Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity 1 1

Review... For the first part of the course, we have discussed: The need for financial risk management How to value fixed cash flows Basic derivative securities Credit derivatives We will now discuss techniques used to evaluate asset and liability risk 2 2

Today An introduction to the asset/liability management (ALM) process What is the goal of ALM? The concepts of duration and convexity Extremely important for insurance enterprises 3 3

Asset/Liability Management As its name suggests, ALM involves the process of analyzing the interaction of assets and liabilities In its broadest meaning, ALM refers to the process of maximizing risk-adjusted return Risk refers to the variance (or standard deviation) of earnings More risk in the surplus position (assets minus liabilities) requires extra capital for protection of policyholders 4 4

The ALM Process Firms forecast earnings and surplus based on “best estimate” or “most probable” assumptions with respect to: Sales or market share The future level of interest rates or the business activity Lapse rates Loss development ALM tests the sensitivity of results for changes in these variables 6 5

ALM of Insurers For insurance enterprises, ALM has come to mean equating the interest rate sensitivity of assets and liabilities As interest rates change, the surplus of the insurer is unaffected ALM can incorporate more risk types than interest rate risk (e.g., business, liquidity, credit, catastrophes, etc.) We will start with the insurers’ view of ALM 5 6

The Goal of ALM If the liabilities of the insurer are fixed, investing in zero coupon bonds with payoffs identical to the liabilities will have no risk This is called cash flow matching Liabilities of insurance enterprises are not fixed Policyholders can withdraw cash Hurricane frequency and severity cannot be predicted Payments to pension beneficiaries are affected by death, retirement rates, withdrawal Loss development patterns change 7 7

The Goal of ALM (p.2) If assets can be purchased to replicate the liabilities in every potential future state of nature, there would be no risk The goal of ALM is to analyze how assets and liabilities move to changes in interest rates and other variables We will need tools to quantify the risk in the assets AND liabilities 8 8

Price/Yield Relationship Recall that bond prices move inversely with interest rates As interest rates increase, present value of fixed cash flows decrease For option-free bonds, this curve is not linear but convex 9 9

Simplifications Fixed income, non-callable bonds Flat yield curve Parallel shifts in the yield curve

Examining Interest Rate Sensitivity Start with two $1000 face value zero coupon bonds One 5 year bond and one 10 year bond Assume current interest rates are 8%

Price Changes on Two Zero Coupon Bonds Initial Interest Rate = 8%

Price Volatility Characteristics of Option-Free Bonds Properties 1 All prices move in opposite direction of change in yield, but the change differs by bond 2+3 The percentage price change is not the same for increases and decreases in yields 4 Percentage price increases are greater than decreases for a given change in basis points Characteristics 1 For a given term to maturity and initial yield, the lower the coupon rate the greater the price volatility 2 For a given coupon rate and intitial yield, the longer the term to maturity, the greater the price volatility

Macaulay Duration Developed in 1938 to measure price sensitivity of bonds to interest rate changes Macaulay used the weighted average term-to-maturity as a measure of interest sensitivity As we will see, this is related to interest rate sensitivity 10 14

Macaulay Duration (p.2) 11 15

Applying Macaulay Duration For a zero coupon bond, the Macaulay duration is equal to maturity For coupon bonds, the duration is less than its maturity For two bonds with the same maturity, the bond with the lower coupon has higher duration 12 16

Modified Duration Another measure of price sensitivity is determined by the slope of the price/yield curve When we divide the slope by the current price, we get a duration measure called modified duration The formula for the predicted price change of a bond using Macaulay duration is based on the first derivative of price with respect to yield (or interest rate) 13 17

Modified Duration and Macaulay Duration i = yield CF = Cash flow P = price 14 18

An Example Calculate: What is the modified duration of a 3-year, 3% bond if interest rates are 5%? 15 19

Solution to Example 16 20

Example Continued What is the predicted price change of the 3 year, 3% coupon bond if interest rates increase to 6%? 17 21

Example Continued What is the predicted price change of the 3 year, 3% coupon bond if interest rates increase to 6%? 17 22

Other Interest Rate Sensitivity Measures Instead of expressing duration in percentage terms, dollar duration gives the absolute dollar change in bond value Multiply the modified duration by the yield change and the initial price Present Value of a Basis Point (PVBP) is the dollar duration of a bond for a one basis point movement in the interest rate This is also known as the dollar value of an 01 (DV01) 23

A Different Methodology The “Valuation…” book does not use the formulae shown here Instead, duration can be computed numerically Calculate the price change given an increase in interest rates of ∆i Numerically calculate the derivative using actual bond prices: 24

A Different Methodology (p.2) Can improve the results of the numerical procedure by repeating the calculation using an interest rate change of -∆i Duration then becomes an average of the two calculations 25

Error in Price Predictions The estimate of the change in bond price is at one point The estimate is linear Because the price/ yield curve is convex, it lies above the tangent line Our estimate of price is always understated 26

Convexity Our estimate of percentage changes in price is a first order approximation If the change in interest rates is very large, our price estimate has a larger error Duration is only accurate for small changes in interest rates Convexity provides a second order approximation of the bond’s sensitivity to changes in the interest rate Captures the curvature in the price/yield curve 27

Computing Convexity Take the second derivative of price with respect to the interest rate 28

Example What is the convexity of the 3-year, 3% bond with the current yield at 5%? 29

Predicting Price with Convexity By including convexity, we can improve our estimates for predicting price 30

An Example of Predictions Let’s see how close our estimates are 31

Notes about Convexity Again, the “Valuation…” textbook computes convexity numerically, not by formula Also, “Valuation…” defines convexity differently It includes the ½ term used in estimating the price change in the definition of convexity 32

Convexity is Good In our price/yield curve, we can see that as interest increases, prices fall As interest increases, the slope flattens out The rate of price depreciation decreases As interest decreases, the slope steepens The rate of price appreciation increases For a bondholder, this convexity effect is desirable 33

Next Time Limitations to duration calculations Effective duration and convexity Other duration measures