Assume a par value of $1,000. Caspian Sea plans to issue a 16.00 year, semi-annual pay bond that has a coupon rate of 7.83%. If the yield to maturity for the bond is 8.24%, what will the price of the bond be?

Question

GPT-4o mini · Answer

To calculate the price of a bond, you can use the present value formula for both the coupon payments and the par value at maturity.

Calculate the semi-annual coupon payment:

$Coupon payment = \frac{Coupon rate \times Par value}{2}$

$Coupon payment = \frac{0.0783 \times 1000}{2} = 39.15$
Determine the total number of periods (n):

Since the bond has a maturity of 16.00 years and pays semi-annually:

$n = 16 \times 2 = 32 periods$
Determine the yield per period (r):

The yield to maturity is annual, so for semi-annual:

$r = \frac{0.0824}{2} = 0.0412$
Calculate the present value of the coupon payments:

The formula for the present value of an annuity (the coupon payments) is:

$P V_{coupons} = C \times (1 - (1 + r)^{- n}) / r$

Where $C$ is the coupon payment and $r$ is the yield per period.

$P V_{coupons} = 39.15 \times (1 - (1 + 0.0412)^{- 32}) / 0.0412$

Calculating the present value of coupons:

$P V_{coupons} = 39.15 \times (1 - (1.0412)^{- 32}) / 0.0412$

First, calculate $(1 + 0.0412)^{- 32}$ :

$(1.0412)^{- 32} \approx 0.2633$

Then plug in the values:

$P V_{coupons} = 39.15 \times (1 - 0.2633) / 0.0412$

$= 39.15 \times 0.7367 / 0.0412 \approx 39.15 \times 17.87 \approx 699.34$
Calculate the present value of the par value:

The formula for the present value of the par value (the face value at maturity) is:

$P V_{par} = \frac{F}{(1 + r)^{n}}$

Where $F$ is the face value.

$P V_{par} = \frac{1000}{(1.0412)^{32}}$

Calculate $(1.0412)^{32}$ :

$(1.0412)^{32} \approx 3.5935$

Then:

$P V_{par} = \frac{1000}{3.5935} \approx 278.19$
Calculate the total price of the bond:

$Bond Price = P V_{coupons} + P V_{par} = 699.34 + 278.19 \approx 977.53$

Therefore, the estimated price of the bond is approximately $977.53.