1. Tutorial. Measuring Interest Rate Risk
Duration
Convexity

2. Contents
A quick review
Some exercises

3. 1. Duration The value of Portfolio A is given by For small Ri
Duration of a portfolio of bond
The value of Portfolio A is given by
For small Ri
If R1= R2=…= RN= R (parallel yield shift), then

4. 1. Duration
If R1=R2=…=RN and m1=m2=…=mN , the duration of Portfolio A, DA, is given by

5. 1.Duration
Duration model MDA: duration of asset portfolio of a FI MDL: duration of liability portfolio of a FI Assume parallel yield shift, R to be small and the same for both asset and liability portfolio.

6. 1. Duration E: equity value of the FI.
k is a measure of the FI’s financial leverage.

7. 1. Duration
If all the bonds in the asset and liability portfolio have the same coupon frequency and equal to 1 and also the yield curve is flat, then
(DA – kDL): leverage adjusted duration gap

8. 2. Convexity Convexity R (%) P(R) R* R*  ΔR R* + ΔR Error
Duration model
P(R) R*
R*  ΔR
R* + ΔR
Error

9. 2. Convexity
CX: convexity of a bond

10. 2. Convexity

11. 2. Convexity Taking the convexity into account,
The convexity of Portfolio A,

12. 2. Convexity With convexity adjustment,
Under the condition of (MDA – kMDL)= 0.
If (CXA – kCXL) > 0, then the value of the equity will be increased irrelevant to the direction of the change of R.
If (CXA – kCXL) < 0, then the value of the equity will be decreased irrelevant to the direction of the change of R.

13. Exercise
Two bonds are available for purchase in the financial markets. The first bond is a two-year, $1,000 bond that pays an annual coupon of 10 percent. The second bond is a 2-year, $1,000, zero-coupon bond.
What is the duration of the coupon bond if the current yield-to-maturity (R) is 8 percent? 10 percent? 12 percent?
How does the change in the current yield to maturity affect the duration of this coupon bond?
Calculate the duration of the zero-coupon bond with a yield to maturity of 8 percent, 10 percent, and 12 percent.
How does the change in the yield to maturity affect the duration of the zero-coupon
Why does the change in the yield to maturity affect the coupon bond differently than the zero-coupon bond?

14. Thanks! Q&A