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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

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

11. 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%

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

13. 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

14. 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

15. Macaulay Duration (p.2) 11 15

16. 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

17. 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

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

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

20. Solution to Example 16 20

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

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

23. 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

24. 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

25. 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

26. 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

28. 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

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

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

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

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

33. 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

34. 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

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