The problem describes a bond issued by Kids Toys with a par value of $1000, a coupon rate of 8% (0.08), and a maturity of 30 years. The yield to maturity (YTM) is 13% (0.13). We need to calculate the current price of the bond. Then, assuming the bond is sold after one year for $850, we need to determine the rate of return.

1. Problem Description

The problem describes a bond issued by Kids Toys with a par value of $1000, a coupon rate of 8$850, we need to determine the rate of return.

2. Solution Steps

Part 1: Calculating the Current Price of the Bond

The current price of the bond can be calculated using the present value formula:

$P = \sum_{t=1}^{n} \frac{C}{(1+YTM)^t} + \frac{FV}{(1+YTM)^n}$

Where:

$P$ = Current Price of the bond

$C$ = Coupon payment per period

$YTM$ = Yield to maturity

$n$ = Number of periods to maturity

$FV$ = Face value (par value) of the bond

In this case:

$FV = 1000$

$C = 0.08 * 1000 = 80$

$YTM = 0.13$

$n = 30$

$P = \sum_{t=1}^{30} \frac{80}{(1+0.13)^t} + \frac{1000}{(1+0.13)^{30}}$

This formula can be simplified using the present value of annuity formula for the coupon payments and the present value of a lump sum for the face value.

Present Value of Annuity = $C * \frac{1 - (1+YTM)^{-n}}{YTM}$

Present Value of Face Value = $\frac{FV}{(1+YTM)^n}$

$P = 80 * \frac{1 - (1+0.13)^{-30}}{0.13} + \frac{1000}{(1+0.13)^{30}}$

$P = 80 * \frac{1 - (1.13)^{-30}}{0.13} + \frac{1000}{(1.13)^{30}}$

$P = 80 * \frac{1 - 0.0255}{0.13} + \frac{1000}{39.155}$

$P = 80 * \frac{0.9745}{0.13} + 25.54$

$P = 80 * 7.496 + 25.54$

$P = 599.68 + 25.54$

$P = 625.22$

Part 2: Calculating the Rate of Return

The rate of return is calculated as:

$Rate\ of\ Return = \frac{Ending\ Value + Coupon\ Payment - Beginning\ Value}{Beginning\ Value}$

Where:

Ending Value = $850

Coupon Payment = $80

Beginning Value = $625.22

$Rate\ of\ Return = \frac{850 + 80 - 625.22}{625.22}$

$Rate\ of\ Return = \frac{304.78}{625.22}$

$Rate\ of\ Return = 0.4875$

Rate of Return = 48.75%

3. Final Answer

Current Price of the Bond: $625.22

Rate of Return: 48.75%