vector and Uncertainty

What do Q and Ω look like?

where

Calculating Ω

There is no best way to calculate Ω. It will depend on how confident you are of your predictions.

Black and Litterman recommend

We have assumed that τ = 1, and so we can ignore it!

The well-known classical Black–Litterman models for asset allocation may be represented by the following two schematic illustrations:

Figures 5.4a and 5.5a.