Question

A 10-year U.S. Treasury bond with a face value of $1,000 pays a coupon of 5.5% (2.75% of face value every six months). The reported yield to maturity is 5.2% (a six-month discount rate of 5.2/2 = 2.6%). What is the present value of the bond? If the yield to maturity changes to 1%, what will be the present value? If the yield to maturity changes to 8%, what will be the present value? If the yield to maturity changes to 15%, what will be the present value?

Answer 1

Answer:

YTM 5.2%  present value: $1,023.1644

YTM 1% present value:      $1,427.2169

YTM 8% present value:       $830.1209

YTM 8% present value:        $515.7617

Explanation:

YTM we will calculate the present value of the coupon payment

andthe maturity at each YTM rate given:

The coupon payment present value will be the present value of an ordinary annuity

$C \times \frac{1-(1+r)^{-time} }{rate} = PV$

Coupon payment 28 (1,000 x 2.75%)

time 20 (10 years x 2 payment per year)

rate          0.026 (YTM over 2 as the payment are semiannually)

$27.5 \times \frac{1-(1+0.026)^{-20} }{0.026} = PV$

PV $424.6800

The present value of the maturity will be the present value of a lump sum:

$\frac{Maturity}{(1 + rate)^{time} } = PV$  

Maturity   1,000.00

time   20.00

rate  0.026

$\frac{1000}{(1 + 0.026)^{20} } = PV$  

PV   598.48

PV c $424.6800

PV m  $598.4843

Total $1,023.1644

Now, we will calculate changin the YTM the concept and formulas are the same, just the rate is diffrent:

If YTM = 1%

$27.5 \times \frac{1-(1+0.005)^{-20} }{0.005} = PV$

$\frac{1000}{(1 + 0.005)^{20} } = PV$  

PV c $522.1540

PV m  $905.0629

Total $1,427.2169

If YTM = 8%

$27.5 \times \frac{1-(1+0.04)^{-20} }{0.04} = PV$

$\frac{1000}{(1 + 0.04)^{20} } = PV$

PV c    $373.7340

PV m   $456.3869

Total    $830.1209

If YTM = 15%

$27.5 \times \frac{1-(1+0.075)^{-20} }{0.075} = PV$

$\frac{1000}{(1 + 0.075)^{20} } = PV$

PV c $280.3485

PV m  $235.4131

Total $515.7617