Question

suppose you have a standard coupon bond with a principal value of 50000 that matures in three years. the coupon rate is 4% and the coupon is paid annually with the first payment due in 12 months from today? ("Standard" refers to a non-callable bond contract.) a) If the YTM is 3%, what is the price of the bond today? b) Suppose the price moves to $48,638.38. What is the new YTM? c) Now suppose the original bond from part a) pays coupon semi-annually instead of annually ($50K principal value, 4% coupon, matures in 3 years, 1st payment due in 6 mths). What is the price of the new bond?

Answer 1

Solution

As provided,
Principal Value (P) = $50000
Maturity period (n) = 3 Years
Coupon Rate = 4%

Therefore, Annual Interest (I) = $50000 x 4% = $2000

(a) Given YTM (r) = 3%, or 0.03, Price of the bond will be,
I (PVIFA_r,n) + P (PVIF_r,n),
Where, PVIFA_3%,3 = {1 - [(1 / 1+0.03)³]} / 0.03 = 2.8286 (Approx.)
and, PVIF_3%,3 = [(1 / 1+0.03)³] = 0.9151 (Approx.)

Therefore, Current Price of the bond = (2000 x 2.8286) + (50000 x 0.9151)
= 5,657.20 + 45,755
= 51412.20

Answer: Current Price of Bond is $51,412.20

(b) Given that Current Market Price is $48,638.38. Therefore YTM should be calculated as,
2000 (PVIFA_r,3) + 50000 (PVIF_r,3) = 48638.38
Now, using Trial-and-Error method, let us find out the solution,
At first, running the trial using 4%,

Year	Cash Flow	PVIF @4%	Discounted Cash Flow
1	$   2,000.00	0.9615	$       1,923.08
2	$   2,000.00	0.9246	$       1,849.11
3	$   2,000.00	0.8890	$       1,777.99
3	$ 50,000.00	0.8890	$     44,449.82
Present Value of Cash Flows			$     50,000.00

Again, running the trial at 5%,

Year	Cash Flow	PVIF @5%	Discounted Cash Flow
1	$   2,000.00	0.9524	$       1,904.76
2	$   2,000.00	0.9070	$       1,814.06
3	$   2,000.00	0.8638	$       1,727.68
3	$ 50,000.00	0.8638	$     43,191.88
Present Value of Cash Flows			$     48,638.38

Therefore, at 5% the value matches accurately.

Answer: At 5% YTM, the value will be $48,638.38.

(c) Using all the information given in Part (a) above,
Principal Value = $50,000
Coupon Rate = 4%
Maturity Period = 3 Years
YTM = 3%
Number of Compounding = 2 (As semi-annual interest payment)

Therefore, Coupon Interest (I) = 50000 x 4% x 1/2 = 1000
Yield (r) = 3% x 1/2 = 1.50%
Term (n) = 3 x 2 = 6

So, Current Market Price of the bond = 1000 (PVIFA_1.50%,6) + 50000 (PVIF_1.50%,6)
Where, (PVIFA_1.50%,6) = {1 - [(1 / 1+0.015)⁶]} / 0.015 = 5.6972 (Approx.)
and, (PVIF_1.50%,6) = [(1 / 1+0.015)⁶] = 0.9145 (Approx.)

Therefore, Current Market Price = (1000 x 5.6972) + (50000 x 0.9145)
= 5,697.20 + 45,725
= 51,422.20

Answer: Price of the new bond will be $51,422.20.