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 Metrics

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 (R p ) = ( σ 2 /n)[1+(n-1) ρ ] R p = 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

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(R i ) = R f + β i [E(R m )-R f ] R i = Return on a specific security R f = Risk free rate R m = Return on the market portfolio β i = Systematic risk = Cov (R i,R m )/ σ m 2

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

Arbitrage Pricing Model (APM) R f ’= Zero systematic risk rate b i,j = Sensitivity factor λ = Excess return for factor j

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 P c = Price of a call option P s = 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

Diffusion Processes Continuous 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

Risk Metrics Interest rate sensitivity –Duration Insurance –Dynamic Financial Analysis (DFA) Finance –Risk profiles –Value at Risk (VaR)

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

Risk Profile w Graphical summary of relationship between two variables w Example: As interest rates increase, S&L value decreases

Risk Profile (Cont.) w NOTE: For S&Ls, this risk profile is apparent from the balance sheet The balance sheet lists long-term vs. short-term assets and liabilities w Economic exposures require more work Example: Construction company will be affected by higher interest rates w Enter correlation analysis

Value at Risk - A Definition Value at risk is a statistical measure of possible portfolio losses –A percentile of the distribution of outcomes Value at Risk (VaR) is the amount of loss that a portfolio will experience over a set period of time with a specified probability Thus, VaR depends on some time horizon and a desired level of confidence

Value at Risk - An Example Let’s use a 5% probability and a one- day holding period VaR is the one day loss that will be exceeded only 5% of the time It’s the tail of the return distribution In the example, the VaR is about $60,000

First - Identify the Market Factors There are three methods to calculate VaR, but the first step is to identify the “market factors” Market factors are the variables that impact the value of the portfolio –Stock prices, exchange rates, interest rates, etc. The different approaches to VaR are based on how the market factors are modeled

Methods of Calculating VaR Historical simulation –Apply recent experience to current portfolio Variance-covariance method –Assume a normal distribution and use the statistical properties to find VaR Monte Carlo Simulation –Generate scenarios to determine changes in portfolio value

Historical Simulation Historical simulation is relatively easy to do –Only requires knowing the market factors and having the historical information Correlations between the market factors are implicit in this method Assumes future will resemble the past

Variance-Covariance Method Assume all market factors follow a multivariate normal distribution The distribution of portfolio gains/losses can then be determined with statistical properties From this distribution, choose the required percentile to find VaR Conceptually more difficult given the need for multivariate analysis Explaining the method to management may be difficult

Monte Carlo Simulation Specify the individual distributions of the future values of the market factors Generate random samples from the assumed distributions Determine the final value of the portfolio Rank the portfolio values and find the appropriate percentile to find VaR Initial setup is costly, but thereafter simulation can be efficient DFA is an example of this approach

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