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Chapter 2: Problem 25

Future value of an annuity Find the future values of the following ordinary annuities: a. FV of \(\$ 400\) paid each 6 months for 5 years at a nominal rate of 12 percent, compounded semiannually. b. FV of \(\$ 200\) paid each 3 months for 5 years at a nominal rate of 12 percent, compounded quarterly. c. These annuities receive the same amount of cash during the 5 -year period and earn interest at the same nominal rate, yet the annuity in part b ends up larger than the one in part a. Why does this occur?

Short Answer

Expert verified
The future value for part a is $5,272.64 and for part b is $5,374.10. Part b is larger due to more frequent compounding.

Step by step solution

01

Understanding the Problem

We are asked to find the future value of two ordinary annuities with different payment frequencies but the same nominal interest rate (12%) over a period of 5 years. An annuity is a series of equal payments made at regular intervals.
02

Identify the Formula

The future value of an ordinary annuity can be calculated using the formula: \( FV = P \frac{(1 + i)^n - 1}{i} \), where \( P \) is the payment, \( i \) is the interest rate per period, and \( n \) is the total number of periods.
03

Calculate Future Value for Part a

For part a, payments are \( \\(400 \) every six months, with a nominal interest rate of 12% compounded semiannually. \[ n = 5 \times 2 = 10 \text{ periods, } i = \frac{12\%}{2} = 6\% = 0.06 \]The future value is calculated as: \[ FV = 400 \times \frac{(1 + 0.06)^{10} - 1}{0.06} \approx 400 \times 13.1816 = \\)5,272.64 \]
04

Calculate Future Value for Part b

For part b, payments are \( \\(200 \) every quarter, with a nominal interest rate of 12% compounded quarterly. \[ n = 5 \times 4 = 20 \text{ periods, } i = \frac{12\%}{4} = 3\% = 0.03 \]The future value is calculated as: \[ FV = 200 \times \frac{(1 + 0.03)^{20} - 1}{0.03} \approx 200 \times 26.8705 = \\)5,374.10 \]
05

Explanation of Different Outcomes for Part c

Parts a and b involve annuities with the same annual nominal rate, but different compounding periods resulting in different future values. Annuity in part b grows larger than part a due to more frequent compounding (quarterly compared to semiannually), leading to interest being added to the principal more often, which increases the total future value.

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Key Concepts

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

Compound Interest
Compound interest is a key concept to understand when calculating the future value of annuities. Unlike simple interest, compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means the interest grows faster than with simple interest.

The formula for compound interest is:
  • \( A = P(1 + r/n)^{nt} \), where
    • \( A \) is the amount of money accumulated after n years, including interest.
    • \( P \) is the principal amount (the initial deposit or loan).
    • \( r \) is the annual interest rate (decimal).
    • \( n \) is the number of times that interest is compounded per unit year.
    • \( t \) is the time the money is invested or borrowed for, in years.

In the context of annuities, each payment is subject to compound interest, which can increase the future value substantially depending on how often interest is compounded.
Annuity Formula
When you want to calculate the future value of an annuity, you rely on the annuity formula. This formula helps to determine how much a series of regular payments will accumulate to in the future, when interest is also taken into account.

For an ordinary annuity, which is a series of equal payments made at regular intervals at the end of each period, the future value is calculated with this formula:
  • \( FV = P \frac{(1 + i)^n - 1}{i} \)
    • \( FV \) is the future value of the annuity.
    • \( P \) is the payment amount per period.
    • \( i \) is the interest rate per period.
    • \( n \) is the total number of payment periods.

The formula takes into account the effect of compound interest over multiple periods to give an accurate prediction of the annuity's future value.
Ordinary Annuity Calculation
An ordinary annuity is characterized by its regular payments made at the end of each period. Understanding the calculation of an ordinary annuity's future value involves knowing the formula and its components.

Using our step-by-step exercise as a guide:
  • Determine the payment amount (\( P \)).
  • Identify the interest rate per compounding period (\( i \)).
  • Calculate the total number of periods (\( n \)).
  • Plug these values into the annuity formula: \( FV = P \frac{(1 + i)^n - 1}{i} \).

The difference in future values of two annuities with different compounding frequencies (as seen in parts a and b) emphasizes the significance of calculating properly. Payments being more frequently compounded tend to grow faster due to interest compounding more often.
Interest Compounding Periods
The frequency of compounding periods significantly affects the future value of an annuity. In the given exercise, the number of times interest is compounded annually plays a crucial role in determining the annuity鈥檚 growth.

There are different compounding intervals, such as annually, semiannually, quarterly, monthly, etc. For any of these periods, the effect of compound interest is as follows:
  • The more frequently the interest is compounded, the higher the future value of the annuity will be.
  • Semiannual compounding results in two interest periods per year.
  • Quarterly compounding divides the year into four periods, allowing the interest to be applied more frequently.

As demonstrated in the exercise, even with the same annual nominal rate, the annuity compounded quarterly (part b) grows to a larger future value than the semiannually compounded annuity (part a) because of more frequent interest calculations. This underscores the importance of considering the compounding frequency when planning investment strategies.

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Most popular questions from this chapter

Building credit cost into prices Your firm sells for cash only, but it is thinking of offering credit, allowing customers 90 days to pay. Customers understand the time value of money, so they would all wait and pay on the 90 th day. To carry these receivables, you would have to borrow funds from your bank at a nominal 12 percent, daily compounding based on a 360 -day year. You want to increase your base prices by exactly enough to offset your bank interest cost. To the closest whole percentage point, by how much should you raise your product prices?

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.

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