Risk Return & The Capital Asset Pricing Model (CAPM)To make good (i.e., value-maximizing) financial decisions, one must understands the relationship between risk and return

We accept the notion that investors like returns and dislike risk

Consider the following proxies for return and risk: Expected return - weighted average of the distribution of possible returns in the future.Variance of returns - a measure of the dispersion of the distribution of possible returns in the future.Chapter 10

Jacoby, Stangeland and Wajeeh, 2000

Expected (Ex Ante) ReturnAn ExampleConsider the following return figures for the following year on stock XYZ under three alternative states of the economy Pk Rk Probability Return in State of Economyof state k state k +1% change in GNP0.25-5%+2% change in GNP0.5015%+3% change in GNP0.2535% 1.00where, Rk = the return in state k (there are S states) Pk = the probability of return k (state k)

Q. Calculate the expected return on stock XYZ for the next year

A.

Expected Returns - An ExampleOr, use the formula:Use the following table Pi Ri Pi Ri Probability Return in State of Economyof state i state i State 1: +1% change in GNP0.25-5%State 2: +2% change in GNP0.5015%State 3: +3% change in GNP0.2535% Expected Return =

Variance and Standard Deviation of ReturnsAn ExampleRecall the return figures for the following year on stock XYZ under three alternative states of the economy Pk Rk Probability Return in State of Economyof state k state k State 1: +1% change in GNP0.25-5%State 2: +2% change in GNP0.5015%State 3: +3% change in GNP0.2535% Expected Return = 15.00%where, Rk = the return in state k (there are S states) Pk = the probability of return k (state k) and s = the standard deviation of the return:

Q. Calculate the variance of and standard deviation of returns on stock XYZA.

Variance & Standard Deviation - An ExampleOr, use the formula:=Standard deviation:Use the following table State of Economy Pk X (Rk - E[R])2 = Pk(Rk - E[R])2 +1% change in GNP0.250.04 +2% change in GNP0.500.00 +3% change in GNP0.250.04 Variance of Return =0.02

Q. Calculate the expected return on assets A and B for the next year, given the following distribution of returns:

A. Expected returns E(RA) = (0.40%0.30) + (0.60%(-0.10)) = 0.06 = 6% E(RB) = (0.40%(-0.05)) + (0.60%0.25) = 0.13 = 13%

State of theProbabilityReturn onReturn on economyof stateasset Aasset BBoom0.4030%-5%Bust0.60-10%25% Portfolio Return and Risk

Jacoby, Stangeland and Wajeeh, 2000

Q. Calculate the variance of the above assets A and B

A. Variances Var(RA) = 0.40%(0.30 - 0.06)2 + 0.60%(-0.10 - 0.06)2 = 0.0384 Var(RB) = 0.40%(-0.05 - 0.13)2 + 0.60%(0.25 - 0.13)2 = 0.0216

Q. Calculate the standard deviations of the above assets A and B

A. Standard DeviationssA = (0.0384)1/2 = 0.196 = 19.6% sB = (0.0216)1/2 = 0.147 = 14.7%

Jacoby, Stangeland and Wajeeh, 2000

Expected Return on a PortfolioThe Expected Return on Portfolio p with N securities

where,E[Ri]= expected return of security i Xi = proportion of portfolio's initial value invested in security i

Example - Consider a portfolio p with 2 assets: 50% invested in asset A and 50% invested in asset B. The Portfolio expected return is given by:

E(RP) = XAE(RA) + XBE(RB) =

Returns and Risk for Portfolios - 2 Assets

Variance of a PortfolioThe variance of portfolio p with two assets (A and B)

where,

Standard Deviation of a PortfolioThe standard deviation of portfolio p with two assets (A and B)

Jacoby, Stangeland and Wajeeh, 2000

Q. Calculate the variance of portfolio p (50% in A and 50% in B)

A. Recall: Var(RA) = 0.0384, and Var(RB) = 0.0216First, we need to calculate the covariance b/w A and B:

The variance of portfolio p

Q. Calculate the standard deviations of portfolio pA. Standard Deviationssp = (0.0006)1/2 = 0.0245 = 2.45%

Note:E(RP) = XAE(RA) + XBE(RB) = 9.5%, but

Var(Rp) =0.0006 < XAVar(RA) + XBVar(RB) = (0.50% 0.0384) + (0.50%0.0216) = 0.03This means that by combining assets A and B into portfolio p, we eliminate some risk (mainly due to the covariance term)

Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments

Two types of Risk:Unsystematic/unique/asset-specific risks - can be diversified awaySystematic or market risks - cant be diversified away

In general, a well diversified portfolio can be created by randomly combining 25 risky securities into a portfolio (with little (no) cost).The Effect of Diversification on Portfolio Risk

Portfolio DiversificationAverage annual standard deviation (%)Number of stocks in portfolioDiversifiable (nonsystematic) riskNondiversifiable (systematic) risk49.223.919.21102030401000

Jacoby, Stangeland and Wajeeh, 2000

Beta and Unique RiskTotal risk = diversifiable risk + market risk

We assume that diversification is costless, thus diversifiable (nonsystematic) risk is irrelevant

Investors should only care about nondiversifiable (systematic) market risk

Market risk is measured by beta - the sensitivity to market changes

Example: Return (%)State of the economy TSE300 BCEGood 18 26 Poor 6 -4

Jacoby, Stangeland and Wajeeh, 2000

Interpretation: Following a change of +1% (-1%) in the market return, the return on BCE will change by +2.5% (-2.5%)NOTE: If the security has a -ve cov w/ TSE 300 => Beta and Market Risk (6%, -4%) (18%, 26%) The Characteristic Line

Beta and Unique RiskMarket Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the TSE300, is used to represent the market

Beta (b )- Sensitivity of a stocks return to the return on the market portfolio

Covariance of security is return with the market returnVariance of market return

Jacoby, Stangeland and Wajeeh, 2000

Markowitz Portfolio TheoryWe saw that combining stocks into portfolios can reduce standard deviation

Covariance, or the correlation coefficient, make this possible:The standard deviation of portfolio p (with XA in A and XB in B):

Note: , orThus,

Jacoby, Stangeland and Wajeeh, 2000

Markowitz Portfolio Theory - An ExampleConsider assets Y and Z, with

Consider portfolio p consisting of both Y and Z. Then, we have:Expected Return of p

Standard Deviation of p

Jacoby, Stangeland and Wajeeh, 2000

Look at the next 3 cases (for the correlation coefficient):

Where

Expected Return of Portfolio

Standard deviation

of a portfolio

Portfolio

YZ = -1

YZ = +1

YZ = 0

1

18.6%

7.38%

7.98%

7.69%

2

17.2

4.52

5.72

5.16

3

15.8

1.66

3.46

2.72

Portfolio

1

2

3

0.75

0.50

0.25

0.25

0.50

0.75

_915964289.unknown

_1013292273.unknown

_1013292287.unknown

_981877052.unknown

_915964288.unknown

. The Shape of the Markowitz Frontier - An ExampleZ. Yr = -1r = 0r = +1

Jacoby, Stangeland and Wajeeh, 2000

Sheet1

Y0.0105-0.0012296341

Z-0.00122963410.000144

(1/N)^2N

BNS110.0105

BCT+BNS0.2520.00027613720.0166173769

BCT+BNS+CM11%30.0009094146

BNS146187145BNS1.461.871.45

BCT187854104BCT1.878.541.04

CM145104289CM1.451.042.89

0.23250.054056250.23250.05405625

0.4070.1656490.4070.165649

36%0.12996025277.1308562536%0.129960252.7713085625

Chart1

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