An Introduction to the Market Price of Interest Rate Risk

Kevin C. Ahlgrim, ASA, MAAA, PhD

Illinois State University

Actuarial Science & Financial Mathematics Seminar

University of Illinois

April 29, 2004

Background

• Modern financial pricing theory uses more and more risk neutral valuation

• Actuarial training historically has not included stochastic processes

• Many financial mathematics texts are inaccessible

• Purpose here: provide some intuition between risk-neutral world and “real” world

Overview

• Some definitions

• Simpler case: Market price of risk for stocks

• Extension to interest rate processes

Risk Neutrality vs. Risk Aversion

• How do investors choose/value securities?

• Risk neutral world– Based on expected values of future payoffs

• Risk averse world– If expected returns are the same, investors choose

securities with less risk– Leads to risk-return tradeoff

Illustration of Risk Aversion

• One year, risk-free rate of interest is 10%

23.64

30

10

Risk Neutral Risk Averse

80%

20%

20

30

10

80%

20%

• Higher risk aversion, lower price

Use of Risk Neutral Paradigm:Pricing Contingent Claims

• Contingent claims are securities whose value is dependent upon outcome of other variables– Bonds depend on interest rates (current and future)– Stock options depend on value of underlying stock

• In general, we would need level of risk aversion of investors to price securities

• Pricing tool – arbitrage

Arbitrage

• “Equivalent” securities (or portfolios of securities) should sell at equivalent prices

• If not, arbitrage attempts to profit from misaligned market prices

• Arbitrage opportunities lead to “law of one price”

• We will use concept of arbitrage for contingent claims pricing

Arbitrage Strategy for Contingent Claims on Stocks

• One-period, risk-free interest rate = 10%

V

15

0

Contingent Claim

20

30

10

80%

20%

Stock

• Set up a portfolio: Buy 0.75 shares of stock, sell one contingent claim

• Strategy has riskless payoff of 7.50 in one period• But risk-free security returns 10%• Arbitrage forces returns to be the same

50.70)75.0(10:

50.715)75.0(30:

DOWN

UP

18.8

50.710.1)75.0(20

V

V

• Probabilities of stock movements were not used in valuing the contingent claim

• But we can imply probabilities from pricing the contingent claim

• Pretend the world is risk-neutral – only probabilities determine the price

8.18

15

0

%60

%10)1%)(100(%33.83

p

ppp

1 - p

•Risk neutral probabilities can now be used to value ANY contingent claim on the stock

Determining Risk-Neutral Future Stock Price

• Expected future stock price under risk-neutral probabilities

20

30

10

60%

40%

22)1(1030 pp

• Return on stock in risk-neutral world is also 10%

Market Price of Risk• Market price of risk () is extra compensation (per

unit of risk) for taking on risk• Stock and claim have same source of risk, but the

risk exposure is higher for the contingent claim• “Real world” risk/return statistics:

Security Exp. Return Risk ()

Stock 30% 40%

Contingent

Claim

46.67% 73.33%

Market Price of Risk (p.2)

frrE

)(

• Market price of risk () is extra compensation per unit of risk for taking on this stock’s risk

• Since both securities have same source of risk, the market price of risk is the same

5.0%33.73

%10%67.46

%40

%10%30

Conclusions from Stock Example

• Risk neutral valuation useful for contingent pricing• Real world returns (we observe) include market price

of risk• Shifting to risk-neutral world eliminates extra return

for accepting risk (usually lowers return)• All securities’ returns are identical in the risk-neutral

world– Equal to the risk-free rate

• All securities dependent on the same underlying return earn the same risk premium, per unit of risk

Applications to Interest Rates

893.012.1

1

10%

12%

8%

“Real World” One-Year Interest Rates

0.820

Risk-Free Bond

1

1

1

926.008.1

1

Returns on Bonds:Real World

• One year risk-free rate is 10%• Like stocks, investors demand “extra” return for

holding risky bonds• Expected one-year return on bond:

%90.10%92.125.0%88.85.0820.0

820.0926.05.0

820.0

820.0893.05.0

Market Price of Interest Rate Risk

• “Amount” or form of interest rate risk is less clear– Measure of risk () is not as clear

• Extra compensation is market price of risk– Constant per year? (=0.90%)– Related to level of interest rates? (r = 0.90% or

≈0.09)

Risk-neutral interest rate process

• Adjust real interest rate process to account for extra return required by investors– Include the market price of risk

• From our “real world” tree, we need to add 0.90% to future rates

10%

12%+0.90%=12.9%

8%+0.90%=8.9%

“Risk Neutral” One-Year Interest Rates

Returns on Bonds:Risk-Neutral World

• Expected one-year return on bond:

%10%125.0%85.0820.0

820.0918.05.0

820.0

820.0886.05.0

886.0129.1

1

0.820

1

1

1

918.0089.1

1

Impact on Interest Rate Models

• Most interest rate processes are based on risk-neutral interest rate dynamics

• Drift term already includes adjustment for market price of interest rate risk– Subtract out assumed form to get real world interest

rate dynamics

tttt dBtrdttradr ),(),(

Conclusion

• Risk-neutral valuation is tool for valuing contingent claims

• Adjust the underlying process (stock price movements or interest rates) so that level of risk aversion is not needed for valuation