Financial Risk Management of
Insurance Enterprises
Introduction to Asset/Liability
Management, Duration & Convexity
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
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
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
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
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
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
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
Price/Yield Relationship
• Recall that bond prices
move inversely with
interest rates
• For option-free bonds,
this curve is not linear
but convex
Price
– As interest rates
increase, present value
of fixed cash flows
decrease
P rice/yield curve
Yield
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%
Principa l
1000
1000
1000
1000
1000
1000
1000
R
0.06
0.07
0.0799
0.08
0.0801
0.09
0.1
5 ye a r Cha nge
747.2582 9.7967%
712.9862 4.7611%
680.8984 0.0463%
680.5832 0.0000%
680.2682 -0.0463%
649.9314 -4.5038%
620.9213 -8.7663%
10 ye a r
Cha nge
558.3948 20.5532%
508.3493 9.7488%
463.6226 0.0926%
463.1935 0.0000%
462.7648 -0.0925%
422.4108 -8.8047%
385.5433 -16.7641%
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 termto-maturity as a measure of interest
sensitivity
• As we will see, this is related to interest rate
sensitivity
Macaulay Duration (p.2)
t  PVCFt
Macaulay Duration = 
t 1 k  PVTCFt
t  index for period
n
n = total number of period
k = number of coupon payments per year
PVCFt  Present value of cash flow in period t
PVTCFt = Total present value of cash flows (price)
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
Percentage change in price 
1

 Macaulay duration  Yield change  100
yield
1
k
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)
Modified Duration and Macaulay
Duration
CFt
P
(1  i )t
t  CFt 1
 P 1
Modified duration  
 

t 1
i P
(1  i )
P
1

 Macaulay duration
(1  i )
i = yield
P = price
CF = Cash flow
An Example
Calculate:
What is the modified duration of a 3-year, 3%
bond if interest rates are 5%?
Solution to Example
Pe rio d
C a sh F lo w
PV
t x PV
1
3
2.86
2.86
2
3
2.72
5.44
3
103
88.98
266.93
T otal
94.55
275.23
275.23
Macaulay duration =
 2.91
94.55
2.91
Modified duration =
 2.77
105
.
Example Continued
• What is the predicted price change of the 3
year, 3% coupon bond if interest rates
increase to 6%?
Example Continued
• What is the predicted price change of the 3
year, 3% coupon bond if interest rates
increase to 6%?
% Price Change =  Modified Duration  Yield Change
= -2.77 .01 = -2.77%
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)
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:
P
Duration = P
i
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
Error in Price Predictions
– The estimate is linear
• Because the price/
yield curve is convex,
it lies above the
tangent line
• Our estimate of price
is always understated
Price
• The estimate of the
change in bond price
is at one point
Yield
Figure 2-A
Present Value of a $1 Million Ten Year Zero Coupon Bond
with Taylor Series Approximations
$1,200,000.00
$1,000,000.00
$800,000.00
Present Value
$600,000.00
Bond Value
First Order
Second Order
Third Order
Fourth Order
$400,000.00
$200,000.00
$0
0.05
0.1
0.15
$(200,000.00)
$(400,000.00)
Interest Rate
0.2
0.25
0.3
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
Computing Convexity
• Take the second derivative of price with
respect to the interest rate
 2 P 1   t  CFt 1 
Convexity 
 
 


2
t 1
i
P  i  (1  i )
P
t  (t  1)  CFt 1


t 2
(1  i )
P
t  (t  1)  CFt
1
1
 

2
P (1  i )
(1  i ) t
Example
• What is the convexity of the 3-year, 3%
bond with the current yield at 5%?
Period
Cash Flow
PV
t x PV
t x (t+1) x PV
1
3
2 .8 6
2 .8 6
5 .7 1
2
3
2 .7 2
5 .4 4
1 6 .3 3
3
103
8 8 .9 8
2 6 6 .9 3
1 ,0 6 7 .7 0
T o ta l
9 4 .5 5
2 7 5 .2 3
1 ,0 8 9 .7 4
1,089.74
Convexity 
 10.46
2
1.05  94.55
Predicting Price with Convexity
• By including convexity, we can improve our
estimates for predicting price
Percentage Price Change =
- Modified Duration  Yield Change
1
+  Convexity  (Yield Change) 2
2
An Example of Predictions
• Let’s see how close our estimates are
Original Bond Price at 5% = 94.55
Prediction at 6% using Duration = 94.55  0.9723 = 91.93
Percentage Change with Convexity = -2.77 .01+ 0.5  10.86.012
 2.72%
Estimated Price with Convexity = 94.55  0.9728 = 91.98
Actual Price at 6% = 91.98
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
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
Next Time
• Limitations to duration calculations
• Effective duration and convexity
• Other duration measures