Purchase Price of Bond

To find the purchase price of the bond, we need to calculate the present value of the future cash flows. The cash flows in this case are the annual dividends and the redemption value at the end of 10 years. The investor wishes to have a yield rate of 8%.

Step 1: Calculate the Present Value of the Annual Dividends

The annual dividends are paid at a rate of 8.5% on a face value of $1000. To calculate the present value of these dividends, we can use the formula for the present value of an annuity:

PV = C × (1 - (1 + r)^-n) / r

Where:
PV = Present Value
C = Cash flow per period
r = Interest rate per period
n = Number of periods

In this case, C = 8.5% of $1000 = $85, r = 8% (yield rate), and n = 10 (years).

PV = $85 × (1 - (1 + 0.08)^-10) / 0.08

Simplifying the equation:

PV = $85 × (1 - 1.08^-10) / 0.08
PV = $85 × (1 - 0.46318) / 0.08
PV = $85 × 0.53682 / 0.08
PV = $567.90

Therefore, the present value of the annual dividends is $567.90.

Step 2: Calculate the Present Value of the Redemption Value

At the end of 10 years, the bond will be redeemed at par, which is $1000. To calculate the present value of this redemption value, we can use the formula for the present value of a single cash flow:

PV = F / (1 + r)^n

Where:
PV = Present Value
F = Future Value
r = Interest rate per period
n = Number of periods

In this case, F = $1000, r = 8% (yield rate), and n = 10 (years).

PV = $1000 / (1 + 0.08)^10

Simplifying the equation:

PV = $1000 / 1.08^10
PV = $1000 / 2.15892
PV = $463.19

Therefore, the present value of the redemption value is $463.19.

Step 3: Calculate the Purchase Price

The purchase price of the bond is the sum of the present values of the annual dividends and the redemption value:

Purchase Price = PV of Annual Dividends + PV of Redemption Value
Purchase Price = $567.90 + $463.19
Purchase Price = $1031.09

Therefore, the purchase price of the bond, with a yield rate of 8%, is $1031.09.