expected returns and risk otto khatamov. 1. client needs: investment policy statement focus:...

Expected Returns and Risk

Otto Khatamov

Portfolio Management

Portfolio Management: Overview

1. Client needs: Investment policy statement• Focus: Investor’s short-term and long-term needs,

expectations

2. Portfolio manager: Examine current and projected financial, economic, political, and social conditions• Focus: Short-term and intermediate-term expected

conditions to use in constructing a specific portfolio

3. Portfolio manager: Implement the plan by constructing the portfolio• Focus: Meet the investor’s needs at minimum risk levels

4. Client/Portfolio manager: Feedback loop• Monitor and update investor needs, environmental

conditions, evaluate portfolio performance

“Which assets or portfolio of assets we should choose to invest?”

Macro expectations◦ Capital market expectations about asset classes

Micro expectations◦ Capital market expectations about individual

assets

Expected returns and Risk…


The risk premium concept Expected return: Risk free rate + Risk premium How do we measure risk?

Expected returns and Risk…

Risk denotes the probability distribution of possible economic outcomes.

◦ Risk is a mix of danger and opportunity◦ Under a mean variance world the distribution is fully characterised by the expected returns and the variance

How can we measure expected returns and variances?

◦ Individual assets risk (i.e. returns/variances) ◦ Portfolio of assets risk (ie. returns/variances)

What is risk?

Formulating the capital market expectations…

Scenario analysis◦ Concepts behind the quantification of risk and return◦ Evaluating investments by using utility score

How to get more realistic estimates of expected returns and risk?

◦ Historical returns approach◦ Problems and pitfalls

Lets consider the following investment:

◦ Is this expected return enough to justify the risk? Depends on the risk premium

Expected Returns and Variance for an asset…

Economic Conditions Probability (p) Rate of Return, r

(%)

Strong Economy (i = 1) 0.15 0.20

Weak Economy (i = 2) 0.15 -0.20

Stable Economy (i = 3) 0.70 0.10

014.0])07.010.0(70.0[])07.020.0(15.0[])07.020.0(15.0[))((

07.0)10.070.0()]20.0(15.0[)20.015.0()(

2223

1

22

3

1

iii

iiir

RErp

rpRE

r

Lets consider an alternative investment in T-bills as a benchmark:

◦ Is this risk premium enough to justify the risk? Utility theory…

Risk Premium for an asset…

Economic Conditions Probability (p) Rate of Return (r)

Strong Economy (i = 1) 0.15 0.20

Weak Economy (i = 2) 0.15 -0.20

Stable Economy (i = 3) 0.70 0.10

Treasury bill 1 0.05

0 05.0r :Investment freeRisk

014.0 07.0)E( :InvestmentRisky

2f

2

f

rrR 20.0r-)E( PremiumRisk f rR

Expected Returns and Variance for a portfolio of assets…

Building on the previous example◦ Allocate 50% of the investment in the risky asset and 50% in the risk free asset

Calculate the expected return and the variance of the portfolio

For N assets

N

i

N

jjijiji

N

iiip

ww

REwRE

p

1 1,

2

1

)()(

Expected Returns and Variance for a portfolio of assets…

Building on the previous example◦ Allocate 50% of the investment in the risky asset and 50% in the risk free asset

Calculate the expected return and the variance of the portfolio

For two assets

◦ Is this a “good” portfolio? How can I select the best among different portfolios?

Utility theory…

0035.000014.0)5.0(2

06.005.0*5.007.05.0)()(

2,21

222

221

2

21

frfrfr

frp

wwww

rwREwRE

p


Evaluating investments by using utility score…

Properties of Utility Functions (1):◦ More is preferred to less… U(x+1) > U(x)

Implicitly we assume that U’(W)>0


Properties of Utility Functions (2):◦ Define investors’ taste for risk…

£2

£0

-£1

p = 0.5

p = 0.5

gamble}fair a is {this E(w) Cost £100.520.5E(w)

Condition Definition

Risk averse Reject a fair gamble

Risk seeking Accept a fair gamble

Risk neutral Indifferent to a fair gamble


Properties of Utility Functions (2):◦ Define investor’s taste for risk…

Graphically

£2

£0

-£1

p = 0.5

p = 0.5

U(0)-U(1)U(1)-U(2)U(0)0.5U(2)0.5 U(1):neutralRisk

U(0)-U(1)U(1)-U(2)U(0)0.5U(2)0.5 U(1):loverRisk

U(0)-U(1)U(1)-U(2)U(0)0.5U(2)0.5 U(1):averseRisk


Properties of Utility Theory (2)◦ Define investor’s taste for risk…

i. Risk averse: U(2) - U(1) < U(1) - U(0) =>U’’(w) < 0ii. Risk lover: U(2) – U(1) > U(1) – U(0) => U’’(w) > 0iii. Risk neutral: U(2) – U(1) = U(1) – U(0) => U’’(w) = 0

◦ (iii) (ii) (i)


Properties of Utility Theory (3)◦ How the wealth invested in risky assets change with wealth?


£20000

> £5000 DARA

= £5000 CARA

< £5000 IARA

£10000 £5000 ???

Wealth Investment Investment Wealth

aversionrisk absolute Increasing 0(W)A If

aversionrisk absoluteConstant 0(W)A If

aversionrisk absolute Decreasing 0(W)A If

(W)U

(W)UA(W)

allyMathematic

Properties of Utility Theory (4)◦ How the percentage of wealth invested in risky assets change with wealth?


£20000

> 50% of Wealth DRRA

= 50% of Wealth CRRA

< 50% of Wealth IRRA

£10000 50% of Wealth ???

Wealth Investment Investment Wealth

aversionrisk relative Increasing 0(W)R If

aversionrisk relativeConstant 0(W)R If

aversionrisk relative Decreasing 0(W)R If

)(R(W)

allyMathematic

WAW

A reasonable Utility function used by the AIMR

Which investment is the best one? Concept of utility and asset/portfolio choice


0 05.0r :Investment freeRisk

014.0 07.0)E( :InvestmentRisky

005.0)(

2f

2

2

f

rrR

ArEU

Utility Score

Risk Aversion (A)

Risky Investment

Risk Free Investment

2 6.99% 5.00%

6 6.96% 5.00%

10 6.93% 5.00%


2005.0)( ArEU

More on the quadratic Utility function…

◦ Implicitly we assumed investors’ are risk averse i.e. reject a zero risk premium fair gamble (A > 0)

◦ A risk neutral investor judge risky projects solely by their expected returns i.e. indifferent to a fair gamble (A = 0)

◦ A risk lover investor adjust the expected returns upward to take into account the “fun” of risk i.e. accept a fair gamble (A < 0)

Historical Rates of Returns and Variance for an asset… (1)

We need both the beginning and the ending value of investment.

Measuring average returns over a period (t=1,…,T)

Measuring variance for historical returns

Value Beginning

Value) Beginning - Value (EndingR

1)]1)...(R(1[R :Mean Geometric )...(

R :Mean Arithmetic1

1G1

A

TT

T RT

RR

T

tARR

T1

22 )(1

Historical Rates of Returns and Variance for an asset: Example… (2)

562.0])25.05.0()25.01[(2

1)(

1

%01))]5.0(1()11[(1)]1)...(R(1[R :Mean Geometric

25%2

(-0.5)][1)...(R :Mean Arithmetic

5.0100

)10050(R 1

50

)50100(R

Value Beginning

Value) Beginning - Value (EndingR

22

1

22

2

11

1G

1A

21

T

tA

TT

T

RRT

R

T

RR

Year Beginning Value

Ending Value

Return, r (%)

1 50 100 1

2 100 50 -0.5

Historical Rates of Returns and Variance for a portfolio of assets…

For N assets

Average returns (arithmetic or geometric) Individual asset variances Correlation (or covariance’s) between assets

N

i

N

jjijiji

N

iiip

ww

RwR

p

1 1,

2

1

The Historical Record: Bills, Bonds and Stocks…

SeriesGeometric Average

Arithmetic Average

Standard Deviation

Small-Company Stocks

11.64% 17.74% 39.30%

Large-Company Stocks

10.01% 12.04% 20.55%

Long-Term Government Bonds

5.38% 5.68% 8.24%

US Treasury Bills 3.78% 3.82% 3.18%Source: BKM Chapter 5 – Sources: Returns on T-bills, large and small stocks – CRSP, T-bonds - RSP for 1926-1995 returns and Lehman Brothers long-term and intermediate indexes for 1996 and later returns.

◦ How bills, bonds and stocks are distributed?

What is the Distribution of Asset Returns…

Modern portfolio theory assumes that asset returns are (log)normally distributed◦ Attractive properties

Symmetric and is described by mean and variance A weighted average of variables that are normally

distributed also will normally distributed

Empirical evidence on asset returns◦ Even if individual asset returns are not exactly

normal, the distribution of returns of large portfolios will be close to normal

What is the Distribution of Asset Returns…

N=1 N=128

Statistic Observed Normal Observed Normal

Minimum -71.1 NA 16.4 NA

5th Percentile -14.4 -39.2 22.7 22.6

50th Percentile 19.6 28.2 28.1 28.2

95th Percentile 96.3 95.6 34.1 33.8

Maximum 442.6 NA 43.1 NA

Mean 28.2 28.2 28.2 28.2

Standard Deviation 41.0 41.0 3.4 3.4

Skewness 255.4 0.0 17.7 0.0

Source: L., Fisher and J., Lorie, 1970, Some Studies of Variability of Returns on Investments in Common Stocks, Journal of Business, 43.

◦ Large portfolio distribution is virtually identical to the hypothetical normally distributed portfolio.

Historical Record and Sample Estimators…

Assuming future returns over the selected time horizon reflect the same probability distribution of past returns then…◦ The sample arithmetic/geometric mean total

return may serve as an estimate of the expected return

◦ The sample variance may serve as an estimate of the variance

◦ The sample correlation (or covariance's) may serve as estimates of correlation (or covariance’s)

Historical Record and Adjustments to Sample Estimators…

Single/Multi-Index Models◦ Explain returns of assets as a function of an

index/indices◦ Estimates of covariance’s using assets’ factor

sensitivities R., Grinold and R., Kahn, 1995, Active Portfolio

Management, Chicago, IL: Probus Publication. R., Michaud, 1989, The Markowitz Optimization

Enigma: Is “Optimised” Optimal? Financial Analysts Journal.

Single-Index models: In practise…


Single-Index Models◦ Let assume the return on a stock can be written as:

◦ Where Rm is the rate of return on the market index (a random variable) and for each security i, α denotes the security’s excess return when the market excess return is zero, b is a constant that measures the expected change in the return of the security given a change in the returns of the market (i.e. systematic risk) and e represents firm-specific risk

imiii eRbaR



◦ Let assume that ei and Rm are uncorrelated.

Cov(ei,Rm) = 0

◦ A final assumption is that ei and ej are uncorrelated.

E(ei,ej) = 0

imiii eRbaR



◦ You may use regression to estimate α, b and e and ensure that the previous assumption about uncorrelated ei and Rm

is met Results

◦ The mean return, ◦ The variance of a security’s returns, ◦ The covariance of returns between securities i and j,

imiii eRbaR

miii RbaR

imiieb 2222

2, mjiji bb


Single-Index Models: Example… Let assume that we estimate α, b and σ2

e (i.e. regression of a stock’s return on the return of the S&P 500 index using five years of monthly returns) and that the expected return of the market is 12.5 and the standard deviation of the returns is 14.9% (in practise we may use the historical risk premium of the market over the period 1926-2008)

Stock αi bi σ2e,i

XYZ 6 1.4 65

ABC 4 0.8 20


Single-Index Models: Example…

◦ The mean return,

RXYZ = 6 + 1.4 * 12.5 = 23.5%

RABC = 4 + 0.8 * 12.5 = 14%

◦ The variance of a security’s returns,

σXYZ = SQRT(1.4 * 1.4 * 14.9 * 14.9 + 65) = 22.36

σABC = SQRT(0.8 * 0.8 * 14.9 * 14.9 + 20) = 12.73

◦ The covariance of returns between securities i and j,

σXYZ,ABC = 1.4 * 0.8 * 14.9 * 14.9 = 249

miii RbaR

imiieb 2222

2, mjiji bb


Shrinkage Estimators◦ Reduce the impact of extreme values in historical

estimates (expected returns, variances, covariance's) Nonrecurring peculiarities of historical record Weighted average of the historical covariance matrix

and an alternative estimator of covariance matrix (i.e. estimated betas) O., Ledoit and M., Wolf, 2003, Improved Estimation of the

Covariance Matrix of Stocks Returns with an Application to Portfolio Selection, Journal of Empirical Finance, 5, 603-621.


Time Series Estimators◦ Estimates of near-term volatility/covariance’s

utilizing the concept of volatility clustering◦ Periods of notably high or low volatility

Autoregressive conditional heteroskedasticity time-series models T., Bollerslev, R., Engle and D., Nelson, 1994, ARCH

Models, Handbook in Econometrics, 4, eds. Amsterdam: Elsevier.

Problems and pitfalls …

Data measurement errors and biases◦ Transcription errors

Data providers: Errors in gathering and recording data

◦ Survivorship bias Data series reflect entities survived to the end of the period

Delisting and share index return

◦ Appraisal data Illiquid assets

Less volatile Lower correlation Lower variance


Limitations of historical estimates◦ Does historical estimates predict future well?

Changing nature of the environment Technology Regulations Wars

Stationary versus Nonstationary data With stationary data use longer data series to increase the precision of estimates The longer the data series the greater the likelihood to capture changes of the environment


Limitations of historical estimates◦ Time-period bias

Starting and ending dates Small stocks outperformed big stocks on average 19.6 % per year during the period 1975-83

◦ Conditioning information

Expectations during bull and bear markets

• Bodie, Kane and Marcus, Chapter 7 (5th edition)

Readings

expected returns and risk otto khatamov. 1. client needs: investment policy statement focus:...

Documents

quantification of risk

individual assets risk

risk free rate risk

risk free asset

risk otto khatamov slide

individual assets slide

portfolio performance

risk premium concept