Coupon Bond Valuation

EXERCISE 01

Face value is 1,000; maturity is 5 years; annual coupon rate is 6.500% and quoted annual YTM is 6.300%. Coupons are annual.

A.Calculate the bond price.
B.At what YTM would it be issued at par?

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.065)/1=65

i=0.063/1=0.063,\qquad N=5\times 1=5

P=65\frac{1-(1.063)^{-5}}{0.063}+\frac{1{,}000}{1.063^{5}}

Price = 1,008.36.

B.

1{,}000=65\frac{1-(1+y)^{-5}}{y}+\frac{1{,}000}{(1+y)^5}

y=C/F=65/1{,}000=6.5\%

YTM = coupon rate = 6.5%.

Coupon Bond Valuation

EXERCISE 02

Face value is 1,000; maturity is 15 years; annual coupon rate is 12.500% and quoted annual YTM is 14.000%. Coupons are annual.

A.Calculate the bond price.

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.125)/1=125

i=0.14/1=0.14,\qquad N=15\times 1=15

P=125\frac{1-(1.14)^{-15}}{0.14}+\frac{1{,}000}{1.14^{15}}

Price = 907.87.

Coupon Bond Valuation

EXERCISE 03

A bond was issued at par one year ago, with a four-year original maturity, face value €100 and an annual coupon rate of 4.00%. Its current yield is 3.00%. Value it just after the first coupon payment.

A.Calculate the current price.

A.

P=C\frac{1-(1+y)^{-n}}{y}+\frac{F}{(1+y)^n}

n=4-1=3,\qquad C=100(0.04)=4

P=4\frac{1-1.03^{-3}}{0.03}+\frac{100}{1.03^3}

€102.83

Coupon Bond Valuation

EXERCISE 04

A bond was issued at par one year ago, with a four-year original maturity, face value €100 and an annual coupon rate of 5.00%. Its current yield is 7.00%. Value it just after the first coupon payment.

A.Calculate the current price.

A.

P=C\frac{1-(1+y)^{-n}}{y}+\frac{F}{(1+y)^n}

n=4-1=3,\qquad C=100(0.05)=5

P=5\frac{1-1.07^{-3}}{0.07}+\frac{100}{1.07^3}

€94.75

Coupon Bond Valuation

EXERCISE 05

A bond was issued at par one year ago, with a four-year original maturity, face value €100 and an annual coupon rate of 3.00%. Its current yield is 5.00%. Value it just after the first coupon payment.

A.Calculate the current price.

A.

P=C\frac{1-(1+y)^{-n}}{y}+\frac{F}{(1+y)^n}

n=4-1=3,\qquad C=100(0.03)=3

P=3\frac{1-1.05^{-3}}{0.05}+\frac{100}{1.05^3}

€94.55

Coupon Bond Valuation

EXERCISE 06

A bond was issued at par one year ago, with a four-year original maturity, face value €100 and an annual coupon rate of 5.00%. Its current yield is 3.00%. Value it just after the first coupon payment.

A.Calculate the current price.

A.

P=C\frac{1-(1+y)^{-n}}{y}+\frac{F}{(1+y)^n}

n=4-1=3,\qquad C=100(0.05)=5

P=5\frac{1-1.03^{-3}}{0.03}+\frac{100}{1.03^3}

€105.66

Coupon Bond Valuation

EXERCISE 07

A bond was issued at par one year ago, with a four-year original maturity, face value €100 and an annual coupon rate of 5.00%. Its current yield is 2.00%. Value it just after the first coupon payment.

A.Calculate the current price.

A.

P=C\frac{1-(1+y)^{-n}}{y}+\frac{F}{(1+y)^n}

n=4-1=3,\qquad C=100(0.05)=5

P=5\frac{1-1.02^{-3}}{0.02}+\frac{100}{1.02^3}

€108.65

Coupon Bond Valuation

EXERCISE 08

A bond has face value €1,000, two years remaining and an 8% annual coupon paid semiannually. Its annual quoted YTM is 6%, compounded semiannually.

A.Convert the annual terms into payment-period terms.
B.Calculate the bond price.

A.

C=1{,}000(0.08)/2=40

i=0.06/2=0.03,\qquad N=2(2)=4

Item	Semiannual value
Coupon	€40
Discount rate	3%
Number of payments	4

Both quoted annual rates are divided by two; maturity is multiplied by two.

B.

P=40\frac{1-(1.03)^{-4}}{0.03}+\frac{1{,}000}{1.03^4}

€1,037.17

If 6% were instead an effective annual yield, the six-month rate would be √1.06 − 1. The quotation convention must be specified.

Coupon Bond Valuation

EXERCISE 09

Face value is 1,000; maturity is 3 years; annual coupon rate is 8.000% and quoted annual YTM is 6.500%. Coupons are paid 2 times per year; YTM is nominal with the same compounding frequency.

A.Calculate the bond price.

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.08)/2=40

i=0.065/2=0.0325,\qquad N=3\times 2=6

P=40\frac{1-(1.0325)^{-6}}{0.0325}+\frac{1{,}000}{1.0325^{6}}

Price = 1,040.29.

Coupon Bond Valuation

EXERCISE 10

Face value is 1,000; maturity is 5 years; annual coupon rate is 5.000% and quoted annual YTM is 6.000%. Coupons are paid 2 times per year; YTM is nominal with the same compounding frequency.

A.Calculate the bond price.

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.05)/2=25

i=0.06/2=0.03,\qquad N=5\times 2=10

P=25\frac{1-(1.03)^{-10}}{0.03}+\frac{1{,}000}{1.03^{10}}

Price = 957.35.

Coupon Bond Valuation

EXERCISE 11

Face value is 1,000; maturity is 3 years; annual coupon rate is 6.000% and quoted annual YTM is 8.000%. Coupons are paid 2 times per year; YTM is nominal with the same compounding frequency.

A.Calculate the bond price.
B.Reprice with annual coupons and 7% YTM.

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.06)/2=30

i=0.08/2=0.04,\qquad N=3\times 2=6

P=30\frac{1-(1.04)^{-6}}{0.04}+\frac{1{,}000}{1.04^{6}}

Price = 947.58.

B.

C=1{,}000(0.06)/1=60

i=0.07/1=0.07,\qquad N=3(1)=3

P=60\frac{1-(1.07)^{-3}}{0.07}+\frac{1{,}000}{1.07^{3}}

Price = €973.76.

Coupon Bond Valuation

EXERCISE 12

Face value is 1,000; maturity is 5 years; annual coupon rate is 4.000% and quoted annual YTM is 6.000%. Coupons are annual.

A.Calculate the bond price.
B.How many bonds must be sold?
C.Reprice with semiannual coupons and the same quoted YTM.

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.04)/1=40

i=0.06/1=0.06,\qquad N=5\times 1=5

P=40\frac{1-(1.06)^{-5}}{0.06}+\frac{1{,}000}{1.06^{5}}

Price = 915.75.

B.

N=\left\lceil\frac{55{,}000{,}000}{40[1-1.06^{-5}]/0.06+1{,}000/1.06^5}\right\rceil

N=\lceil 60{,}059.88139\rceil=60{,}060

60060 bonds.

Round up to a whole number to raise at least €55 million.

C.

C=1{,}000(0.04)/2=20

i=0.06/2=0.03,\qquad N=5(2)=10

P=20\frac{1-(1.03)^{-10}}{0.03}+\frac{1{,}000}{1.03^{10}}

Price = €914.70.

Coupon Bond Valuation

EXERCISE 13

Face value is 1,000; maturity is 20 years; annual coupon rate is 12.000% and quoted annual YTM is 10.000%. Coupons are paid 2 times per year; YTM is nominal with the same compounding frequency.

A.Calculate the bond price.
B.Reprice if YTM rises to 14%.

A.

P=\frac{Fc}{m}\frac{1-(1+y/m)^{-mn}}{y/m}+\frac{F}{(1+y/m)^{mn}}

C=1{,}000(0.12)/2=60

i=0.1/2=0.05,\qquad N=20\times 2=40

P=60\frac{1-(1.05)^{-40}}{0.05}+\frac{1{,}000}{1.05^{40}}

Price = 1,171.59.

B.

C=1{,}000(0.12)/2=60

i=0.14/2=0.07,\qquad N=20(2)=40

P=60\frac{1-(1.07)^{-40}}{0.07}+\frac{1{,}000}{1.07^{40}}

New price = 866.68.

The bond moves from above par to below par.

Coupon Bond Valuation

EXERCISE 14

The firm has zero-coupon debt with face value €1,000 due in two years and coupon debt with face value €2,000, a 5% annual coupon and five years remaining. Both have the same YTM, but that yield is missing from the supplied question.

A.Express the debt value and identify the missing input.

A.

D(y)=\frac{1000}{(1+y)^2}+100\frac{1-(1+y)^{-5}}{y}+\frac{2000}{(1+y)^5}

A unique numerical answer requires the common yield. If y = 5%, the coupon bond is at par and D = 1,000/1.05² + 2,000 = €2,907.03. That matches the printed answer, but 5% is an inferred assumption, not a supplied input.