Respuesta :
Answer: The Present Value of the bond is $10,231.64.
We have
Face Value of the bond $10000
Coupon rate per year 5.5%
Frequency of int payments Semi-Annual (two periods in a year)
Discount rate per year 5.2%
No. of years to maturity 10 years
First we calculate the coupon interest per period
[tex]C = Face Value * \frac{interest rate}{2}[/tex]
[tex]C = 10000 * \frac{0.055}{2}[/tex]
[tex]C = 10000 * \frac{0.055}{2}[/tex]
[tex]C = 10000 * 0.0275[/tex]
[tex]C = 275[/tex]
We can think of a bond as an instrument have types of cash flows.
One is the coupons we receive from a bond, where we receive a fixed amount per period for a stated number of periods.
An instrument that gives a fixed amount per period for a stated number of periods is known as an annuity.
Hence we can treat the coupon from the bonds as an annuity.
The Present Value formula for an annuity is:
[tex]PV_{Coupons} = C\left \{ \frac{1-(1+i)^{-n}}{i} \right\}[/tex]
where
C = Coupon per period
i = discount rate per period
n = number of periods
In this question, we'll get [tex]2*10 =20[/tex] coupon payments, so the number of periods, n = 20.
The discount rate per period (i) is [tex]\frac{0.052}{2} = 0.026[/tex] or 2.6% per period.
Applying these values to the equation above we can find the PV of Coupons as:
[tex]PV_{Coupons} = 275\left \{ \frac{1-(1+0.026)^{-20}}{0.026} \right\}[/tex]
[tex]PV_{Coupons} = 275\left \{ \frac{1-(1+0.026)^{-20}}{0.026} \right\}[/tex]
[tex]PV_{Coupons} = 275\left \{ \frac{0.598484331}{0.026} \right\}[/tex]
[tex]PV_{Coupons} = 275 * 15.44291035[/tex]
[tex]PV_{Coupons} = 4246.800346[/tex]
In addition to the coupon, we also get back the bond's face value at the end of the bond's life. We can treat this as a lump-sum amount we will get back at the end of a stated number of periods. We can find the Present Value of the lumpsum as follows:
[tex]PV = \frac{Face Value}{(1+i)^{n}}[/tex]
Substituting the values we get,
[tex]PV = \frac{10000}{(1+0.026)^{20}}[/tex]
[tex]PV = \frac{10000}{1.670887521}[/tex]
[tex]PV_{lump sum} = 5984.843309[/tex]
Finally, we compute the Present Value of the bond as follows:
[tex]PV_{bond} = PV_{Coupons} + PV_{lump sum}[/tex]
[tex]PV_{bond} = 4246.800346 + 5984.843309[/tex]
[tex]PV_{bond} = 10231.64366[/tex]
