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A U.S. treasury bond (selling at a par value of $1,000) that matures at the end of five years is said to have a coupon rate of 6% if, after paying $1,000, the purchaser receives $30 at the end of each of the following nine six-month periods and then receives $1,030 at the end of the the tenth period. That is, the bond pays a simple interest rate of 3% per six-month period, with the principal repaid at the end of five years. Assuming a continuously compounded interest rate of 5%, find the present value of such a stream of cash payments.

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jepessoa

Answer:

$1,042.04

Explanation:

to calculate the present value using a continuously compounded interest rate, we can use the following 2 formulas:

1) present value = cash flow / eⁿˣ

  • e = 2.71828
  • x = 5% / 2 = 2.5%
  • n = 10
  • cash flow = $1,030

present value = $1,030 / 2.71828¹⁰ˣ⁰°⁰²⁵ = $1,030 / 1.284 = $802.16

2) present value of an annuity = payment [(1 - e⁻ⁿˣ) / (eˣ - 1)]

  • payment = $30
  • x = 2.5%
  • n = 9
  • e = 2.71828

present value = $30 [(1 - 2.71828⁻⁹ˣ⁰°⁰²⁵) / (2.71828⁰°⁰²⁵ - 1)] = $30 [(1 - 2.71828⁻⁹ˣ⁰°⁰²⁵) / (2.71828⁰°⁰²⁵ - 1)] = $30(0.2015 / 0.0252) = $239.88

present value of the stream of cash flows = $802.16 + $239.88 = $1,042.04

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