Answer:
16 years to maturity
Explanation:
We will calculate time:
The bonds present value is 896.23
YTM 10.34
and the face value is 1,000
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
Coupon Payment: 45 (1,000 x 9% / 2 payment per year)
time n
rate 0.0517 (10.34/2 payment per year
[tex]45 \times \frac{1-(1+0.0517)^{-n} }{0.0517} = PV\\[/tex]
PVc
[tex]\frac{Maturity}{(1 + rate)^{time} } = PV[/tex]
Maturity 1,000.00
time n
rate 0.0517
[tex]\frac{1000}{(1 + 0.0517)^{n} } = PV[/tex]
PVm
PV c + PV m = 896.23
[tex]45 \times \frac{1-(1+0.0517)^{-n} }{0.0517} + \frac{1000}{(1 + 0.0517)^{n} } = 896.23[/tex]
We rearrenge:
[tex](1-(1+0.0517)^{-n})\times \frac{45}{0.0517}+(1+0.0517)^{-n} \times 1,000 = 896.23[/tex]
We solve to clear the expression 1.0517 power -n:
[tex]870.4061896 - 870.4061896(1.0517)^{-n}) + 1000(1.0517)^{-n} = 896.23[/tex]
[tex] 129.59(1.0517)^{-n} = 896.23 - 870.4061896[/tex]
[tex](1.0517)^{-n} = 25.82381044 \div 129.59 [/tex]
We now use logarithmics properties:
[tex]log0.20 \div log1.0517 = -n[/tex]
n = 32.00
These are semianual payment, so we will divide by two to get the time expressses in years:
32/2 = 16 years to maturity
