The problem describes a bond with a par value of $1000, a coupon rate of 0.075, and a maturity of 28 years. The yield to maturity is 0.15. We need to find the current price of the bond and the rate of return if the bond is sold after one year for $850.

1. Problem Description

The problem describes a bond with a par value of $1000, a coupon rate of 0.075, and a maturity of 28 years. The yield to maturity is 0.

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5. We need to find the current price of the bond and the rate of return if the bond is sold after one year for $

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0.

2. Solution Steps

First, let's find the current price of the bond. The current price of a bond is the present value of all future cash flows, which include coupon payments and the par value at maturity, discounted at the yield to maturity.

The annual coupon payment is:

$Coupon = Par Value * Coupon Rate =$1000 * 0.075 = $75$

The present value of the bond can be calculated as:

$Bond Price = \sum_{t=1}^{n} \frac{Coupon}{(1+YTM)^t} + \frac{Par Value}{(1+YTM)^n}$

Where:

$Coupon =$75$

$YTM = 0.15$

$Par Value =$1000$

$n = 28$

We can also use the following formula:

$Bond Price = Coupon * \frac{1 - (1 + YTM)^{-n}}{YTM} + \frac{Par Value}{(1+YTM)^n}$

$Bond Price = 75 * \frac{1 - (1 + 0.15)^{-28}}{0.15} + \frac{1000}{(1+0.15)^{28}}$

$Bond Price = 75 * \frac{1 - (1.15)^{-28}}{0.15} + \frac{1000}{(1.15)^{28}}$

$Bond Price = 75 * \frac{1 - 0.01835}{0.15} + \frac{1000}{54.517}$

$Bond Price = 75 * \frac{0.98165}{0.15} + 18.343$

$Bond Price = 75 * 6.54433 + 18.343$

$Bond Price = 490.82 + 18.343$

$Bond Price = 509.163$

$Bond Price \approx 509.16$

Now, let's calculate the rate of return if the bond is sold after one year at $

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0. The rate of return is calculated as:

$Return = \frac{Selling Price - Purchase Price + Coupon}{Purchase Price}$

The purchase price is the current bond price, which is $509.

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6. The selling price is $

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0. The coupon payment received is $

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5.

$Return = \frac{850 - 509.16 + 75}{509.16}$

$Return = \frac{415.84}{509.16}$

$Return = 0.8167$

$Return = 81.67\%$

3. Final Answer

The current price of the bond is approximately $509.

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6. The rate of return if the bond is sold after one year at $850 is approximately 81.67%.