Actuarial Science Meets Financial Economics

Buhlmann’s classifications of actuaries

Actuaries of the first kind - Life

Deterministic calculations

Actuaries of the second kind - Casualty

Probabilistic methods

Actuaries of the third kind - Financial

Stochastic processes

Similarities

Both Actuaries and Financial Economists:

Are mathematically inclined

Address monetary issues

Incorporate risk into calculations

Use specialized languages

Different Approaches

Risk

Interest Rates

Profitability

Valuation

Risk

Insurance

Pure risk - Loss/No loss situations

Law of large numbers

Finance

Speculative risk - Includes chance of gain

Portfolio risk

Portfolio Risk

Concept introduced by Markowitz in 1952

Var (Rp) = (σ2/n)[1+(n-1)ρ]

Rp = Expected outcome for the portfolio

σ = Standard deviation of individual outcomes

n = Number of individual elements in portfolio

ρ = correlation coefficient between any two

elements

Portfolio Risk

Diversifiable risk

Uncorrelated with other securities

Cancels out in a portfolio

Systematic risk

Risk that cannot be eliminated by diversification

Interest Rates

Insurance

One dimensional value

Constant

Conservative

Finance

Multiple dimensions

Market versus historical

Stochastic

Interest Rate Dimensions

Ex ante versus ex post

Real versus nominal

Yield curve

Risk premium

Yield Curves

0

2

4

6

8

10

12

1 5 10 20

Years to Maturity

Percent

UpwardSlopingInverted

Profitability

Insurance

Profit margin on sales

Worse yet - underwriting profit margin that ignores investment income

Finance

Rate of return on investment

Valuation

Insurance

Statutory value

Amortized values for bonds

Ignores time value of money on loss reserves

Finance

Market value

Difficulty in valuing non-traded items

Current State of Financial Economics

Valuation

Valuation models

Efficient market hypothesis

Anomalies in rates of return

Asset Pricing Models

Capital Asset Pricing Model (CAPM)

E(Ri) = Rf + βi[E(Rm)-Rf]

Ri= Return on a specific security

Rf = Risk free rate

Rm = Return on the market portfolio

βi = Systematic risk

= Cov (Ri,Rm)/σm2

Empirical Tests of the CAPM

Initially tended to support the model

Anomalies

Seasonal factors - January effect

Size factors

Economic factors

Systematic risk varies over time

Recent tests refute CAPM

Fama-French - 1992

Arbitrage Pricing Model (APM)

Rf’ = Zero systematic risk rate

bi,j = Sensitivity factor

λ = Excess return for factor j

E R R bi f i j j

j

n

( ) ' ,

1

Empirical Tests of APM

Tend to support the model

Number of factors is unclear

Predetermined factors approach

Based on selecting the correct factors

Factor analysis

Mathematical process selects the factors

Not clear what the factors mean

Option Pricing Model

An option is the right, but not the obligation, to buy or sell a security in the future at a predetermined price

Call option gives the holder the right to buy

Put option gives the holder the right to sell

Black-Scholes Option Pricing Model

Pc = Price of a call option

Ps = Current price of the asset

X = Exercise price

r = Risk free interest rate

t = Time to expiration of the option

σ = Standard deviation of returns

N = Normal distribution function

P P N d Xe rt N dc s ( ) ( )1 2

d P X r t t

d d t

s12 1 2

2 11 2

2

[ln( / ) ( / ) ] / /

/

Diffusion ProcessesContinuous time stochastic process

Brownian motion

Normal

Lognormal

Drift

Jump

Markov process

Stochastic process with only the current value of variable relevant for future values

Hedging

Portfolio insurance attempted to eliminate downside investment risk - generally failed

Asset-liability matching

Duration

D = -(dPV(C)/dr)/PV(C)

d = partial derivative operator

PV(C) = present value of stream of cash flows

r = current interest rate

Duration Measures

Macauley duration and modified duration

Assume cash flows invariant to interest rate changes

Effective duration

Considers the effect of cash flow changes as interest rates change

Applications of Financial Economics to Insurance

Pensions

Valuing PBGC insurance

Life insurance

Equity linked benefits

Property-liability insurance

CAPM to determine allowable UPM

Discounted cash flow models

Conclusion

Need for actuaries of the third kind

Financial guarantees

Investment portfolio management

Dynamic financial analysis (DFA)

Financial risk management

Improved parameter estimation

Incorporate insurance terminology