Question

What is the price of the following US T-Bond? (Use any method you prefer)

Face value: $100

Maturity: 7 years

Coupon rate 2.5% (paid annually)

Yield = 7.5%

Answer : 73.52

Suppose you observe that the above bond is trading at $83.00. What is the yield?

Yield:

Calculate the price, duration, and modified duration of this bond when the yield is 9% (Enter all answers with two decimal places).

Price:

Duration:

Modified Duration:

Suppose the yield for the bond from the previous question increases by 1½ percentage points. Without re-calculating the price, what is the expected change (in %) in the bond’s price? That is, what is the expected percentage change using the Duration approximation? What would be the new price predicted by (modified) duration?

Expected change:

New price:

Answer 1

What is the price of the following US T-Bond? (Use any method you prefer)

Face value: $100

Maturity: 7 years

Coupon rate 2.5% (paid annually)

Yield = 7.5%

The initial bond price

Bond price P0 = C* [1- 1/ (1+i) ^n] /i + M / (1+i) ^n

Where

M = value at maturity, or par value = $ 100

C = coupon payment = 2.5% of $100 = $2.50

n = number of payments = 7

i = interest rate, or required yield = 7.5% or 0.075

Bond Price = $2.5 * [1 – 1 / (1+0.075) ^7] /0.075 + $100 / (1+0.075) ^7

= $13.24 + $60.28

= $73.52

Suppose you observe that the above bond is trading at $83.00. What is the yield?

Now we have following formula for calculation of bond’s yield to maturity (YTM)

Bond price P0 = C* [1- 1/ (1+YTM) ^n] /YTM + M / (1+YTM) ^n

Where,

M = value at maturity, or par value = $ 100

P0 = the current market price of bond = $83.00

C = coupon payment = 2.5% of $100 = $2.5

n = number of payments (time remaining to maturity) = 7 years

YTM = interest rate, or yield to maturity =?

Now we have,

$83 = $2.50 * [1 – 1 / (1+YTM) ^7] /YTM+ 100 / (1+YTM) ^7

By trial and error method we can calculate the value of YTM = 5.49%

[Or you can use excel function for YTM calculation in following manner

“= Rate(N,PMT,PV,FV)”

“Rate(7,-2.5,83,-100)” = 5.49%]

Calculate the price, duration, and modified duration of this bond when the yield is 9% (Enter all answers with two decimal places).

Year (t)	Cash Flow from coupon payments (2.5% of $100)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 5% [=CF/(1+9%)^t]	PV *t
1	$2.50		$2.5	$2.29	$2.29
2	$2.50		$2.5	$2.10	$4.21
3	$2.50		$2.5	$1.93	$5.79
4	$2.50		$2.5	$1.77	$7.08
5	$2.50		$2.5	$1.62	$8.12
6	$2.50		$2.5	$1.49	$8.94
7	$2.50	$100.0	$102.5	$56.07	$392.50
sum				$67.29	$428.94
				Bond's Price↑
			Macaulay Duration = sum of (PV*t)/sum of PVs =	$428.94/ $100	6.37	Years

Price: $67.29

Duration: 6.37 years

Modified Duration = Macaulay Duration / (1 + YTM/n)

Where,

Macaulay Duration = 6.37 years

Yield to maturity, YTM = 9% per year

Number of discounting periods in year, n = 1 (for annual coupon payments)

Therefore,

Modified Duration = 6.37/ (1+ 9%)

= 5.85 years

Suppose the yield for the bond from the previous question increases by 1½ percentage points. Without re-calculating the price, what is the expected change (in %) in the bond’s price? That is, what is the expected percentage change using the Duration approximation? What would be the new price predicted by (modified) duration?

Percentage change in the Price of bond = - duration of bond *Change in the yield

Where,

Duration of bond = 6.37 years

Change in the yield = 1.5%

Percentage change in the Price of bond = - 6.37 * 1.5% = - 9.56%

The new price of bond will reduce by 9.56%

Therefore new price of bond = $67.29* (1- 9.56%) = $60.85

The new price predicted by (modified) duration -

New price of bond can be calculated with following formula

% Change in bond’s Price = -1 * Modified Duration * Change in interest rate

= -1 *5.85 * (1.50%) = - 8.77%

New price of bond = current price * % Change in bond’s Price

Where, Current price of the bond = $67.29

New price of bond = $67.29 * (1- 8.77%) = $61.38

Therefore new price of bond is approx. $61.38