Question

An investor is contemplating the purchase of a 20-year bond that pays $50 interest every six months. the investor plans to hold the bond only for 10 years, at which time she will sell it in the marketplace. she requires a 12 percent annual return but believes the market will require only an 8 percent return when she sells the bond 10 years from now. assuming she is a rational investor, how much should she be willing to pay for the bond today

Answer 1

Answer: The investor should be willing to pay $927.68 for the bond today.

We in need to compute the price at which the investor can sell the bond in year 10 (Y10).

The price of the bond in year 10 will be the present value of the coupons over the remaining life of the bond and the maturity value of the bond after 20 years.

We have

Coupon  Value (C )                     $50.00

No. of coupons remaining (n)           20

Expected YTM in year 10                 0.08

Expected semi annual  YTM in year 10      $\frac{0.08}{2} =0.04$

Face (Maturity) Value of the bond (MV)    $1,000.00

The bond price in year 10 will be

$\mathbf{Bond Price_{Y10}=C*\left ( \frac{1-(1+r)^{-n}}{r}\right )+\frac{MV}{(1+r)^{n}}}$

Substituting the values we get,

$Bond Price_{Y10}=50*\left ( \frac{1-(1+0.04)^{-20}}{0.04}\right )+\frac{1000}{(1+0.04)^{20}}$

$Bond Price_{Y10}=50*\left (13.59\right )+\frac{1000}{2.19}$

$\mathbf{Bond Price_{Y10}= 679.52+ 456.39 = 1,135.90}$

Hence the investor can expect to sell the bond in year 10  at $1,135.90.

Now, we'll calculate the price the investor is willing to pay for the bond. The investor can expected to pay the Present Value of the coupons she'll receive over 10 years and the selling price of the bond 10 years from now. We discount the cash flows at the rate of return the investor expects.

We have

Coupon  Value (C )                     $50.00

No. of coupons remaining (n)           20

Expected rate of return                          0.12

Expected semi annual  rate of return          $\frac{0.12}{2} =0.06$

Selling Price of the bond (SP)                $1,135.90

$\mathbf{Bond Price=C*\left ( \frac{1-(1+r)^{-n}}{r}\right )+\frac{SP}{(1+r)^{n}}}$

Substituting the values we get,

$Bond Price=50*\left ( \frac{1-(1+0.06)^{-20}}{0.06}\right )+\frac{1000}{(1+0.06)^{20}}$

$Bond Price=50*\left (11.47\right )+\frac{1000}{3.21}$

$\mathbf{Bond Price= 573.50+ 354.18 = 927.80}$