Question

Lucky Star Inc. just issued a bond with the following characteristics:

Maturity = 3 years

Coupon rate = 8%

Face value = $1,000

YTM = 10%

Interest is paid annually and the bond is noncallable.

1. Calculate the bond’s Macaulay duration （Round "Present value" to 2 decimal places and "Duration" to 4 decimal place.）
2. Calculate the bond’s modified duration
3. Assuming the bond’s YTM goes from 10% to 9.5%, calculate an estimate of the price change without considering convexity.
4. Calculate the convexity of the bond.

Problem 2: Evaluate the following pure-yield pickup swap: You currently hold a 20-yearm AA-rated, 9% coupon rate bond with yield to maturity of 11.0%. As a swap candidate, you are considering a 20-yearm AA-rated, 11% coupon rate bond with yield to maturity of 11.5%, assume reinvestment rate is 11.5% and coupon are paid semi-annually, please fill out the table below and you must show (explain) clearly how numbers in the table are obtained.

	Current Bond	Candidate bond
Dollar investment
Coupon
i on one coupon
Principal value at year end
Total accrued
Realized compound yield
Value of swap:		basis point in one year

Answer 1

From information given above about the LUCKY STAR INC. Bond:

YTM = 10%

Coupon Rate = 8%

Facevalue = $1000

Maturity = 3 years

(a). Macaulay duration calculation:

Time period	Cashflows	Present Value of Cashflow	PV of Time weighted Cashflows
1	80	72.73	72.73
2	80	66.12	132.24
3	80 + 1000(Coupon + principal value at maturity)	811.42	2434.26
	Total	890.27	2639.23

From the above table:

Here, we have taken YTM = 10% as the discount rate.

Cashflows = Facevalue * coupon rate = 1000 * 8% = 80

PV of Cashflows = Discount factor * cashflows = 80 * 1/(1+YTM)^n = 80 * 1/(1.1)^1 = 72.73

PV of time weighted cashflows = PV of cashflows * time period = 72.73 * 1 = 72.73

Formula to calculate Macaulay Duration : MacD =  PV of Time weighted cashflows / PV of cashflows

Therefore, Macaulay Duartion = 2639.23 / 890.27 = 3.3015 years

(b). Modified Duration : Macaulay Duration / (1+YTM)

Modified Duration = 3.3015 / (1 + 0.1) = 3.0014 years

(c). Here, we have to calculate price of a bond at 9.5% YTM

we should remember that price of a bond at any given time is the sum of the present value of future cashflows at a given discount rate.

Previously, we have taken 10% discount rate and calculated PV of cash flows, now we have to take 9.5% as discount rate and calculate Price of bond.

Time period	Cashflows	Discount Factor @9.5%	PV of cashflows
1	80	1/(1+0.095)^1 = 0.9132	73.056
2	80	1/(1+0.095)^2 = 0.8340	66.72
3	80 + 1000	1/(1+0.095)^3 = 0.7616	822.528
	Total		962.304

Price of the bond at 9.5% YTM is $962.304 whereas at 10% it was $890.27

Bond price change = (Price at lower rate - price at upper rate) / price at lower rate =( 962.304 - 890.27) / 962.304 = 0.0748 or 7.48%

(d) Convexity of the bond:

Convexity (C) of a bond = (price at higher rate + price at lower rate - 2*initial price) / (2 * initial price ) * (change in price of the bond)^2

here, let us assume that the initial rate is 10%, lower rate is 9.5% and higher rate is 10.5%.

so, we have price at 10%(initial rate ) as 890.27

Price at lower rate 9.5% = 962.304

we have to calculate price at 10.5%

price of a bond at 10.5 % = present value of all future cashflows at 10.5%

time period	Cashflows	discount rate @10.5%	PV of Cashflows
1	80	1/(1.105)^1 = 0.905	72.40
2	80	1/(1.105)^2 = 0.819	65.52
3	1000 + 80	1/(1.105)^3 = 0.741	800.455
	Total (Price)		938.375

Hence price of the bond at 10.5% rate is 938.375

Convexity (C) =( 962.304 + 938.375 - 2 * 890.27) / (2* 890.27 ) * (962.304 - 938.375) ^2 = 120.139 / 1780.54 * 572.59 = 0.0118