Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management,...
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Transcript of Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management,...
![Page 1: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/1.jpg)
Financial Risk Management of Insurance Enterprises
Introduction to Asset/Liability Management, Duration & Convexity
![Page 2: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/2.jpg)
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
![Page 3: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/3.jpg)
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
![Page 4: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/4.jpg)
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
![Page 5: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/5.jpg)
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
![Page 6: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/6.jpg)
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
![Page 7: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/7.jpg)
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
![Page 8: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/8.jpg)
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
![Page 9: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/9.jpg)
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
Price/yield curve
Yield
Pri
ce
![Page 10: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/10.jpg)
Simplifications
• Fixed income, non-callable bonds
• Flat yield curve
• Parallel shifts in the yield curve
![Page 11: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/11.jpg)
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%
![Page 12: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/12.jpg)
Price Changes on Two Zero Coupon Bonds
Initial Interest Rate = 8%
Principal R 5 year Change 10 year Change1000 0.06 747.2582 9.7967% 558.3948 20.5532%1000 0.07 712.9862 4.7611% 508.3493 9.7488%1000 0.0799 680.8984 0.0463% 463.6226 0.0926%1000 0.08 680.5832 0.0000% 463.1935 0.0000%1000 0.0801 680.2682 -0.0463% 462.7648 -0.0925%1000 0.09 649.9314 -4.5038% 422.4108 -8.8047%1000 0.1 620.9213 -8.7663% 385.5433 -16.7641%
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Price Volatility Characteristics of Option-Free BondsProperties
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
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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
![Page 15: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/15.jpg)
Macaulay Duration (p.2)
Macaulay Duration =
index for period
= total number of period
= number of coupon payments per year
Present value of cash flow in period t
= Total present value of cash flows (price)
t PVCF
k PVTCF
t
n
k
PVCF
PVTCF
t
tt
n
t
t
1
![Page 16: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/16.jpg)
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
Macaulay duration Yield change 100
1
1yieldk
![Page 17: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/17.jpg)
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)
![Page 18: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/18.jpg)
Modified Duration and Macaulay Duration
durationMacaulay )1(
1
1
)1(
1
i
duration Modified
)1(
1
i
Pi
CFt
P
P
i
CFP
tt
tt
i = yield CF = Cash flow
P = price
![Page 19: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/19.jpg)
An Example
Calculate:What is the modified duration of a 3-year, 3%
bond if interest rates are 5%?
![Page 20: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/20.jpg)
Solution to ExamplePeriod Cash Flow PV t x PV
1 3 2.86 2.86 2 3 2.72 5.44 3 103 88.98 266.93
Total 94.55 275.23
Macaulay duration =
Modified duration =
275 23
94 552 91
2 91
1052 77
.
..
.
..
![Page 21: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/21.jpg)
Example Continued
• What is the predicted price change of the 3 year, 3% coupon bond if interest rates increase to 6%?
![Page 22: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/22.jpg)
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%
![Page 23: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/23.jpg)
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)
![Page 24: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/24.jpg)
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:
Duration = -Pi
P
![Page 25: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/25.jpg)
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
![Page 26: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/26.jpg)
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 Yield
Pri
ce
![Page 27: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/27.jpg)
Figure 2-A Present Value of a $1 Million Ten Year Zero Coupon Bond
with Taylor Series Approximations
$(400,000.00)
$(200,000.00)
$-
$200,000.00
$400,000.00
$600,000.00
$800,000.00
$1,000,000.00
$1,200,000.00
0 0.05 0.1 0.15 0.2 0.25 0.3
Interest Rate
Pre
sen
t V
alu
e
Bond ValueFirst OrderSecond OrderThird OrderFourth Order
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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
![Page 29: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/29.jpg)
Computing Convexity
• Take the second derivative of price with respect to the interest rate
tt
tt
tt
i
CFtt
iP
Pi
CFtt
Pi
CFt
P
PConvexity
)1(
)1(
)1(
11
1
)1(
)1(
1
)1(i
1
i
2
2
12
2
![Page 30: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/30.jpg)
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 PV1 3 2.86 2.86 5.71 2 3 2.72 5.44 16.33 3 103 88.98 266.93 1,067.70
Total 94.55 275.23 1,089.74
46.1055.9405.1
74.089,12
Convexity
![Page 31: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/31.jpg)
Predicting Price with Convexity
• By including convexity, we can improve our estimates for predicting price
Percentage Price Change =
- Modified Duration Yield Change
+1
2Convexity (Yield Change)2
![Page 32: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/32.jpg)
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 .9723 = 91.93
Percentage Change with Convexity = -2.77 .01+ 0.5
Estimated Price with Convexity = 94.55 .9728 = 91.98
Actual Price at 6% = 91.98
0
10 86 01
2 72%
0
2. .
.
![Page 33: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/33.jpg)
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
![Page 34: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/34.jpg)
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
![Page 35: Financial Risk Management of Insurance Enterprises Introduction to Asset/Liability Management, Duration & Convexity.](https://reader035.fdocuments.in/reader035/viewer/2022062315/5697bffa1a28abf838cc0345/html5/thumbnails/35.jpg)
Next Time
• Limitations to duration calculations
• Effective duration and convexity
• Other duration measures