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Chapter 6: Problem 30

An investor buys an annuity with payments of principal and interest of $\$ 500\( per quarter for 10 years. Interest is at the effective rate of \)8 \%$ per annum. How much interest does the investor receive in total over the 10 -year period? Answer to the nearest dollar.

Short Answer

Expert verified
The present value of the annuity is calculated as \( PV = 500 * \frac{1 - (1 + 0.02)^{-40}}{0.02} = \$14,379 \). The total paid over 10 years is \( 500 * 40 = \$20,000 \). Therefore, the total interest received over the 10-year period is \( \$20,000 - \$14,379 = \$5,621 \), rounded to the nearest dollar.

Step by step solution

01

Calculate the present value of the annuity

To determine the total interest received, we need to find out how much the annuity is worth in today's dollars. To do this, we will use the Present Value of an Annuity formula: \(PV = PMT * \frac{1 - (1 + r)^{-n}}{r}\) where: - PV is the present value of the annuity - PMT is the regular payment amount ($$500 per quarter) - r is the interest rate per period (8% per annum, but we need to convert it to a quarterly rate: \(0.08 / 4\)) - n is the total number of payments (10 years * 4 quarters)
02

Calculate the total paid over 10 years

We also need to find out how much the investor paid throughout the entire 10-year period. To do this, we simply multiply the payment amount per quarter by the total number of payments: Total paid = PMT * n
03

Calculate the total interest received

Now that we have the present value and the total paid, we can find the total interest earned by subtracting the present value from the total paid: Total interest = Total paid - Present value Lastly, we'll round the answer to the nearest dollar as instructed. Now, let's calculate.

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Most popular questions from this chapter

A loan is being repaid with level payments. If \(B_{t}, B_{t+1}, B_{t+2},\) and \(B_{t+3}\) are four successive outstanding loan balances, show that: a) \(\quad\left(B_{t}-B_{t+1}\right)\left(B_{t+2}-B_{t+3}\right)=\left(B_{t+1}-B_{t+2}\right)^{2}\) b) \(\quad B_{t}+B_{t+3} < B_{t+1}+B_{t+2}\)

a) Show that if a loan is amortized with \(n\) level payments of \(R\) b) Verbally interpret the result obtained in \((a)\)

A loan of \(\$ 1000\) is to be amortized with quarterly installments of \(\$ 100\) for as long as necessary plus a smaller final payment one quarter after the last regular payment. Interest is computed at \(12 \%\) convertible quarterly on the first \(\$ 500\) of outstanding loan balance and at \(8 \%\) convertible quarterly on any excess. a) Find the principal repaid in the fourth installment. b) Show that prior to the "crossover" point, the successive principal repayments plus a constant form a geometric progression, i.e. $$\frac{P_{t+1}+k}{P_{t}+k}=1+j \text { for } t=1,2, \ldots, a-1$$ (1) Find k. (2) Find \(j\)

An investor is making level payments at the beginning of each year for 10 years to accumulate \(\$ 10,000\) at the end of the 10 years in a bank which is paying \(5 \%\) effective. At the end of five years the bank drops its interest rate to \(4 \%\) effective. a) Find the annual deposit for the first five years. b) Find the annual deposit for the second five years.

A borrows \(\$ 12,000\) for 10 years and agrees to make semiannual payments of \(\$ 1000\) The lender receives \(12 \%\) convertible semiannually on the investment each year for the first 5 years and \(10 \%\) convertible semiannually for the second 5 years. The balance of each payment is invested in a sinking fund earning \(8 \%\) convertible semiannually. Find the amount by which the sinking fund is short of repaying the loan at the end of the 10 years. Answer to the nearest dollar.
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