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Chapter 4: Problem 10

Find the periodic payment \(R\) required to accumulate a sum of \(S\) dollars over \(t\) yr with interest earned at the rate of \(r \% /\) year compounded \(m\) times a year. $$ S=40,000, r=4, t=9, m=4 $$

Short Answer

Expert verified
The periodic payment (R) required to accumulate a sum of $40,000 over 9 years with a 4% interest rate compounded quarterly is approximately \(939.16\).

Step by step solution

01

Future Value of Annuity Formula

The future value of annuity formula is used to calculate the accumulated sum (S) after making a series of periodic payments (R) at the end of each period with a specified interest rate (r) and compounding frequency (m). The formula is: \(S = R \cdot \frac{(1+i)^{mt} - 1}{i}\), where S = accumulated sum R = periodic payment i = periodic interest rate (r/m) t = time duration in years m = compounding frequency per year Step 2: Calculate the periodic interest rate
02

Calculate Periodic Interest Rate

We need to find the periodic interest rate (i) by dividing the annual interest rate (r) by the compounding frequency (m). Here, r = 4% and m = 4. So, \(i = \frac{r}{m} = \frac{0.04}{4} = 0.01\). Step 3: Calculate the future value factor
03

Calculate Future Value Factor

Next, we need to calculate the future value factor (FVF) by using the periodic interest rate (i) and the time duration (t) in the formula: \((1+i)^{mt}\) Here, i = 0.01 and t = 9 years So, mt = 9 脳 4 = 36. So, FVF = \((1 + 0.01)^{36} = 1.425992\). Step 4: Calculate the periodic payment (R)
04

Calculate Periodic Payment

Now, we will use the future value of annuity formula to calculate the periodic payment (R): \( S = R \cdot \frac{(1+i)^{mt} - 1}{i} \) Rearrange the formula to solve for R: \(R = \frac{S \times i}{(1+i)^{mt} - 1}\), Plug in the values: S = $40,000 i = 0.01 FVF = 1.425992 \(R = \frac{40000 \times 0.01}{1.425992 - 1} = \frac{400}{0.425992} = 939.16\). The periodic payment (R) required to accumulate a sum of \(40,000 over 9 years with a 4% interest rate compounded quarterly is approximately \)939.16.

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Key Concepts

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

Periodic Payment Calculation
Understanding how to calculate periodic payments for an annuity is crucial when planning investments or savings. Let's consider you want to save a certain amount of money over a period. Based on the future value you desire, the interest rate available to you, and how often that interest is compounded, you can determine the payments you need to make regularly. In our example, an individual wants to save $40,000 over 9 years with an interest rate of 4% compounded quarterly. To calculate the periodic payment, we must rearrange the future value of an annuity formula to solve for the payment amount (R). This involves deciphering the periodic interest rate and the future value factor鈥攌ey variables in this calculation.
Compound Interest
Compound interest is the phenomenon where the interest you earn also earns interest over time. It's like a snowball effect for your investment or savings account: in addition to earning interest on the principal amount, you also earn interest on the interest that was added previously. This concept is powered by the frequency of compounding; the more frequent the compounding, the greater the accumulation of wealth. In our example with a 4% annual interest rate compounded quarterly, interest is added to the principal four times a year, which accelerates the growth of the investment due to compound interest.
Time Value of Money
The time value of money is a financial concept that represents the idea that money available today is worth more than the same amount in the future due to its potential earning capacity. Essentially, given the choice between receiving a dollar today or a dollar in the future, the rational decision is to take the money now. This principle underpins the idea of compound interest and is the reason why we discount future cash flows to get their present value. It also plays a significant role when calculating the future value of annuity, because the annuity formula takes into consideration not just the amount of periodic payments, but also the time period over which the payments are made and interest is accumulated.
Annuity Formula
The annuity formula allows us to calculate the future value of a series of equal payments made at regular intervals. The formula incorporates several factors: the size of the periodic payments, the interest rate per period, and the total number of payments. When interest is compounded, each payment itself starts to earn interest, which is why the annuity formula must account for compounding to give you the correct future value of your annuity. In context, once we acquire the periodic interest rate and apply it to the number of periods the money is compounded, we can use the annuity formula to find out our required periodic payment or the future value of the payments we plan to make.

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Most popular questions from this chapter

A young man is the beneficiary of a trust fund established for him 21 yr ago at his birth. If the original amount placed in trust was $$\$ 10,000$$, how much will he receive if the money has earned interest at the rate of \(8 \% /\) year compounded annually? Compounded quarterly? Compounded monthly?

Four years ago, Emily secured a bank loan of $$\$ 200,000$$ to help finance the purchase of an apartment in Boston. The term of the mortgage is \(30 \mathrm{yr}\), and the interest rate is \(9.5 \% /\) year compounded monthly. Because the interest rate for a conventional 30 -yr home mortgage has now dropped to \(6.75 \%\) /year compounded monthly, Emily is thinking of refinancing her property. a. What is Emily's current monthly mortgage payment? b. What is Emily's current outstanding principal? c. If Emily decides to refinance her property by securing a 30 -yr home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 6.75\%/year compounded monthly, what will be her monthly mortgage payment? d. How much less would Emily's monthly mortgage payment be if she refinances?

The inflation rates in the U.S. economy for 2003 through 2006 are \(1.6 \%, 2.3 \%, 2.7 \%\), and \(3.4 \%\), respectively. What was the purchasing power of a dollar at the beginning of 2007 compared to that at the beginning of \(2003 ?\)

Maxwell started a home theater business in 2005 . The revenue of his company for that year was $$\$ 240,000$$. The revenue grew by \(20 \%\) in 2006 and by \(30 \%\) in 2007 . Maxwell projected that the revenue growth for his company in the next 3 yr will be at least \(25 \% /\) year. How much does Maxwell expect his minimum revenue to be for \(2010 ?\)

Steven purchased 1000 shares of a certain stock for $$\$ 25,250$$ (including commissions). He sold the shares 2 yr later and received $$\$ 32,100$$ after deducting commissions. Find the effective annual rate of return on his investment over the 2 -yr period.
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