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Some Important Concepts Involving Expected Investment Returns

1. Investors perform two functions for capital markets:

- Commit Financial Capital

- Assume Risk

so,

E(R) = (Risk-Free Rate) + (Risk Premium)

2. The expected return (i.e., E(R)) of an investment has a number of alternative names: e.g., discount rate, cost of capital, cost of equity, yield to maturity. It can also be expressed as:

k = (Nominal RF) + (Risk Premium)

= [(Real RF) + E(Inflation)] + (Risk Premium)

where:

Risk Premium = f(business risk, liquidity risk, political risk, financial risk)

3. Investors can be compensated in two ways:

- Period Cash Flows

- Capital Gain

so,

E(R) = E(Cash Flow) + E(Capital Gain)

Measuring Expected Returns: Overview

Risk Premium

Rt = (1 + Rft) (1+ RPt) – 1

or

Rt = (1 + Inft) (1 + RRft) (1 + RPt) – 1

where Rt = return on asset class for year t,

Inft = inflation rate

Rft = risk free rate

RRft = real risk free rate

RPt = risk premium

RPt =

where RRt = real asset class return

1 + Rt

1 + RRt

- 1 =

- 1

1 + RRft

1 + Rft

Methods for Estimating the Equity Risk Premium

1. Historical Evidence

2. Fundamental Estimates

3. Economic Estimates

4. Surveys

Estimating the Equity Risk Premium

1. Historical Evidence: Representative Work

- Ibbotson Associates – US Markets (2004)
- Fidelity Investments - Global Markets (2004)
- Jorion and Goetzmann (Journal of Finance, 1999)
- Siegel (Financial Analysts Journal, 1992)
- Dimson, Marsh and Staunton (Business Strategy Review, 2000)

Historical Real Returns, 1954-2003: The Global Experience

Chile: Returns 1/54 – 6/03

Chile*: Returns 1/54 – 12/71; 1/76 – 6/03

Source: Global Financial Data

Estimating the Equity Risk Premium (cont.)

2. Fundamental Estimates: Representative Work

- Fama and French (University of Chicago, 2000)
- Ibbotson and Chen (Yale University, 2001)
- Claus and Thomas (Journal of Finance, 2001)
- Arnott and Bernstein (Financial Analysts Journal, 2002)

Fundamental Risk Premium Estimates: An Overview

- One potential problem with using historical averages to estimate future expected returns is that there is no way to control for the possibility that the past data sample you selected produced averages that are “abnormal” (i.e., too high or too low) in some way.
- Another problem we have seen is that historical average returns tend to be fairly unstable (i.e., they are extremely sensitive to the time period chosen in the analysis).
- Fundamental risk premium estimates attempt to objectively forecast the expected returns that would normally occur, given the fundamental relationships that tend to exist in the capital markets. In other words, fundamental forecasts attempt to link return expectations to the economic conditions likely to pertain in the market during the forecast interval.

Fama and French: The Equity Risk Premium

- Main Idea: Use dividend and earnings growth rates to measure the expected rate of capital gains for equity investments. This process creates two ways of then estimating real (i.e., inflation-adjusted) expected equity returns:
- E(R) = E(Div Yld) + E(Real Growth Rate of Dividends) = RD
- E(R) = E(Div Yld) + E(Real Growth Rate of Earnings) = RY
- Notice that the intuition behind this approach is simply that it is possible to compensated investors in two ways: cash flow and capital gain.
- Real Equity Risk Premium can then be estimated by subtracting short-term commercial paper yields from RD and RY, which leaves RXD and RXY, respectively.
- Main Result: Using data from the period 1951 to 2000 for the US market (i.e., S&P 500), they find that:
- RXD = 2.55%
- RXY = 4.32%
- Notice that both of these fundamental risk premium estimates are well below the average historical risk premium during the period (i.e., 7.43%), leading the authors that future expected returns to equity investments are unlikely to match the high levels of the recent past.

Claus and Thomas: Equity Risk Premia in US and International Markets

- Main Idea: Based on the notion that the fundamental value of an equity investment can be described by its book value plus the present value of future abnormal earnings.
- This valuation can be estimated by a modified version of the multi-stage growth model:

where the discount rate k (= rf + rp) is the equity expected return.

- Main Results: Using observed market data (e.g., p, bv) and analyst forecasts (e.g., g) for the other inputs over 1985-1998, the authors calculate the values of the equity risk premium (rp) that solve the model:
- US: 3.40%
- Japan: 0.21%
- UK: 2.81%
- France: 2.60%
- Canada: 2.23%

Arnott and Bernstein: What Risk Premium is “Normal”?

- Main Idea: The risk premium for stocks relative to bonds can be forecast as the difference between the expected real stock return and the expected real bond return
- The real return to stocks consists of three components:
- Dividend yield
- Growth rate in the real dividend
- Change in equity valuation level (e.g., change in market P/E)
- The real return to bonds consists of three components:
- Nominal yeld
- Inflation
- Change in yield times duration (i.e., reinvestment)
- Main Conclusions:
- Historical real stock returns and the excess return for stocks relative to bonds over the past century have extraordinarily high (due to rising valuation multiples) and unlikely to be repeated in the future. The fundamental expected risk premium estimate over this past period would have been 2.4%.
- Future expectations should be based on tractable fundamental relationships and indicate a real risk premium of near 0%.

Estimating the Equity Risk Premium (cont.)

3. Economic Estimates: Representative Work

- Black and Litterman (1992)
- Asset Class-Specific Risk Premia
- Ennis Knupp Associates (2005)

Implied Returns and the Black-Litterman Forecasting Process

- The Black-Litterman (BL) model uses a quantitative technique known as reverse optimization to determine the implied returns for a series of asset classes that comprise the investment universe.
- The main insight of the BL model is that if the global capital markets are in equilibrium, then the prevailing market capitalizations of these asset classes suggest the investment weights of an efficient portfolio with the highest Sharpe Ratio (i.e., risk premium per unit of risk) possible.
- These investment weights can then be used, along with information about asset class standard deviations and correlations, to transform the user’s forecast of the global risk premium into asset class-specific risk premia (and expected returns) that are consistent with a capital market that is in equilibrium.
- These equilibrium expected returns for the asset classes can then be used as inputs in a mean-variance portfolio optimization process or adjusted further given the user’s tactical views on asset class performance.

The Black-Litterman Process: An Example

- Consider an investable universe consisting of the following five asset classes:
- US Bonds
- Global Bonds-ex US
- US Equity
- Global Equity-ex US
- Emerging Market Equity
- As of September 2005, these asset classes had the following market capitalizations (in USD millions):
- US Bonds $ 8,607,149 (17.71%)
- Global Bonds-ex US 12,426,562 (25.57%)
- US Equity 13,776,249 (28.35%)
- Global Equity-ex US 12,266,988 (25.24%)
- Emerging Market Equity 1,521,275 ( 3.13%)

Total: $48,598,223

Black-Litterman Example (cont.)

- Consider also the following historical return standard deviations (October 2000 – September 2005):

susb = 4.02% sgb = 8.81% suss = 15.62%

sgs = 15.35% sems = 21.29%

- The historical correlation matrix, measured using all available pairwise historical return data:

rusb,gb = 0.41 rgb,gs = 0.21

rusb,uss = -0.14 rgb,ems = -0.00

rusb,gs = -0.16 russ,gs = 0.65

rusb,ems = -0.20 russ,ems = 0.69

rgb,uss = -0.02 rgs,ems = 0.73

Black-Litterman Example (cont.)

- The remaining inputs that the user must specify are: (i) the global risk premium of the investment universe, and (ii) the risk-free rate. Using current market data we have:
- Global Risk Premium: 3.55%(10-yr Global Balanced)
- Risk-Free Rate: 4.30%(10-yr US Treasury)
- The heart of the BL process is to then calculate the implied excess return for each asset class, using the following (stylized) formula:

[Risk Aversion Parameter] x [Covariance Matrix] x [Market Cap Weight Vector]

Black-Litterman Example (cont.)

- The risk aversion parameter is the rate at which more return is required as compensation for more risk. It is calculated as:

RAP = [Global Risk Premium] / [Market Portfolio Variance]

It can be shown in this example that the market portfolio variance is (8.57%)2 = 0.734%, so that:

RAP = (0.0355)/(0.0073) = 4.84

- The covariance between two asset classes (Y and Z) is given by the formula:

Cov(Y,Z) = ry,z x sy x sz

For instance, the covariance between US Equity and Global Equity-ex US is: (15.62%) x (15.35%) x (0.65) = 0.016

Black-Litterman Example (cont.)

- The implied excess return (IER) for US Equity can then be computed as follows:

IERuss = (RAP) x {[Cov(uss,usb) x wusb] + [Cov(uss,gb) x wgb] +

… + [Cov(uss,ems) x wems]}

= (4.84) x {(-0.001)(.1771) + … + (0.023)(.0313)} = 5.49%

- More formally, the solution for the entire asset class implied excess return vector is given by:

Black-Litterman Example (cont.)

- The total expected return for US Equity is then simply the IER plus the risk-free rate:

4.30% + 5.49% = 9.79%

- The excess and total expected returns for the other asset classes in this example are:

ExcessTotal

- US Bonds: 0.05% 4.35%
- Global Bonds: 1.39% 5.69%
- Global Equity: 5.64% 9.94%
- Emerging Equity: 6.59% 10.89%

Ennis Knupp Associates (EKA) – July 2005

- EKA uses a similar process to the BL methodology in that they develop asset class expected return forecasts that are grounded in the notion that the global capital markets are in equilibrium.
- Specifically, EKA estimates asset class expected returns to be consistent with a global Capital Asset Pricing Model (CAPM). Two expected return “anchors” are used as a starting point:
- US Equity = 9.1%: Total return is divided into three components: dividend yield (1.7%), nominal growth rate of corporate earnings (7.4%), and change in valuation levels (0%)
- US Bonds = 5.4%: Based on two components: current yield and simulated future changes in yields (based on forecasts of expected inflation, inflation risk premium, and real yields)

Other asset class expected returns are then estimated relative to these anchors using the global CAPM. Specifically, expected returns on the various asset classes are proportional to their systematic risk levels relative to the global market portfolio, shown at the right.

For example, the ratio of the US Bonds beta (0.40) to the US Equity beta (1.71) is 0.23. This implies that the ratio of US Bond risk premium to US Equity risk premium should also be 23%.

Therefore:

(5.4 – RF)/(9.1 – RF) = 0.23

which results in an implied risk-free rate of 4.3%. The risk premium of each asset class is then calculated so that it is directly proportional to its systematic risk, given these two anchors

EKA Fundamental Expected Return Estimates (cont.)Estimating the Equity Risk Premium (cont.)

4. Surveys: Representative Work

- Graham and Harvey (Duke University, 2005)
- UTIMCO (2005)
- Ennis Knupp: Managers & Consultants (2005)
- Burr (Pensions and Investments, 1998)
- Welch (Journal of Business, 2000)

Survey of Asset Class Return & Risk Expectations:UTIMCO Staff & External Expert Opinions – March 2005

March, 2005

20

Survey of Asset Class Return & Risk Expectations (cont.):UTIMCO Staff & External Expert Opinions – March 2005

March, 2005

21

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