Models of Risk

2003,3,24

What is risk?

Risk: to expose to hazard or danger Risk = riscare (dare to do something) Risk = 危機 (danger and opportunity)

Goal of finance (investment)

To ensure that when an investor is exposed to risk, there will be an “appropriate” reward for taking the risk.

Characteristics of risk and reward models

Provide a measure of risk that applies to all assets and not be asset-specific

Delineate those type of risk that are rewarded and those that are not, and should give a rationale for the delineation

Based on standardized risk measures ( to conclude the asset involve above-average or below-average risk)

Translate the measure of risk into a rate of return that the investor can expect or demand as compensation for bearing the risk

It should work well not only as an explanation of past returns, but also as a prediction of future expected returns.

How risk is viewed in finance

Defined the risk in terms of the distribution of actual returns around an expected return

differentiate between risk that is specific to one investment or a few investments, and risk that affects a much wider cross section of investment.

Measure risk

Riskless investment: default-free bond, the actual return is the expected return.

Risk investment: stock, the actual return will almost certainly NOT be equal to the expected return.

Symmetric?

The spread of actual returns around the expected return is captured by the variances or standard deviation of the distribution.

The bias toward positive or negative returns is captured by the skewness of the distribution.

The shape of the tails of the distribution is measured by the kurtosis of the distribution. (fatter tails lead to higher kurtosis)

Symmetric and normal: expected return and variance

Symmetric : same standard deviation, chose higher expected return

Neither symmetric nor normal : expected return, variance, positive skewness, and then lower kurtosis (lower likelihood of jumps) than higher kurtosis (higher likelihood of jumps).

Investors tend to trade off good moment (higher expected returns and more positive skewnwess) than bad moment (higher variance and higher kurtosis)

Capital asset pricing model (CAPM) explicitly requires that choices be made only in terms of expected returns and variances.

Utility function =?

Disney (Jan 1992 – Dec 1996)

SD in monthly returns = 6.14% Variance in monthly returns = 37.66% (?) annualized SD = (?) annualized variance =

6.14 12 21.26 37.66 12 452

Rewarded and Unrewarded Risk

Firm-specific risk Market risk

The components of Risk

Project-specific risk competitive risk Industry-specific risk: technology risk, legal

risk, commodity risk International risk: currency Market risk: macroeconomic factors

Diversification

Reduces or eliminates firm-specific risk

diversification

Why marginal investor is assumed to be diversified?

Risk increase if undiversified. Diversified investors are willing to pay higher

price to reduce risk. The asset will, over time, will end up being

held by diversified investors.

Measuring Market Risk

Risk comes from the distribution of actual returns around expected return.

Risk should be measured from the perspective of marginal investor who is well diversified. (spot price)

Methods for measuring risk (Asset Pricing)

Capital asset pricing theorem (CAPM) Arbitrage pricing theory (APT) Multi-factor models Regression models

CAPM

Assumptions: diversification is attractive Real world: tractions cost and monitoring

cost, reduced risk is smaller when the number of stock is larger. We believe we can find under-valued stock.

Market portfolio: A portfolio that contains all traded assets in the market.

CAPM

Implications for investors: If all investors hold the identical market portfolio, how do investors r? Answer: Asset Allocation on riskless and risky asset.

Measuring the market risk of an individual Asset: Beta of an asset i = Covariance of asset I with

market portfolio/variance of portfolio Beta of market portfolio=1

Beta

Expected return

( ) [ ( ) ]i f i m fE R R E R R

The CAPM in Practice (three inputs)

Riskless asset: may dependent on time period

risk premium: difference of expected return (historical return) of risky asset and riskless return

Beta: Covariance of risky asset and market portfolio divided by the variance of market portfolio (estimated by historical data)

The Arbitrage Pricing Model (APM)

Assumption: If there are two portfolio has the same exposure of risk but offer different expected returns, the investor will buy the portfolio with higher return. This action will adjust the expected returns to equilibrium.

Risk=Market +Firm-specific

m is the marketwise component of unanticipated risk

is the firm-specific component Return model:

( )r E R m

Market Risk

In CAPM, the market risk is capture by market portfolio.

In APM, the market risk is coming from economical fundamentals, gross national product, interest rates, and inflation, and it measures the sensitivity of investments to these changes with factor betas.

Market Risk

Where the sensitivity of the investment to unanticipated changes in factor j

unanticipated changes in factor j

1 1 2 2 n nm F F F

j

jF

The Effects of Diversification

The returns of portfolio will not have firm-specific component of unanticipated returns.

1 1 1 1,1 1, 1

1 ,1 ,

( ) ( )

( )

p n n n n

n n n n n

R w R w R w w F

w w F

Expected Returns and Betas

The beta of a portfolio is the weighted average of the betas of the assets in the portfolio, in conjunction with absence of arbitrage, leads to the conclusion that expected returns should be linearly related to betas.

Expected returns and Betas

Portfolio Expected return Beta

A 20% 2.0

B 12% 1.0

C 14% 1.5

can be viewed as risk premium of factor j

1 1( ) [ ( ) ]

[ ( ) ]

f f

n n f

E R R E R R

E R R

( )j fE R R

APM in practice

Estimates of each of the factors betas and factor risk premium.