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Chapter 2: Problem 6

Future value: annuity versus annuity due What's the future value of a 7 percent, 5 -year ordinary annuity that pays \(\$ 300\) each year? If this were an annuity due, what would its future value be?

Short Answer

Expert verified
The future value of the ordinary annuity is $1,725.21; for the annuity due, it's $1,846.00.

Step by step solution

01

Understand Ordinary Annuity

An ordinary annuity is a series of equal payments made at the end of each period. For this problem, we have an ordinary annuity with payments of $300 per year, an interest rate of 7%, and a duration of 5 years.
02

Use Future Value of Ordinary Annuity Formula

The future value of an ordinary annuity is given by the formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where \( P = 300 \), \( r = 0.07 \), and \( n = 5 \).
03

Calculate Future Value of Ordinary Annuity

Substitute the given values into the formula: \[ FV = 300 \times \frac{(1 + 0.07)^5 - 1}{0.07} \]. First calculate \((1 + 0.07)^5\): \((1.07)^5 = 1.40255\). Then substitute back into the equation: \[ FV = 300 \times \frac{1.40255 - 1}{0.07} \].
04

Simplify the Equation for Ordinary Annuity

Continue to solve: \(FV = 300 \times \frac{0.40255}{0.07} \approx 300 \times 5.75071 \). Thus, \( FV \approx 1,725.213 \). The future value of the ordinary annuity is approximately $1,725.21.
05

Understand Annuity Due

An annuity due is a series of equal payments made at the beginning of each period. It differs from an ordinary annuity in that each payment is made one period earlier.
06

Calculate Future Value of Annuity Due

The future value of an annuity due is calculated by modifying the ordinary annuity formula as follows: \[ FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + r) \]. Use the previously calculated \(FV_{\text{ordinary}} = 1,725.21\) and \(r = 0.07\).
07

Simplify the Equation for Annuity Due

Substitute the known values: \[ FV_{\text{due}} = 1,725.21 \times 1.07 \approx 1,845.9757 \]. Round to nearest cent: \( FV_{\text{due}} \approx 1,846.00 \). Therefore, the future value of the annuity due is approximately $1,846.00.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordinary Annuity
An ordinary annuity consists of a sequence of equal cash flows made at the end of each period. Imagine paying for a subscription service that you renew at the end of every month. In our original exercise, the situation is similar where you invest \(300 annually, but these payments are recurrently made at the end of each year.
This type of annuity assumes that each payment starts at the end of the first period, thus granting one less cycle of interest compounding compared to its counterpart, the annuity due.
To compute the future value of an ordinary annuity, you use the formula:
  • \[FV = P \times \frac{(1 + r)^n - 1}{r}\]
Here:
  • \( P \) is the payment amount, \)300 in our case.
  • \( r \) is the interest rate, which is 7%, or 0.07 in decimal form.
  • \( n \) is the number of periods, which is 5 years for us.
The calculations showed the future value of this ordinary annuity to be approximately $1,725.21.
Annuity Due
An annuity due shifts the cash flow timing forward so that payments occur at the beginning of each period. Imagine those subscription payments are now due at the beginning of every month.
This creates a series of payments that experience an extra period of interest growth, compared to an ordinary annuity. Thus, an annuity due generally results in a higher future value.
We find the future value by adjusting the ordinary annuity's result with a simple tweak:
  • \[FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + r)\]
For our example, where the ordinary annuity's future value is \(1,725.21 and the interest rate is 7%, the annuity due's future value was calculated as approximately \)1,846.00.
Financial Formulas
Understanding financial formulas is crucial when dealing with investments and savings. They help in making informed decisions and projections for future savings and investments.
The formulas for both the future value of ordinary annuities and annuities due are key financial tools. They project how much a series of periodic investments will be worth after a certain number of periods, considering a fixed interest rate.
These equations involve basic algebra and arithmetic operations, and they rely heavily on understanding the effect of time and interest on the value of money.
When utilizing these formulas, remember to:
  • Convert percentages to decimal form by dividing by 100.
  • Follow the order of operations (parentheses, exponents, multiplication/division, addition/subtraction).
  • Round your final answer appropriately, especially if dealing with currency.
Mastering these calculations equips you with the ability to assess various financial scenarios with confidence.

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Most popular questions from this chapter

Effective rate of interest Find the interest rates earned on each of the following: a. You borrore \(\$ 700\) and promise to pay back \(\$ 749\) at the end of 1 year. b. You lend \(\$ 700\) and the borrower promises to pay you \(\$ 749\) at the end of 1 year. c. You borrow \(\$ 85,000\) and promise to pay back \(\$ 201,229\) at the end of 10 years. d. You borrow \(\$ 9,000\) and promise to make payments of \(\$ 2,684.80\) at the end of each year for 5 years.

Present value of an annuity Find the present ralues of these ordinary anmuities. Discounting occurs once a year. a. \(\quad \$ 400\) per year for 10 years at 10 percent. b. \(\$ 200\) per year for 5 years at 5 percent. c. \(\$ 400\) per year for 5 years at 0 percent. d. Rework parts a, b, and c assuming that they are annuities due.

Time value of money Answer the following questions: a. Find the FV of \(\$ 1,000\) after 5 years earning a rate of 10 percent annually. b. What would the investment's FV be at rates of 0 percent, 5 percent, and 20 percent after \(0,1,2,3,4,\) and 5 years? c. Find the PV of \(\$ 1,000\) due in 5 years if the discount rate is 10 percent. d. What is the rate of return on a security that costs \(\$ 1,000\) and returns \(\$ 2,000\) after 5 years? e. Suppose California's population is 30 million people, and its population is expected to grow by 2 percent annually. How long would it take for the population to double? f. Find the PV of an ordinary annuity that pays \(\$ 1,000\) each of the next 5 years if the interest rate is 15 percent. What is the annuity's FV? g. How would the \(P V\) and \(F V\) of the above annuity change if it were an annuity due? h. What would the \(\mathrm{FV}\) and the \(\mathrm{PV}\) be for \(\$ 1,000\) due in 5 years if the interest rate were 10 percent, semiannual compounding? i. What would the annual payments be for an ordinary annuity for 10 years with a PV of \(\$ 1,000\) if the interest rate were 8 percent? What would the payments be if this were an annuity due? j. Find the PV and the FV of an investment that pays 8 percent annually and makes the following end-of-year payments: $$\begin{array}{cccc} 0 & 1 & 2 & 3 \\ \hline & \$ 100 & \$ 200 & \$ 400 \end{array}$$ k. Five banks offer nominal rates of 6 percent on deposits, but A pays interest annually, B pays semiannually, C quarterly, D monthly, and E daily. (1) What effective annual rate does each bank pay? If you deposited \(\$ 5,000\) in each bank today, how much would you have at the end of 1 year? 2 years? (2) If the banks were all insured by the government (the FDIC) and thus equally risky, would they be equally able to attract funds? If not, and the TVM were the only consideration, what nowtinal nate would cause all the banks to provide the same effective annual rate as Bank A? (3) Suppose you don't have the \(\$ 5,000\) but need it at the end of 1 year. You plan to make a series of deposits, annually for A, semiannually for B, quarterly for \(C\), monthly for \(D,\) and daily for \(E,\) with payments beginning today. How large must the payments be to each bank? (4) Even if the 5 banks provided the same effective annual rate, would a rational investor be indifferent between the banks? 1\. Suppose you borrowed \(\$ 15,000\). The loan's annual interest rate is 8 percent, and it requires 4 equal end-of-year payments. Set up an amortization schedule that shows the annual payments, interest payments, principal repayments, and beginning and ending loan balances.

Nonannual compounding a. You plan to make 5 deposits of \(\$ 1,000\) each, one every 6 months, with the first payment being made in 6 months. You will then make no more deposits. If the bank pays 4 percent nominal interest, compounded semiannually, how much would be in your account after 3 years? b. One year from today you must make a payment of \(\$ 10,000\). To prepare for this payment, you plan to make 2 equal quarterly deposits, in 3 and 6 months, in a bank that pays 4 percent nominal interest, compounded quarterly. How large must each of the 2 payments be?

Amortization schedule a. Set up an amortization schedule for a \(\$ 25,000\) loan to be repaid in equal installments at the end of each of the next 3 years. The interest rate is 10 percent, compounded annually. b. What percentage of the payment represents interest and what percentage represents principal for each of the 3 years? Why do these percentages change over time?
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