Question

Find the duration of a 8% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 7%.

Find the bond price.

If the market interest rates decrease by .5% per year (i.e. YTM becomes 6.5%). Use duration formula to find how such interest rate change will affect the bond price?

Find the new bond price using a financial calculator.

Compare actual and duration predicted bond price changes.

Which change is larger? What role does bond price convexity play here?

If the market interest rates decrease by .5% per year (i.e. YTM becomes 6.5%).

Use the duration formula to find how such interest rate change will affect the bond price?

Find the new bond price?

Compare actual and duration predicted bond price changes. Which change is larger? What role does bond price convexity play here?

Answer 1

a.) For the bond given in the question, duration D is found as

Alternatively, you can solve in excel

Period			PV of cash flows	Period* Cash flow	PV of period*cash flows
1	80	0.934579	74.76635514	80	74.76635514
2	80	0.873439	69.87509826	160	139.7501965
3	80	0.816298	65.30383015	240	195.9114905
4	1080	0.762895	823.926829	4320	3295.707316
			1033.872113		3706.135358
			Duration	3.584713538	PV of cash flows/ PV of period*cash flows

D= 3.584713

Bond price can be found out by discounting all the cash flow by the market rate or yield to maturity

Period			PV of cash flows
1	80	0.934579	74.76635514
2	80	0.873439	69.87509826
3	80	0.816298	65.30383015
4	1080	0.762895	823.926829
			1033.872113

Bond price = $1033.87

c.) Change in bond price = -D* change in yield* Bond price prior to yield increase

Change in bond price = -3.584713*(-0.5/100)*1033.872113

Change in bond price =18.5306

Hence, the new bond price (by duration) = 1033.872113+18.5306 =$1052.4027

d) New bond price using a calculator

FV=1000

I/Y = 6.5%

PMT=80

N=4

Solving we get, PV = $1051.3869

e.) New bond price (by duration) = $1052.4027

Actual new bond price = $1051.3869

The difference between bond value by duration and actual bond value is $1052.4027-$1051.3869 =$1.0158

f.) The change in duration based calculated bond price is larger by $1.0158. By inclusion of convexity, the bond price ( calculated using duration and convexity) becomes more accurate, since convexity takes into account the curvature of the relationship between bond prices and bond yields).In this case, since the yields have decreased, convexity (by taking in consideration curvature) would have reduced the bond prices by some smount, which would had made the calculation more accurate ( closer to the calculated bond price using calculator).Hence, the bond prices calculated through combined effects of duration and convexity both are more accurate.