Problem 18
Question
Suppose payments will be made for \(9 \frac{1}{4}\) yr at the end of each month into an ordinary annuity earning interest at the rate of \(6.25 \% /\) year compounded monthly. If the present value of the annuity is $$\$ 42,000$$, what should be the size of each payment?
Step-by-Step Solution
Verified Answer
The size of each monthly payment for the ordinary annuity should be approximately \$400.98.
1Step 1: Understand the variables and formula
In this problem, we are given the present value (PV) of the annuity, which is \$42,000, and we need to find the annuity payment (PMT). The interest rate per year is 6.25%, and it is compounded monthly, so we need to find the monthly interest rate (i). The term of the annuity is 9 and 1/4 years, so we need to find the total number of payments (n).
The present value of the annuity formula is:
PV = PMT × (1 - (1 + i)^(-n)) / i
We can use this formula to solve for PMT.
2Step 2: Convert the annual interest rate to a monthly rate
The annual interest rate is 6.25%, so to find the monthly rate (i), we divide it by 12 (the number of months in a year):
i = 0.0625 / 12
i ≈ 0.00520833
3Step 3: Calculate the total number of payments
The term of the annuity is 9 and 1/4 years, so we need to multiply this by 12 (the number of months in a year) to find the total number of payments (n):
n = (9 + 1/4) × 12
n = 9.25 × 12
n = 111
4Step 4: Rearrange the formula and solve for PMT
Now, we can rearrange the present value of annuity formula and plug in the values we found for PV, i, and n:
PMT = PV × i / (1 - (1 + i)^(-n))
PMT = 42000 × 0.00520833 / (1 - (1 + 0.00520833)^(-111))
PMT ≈ 42000 × 0.00520833 / (1 - 0.453127)
PMT ≈ 42000 × 0.00520833 / 0.546873
PMT ≈ 400.982
5Step 5: Round and state the final answer
We can round the result to the nearest cent to get the size of each payment:
PMT ≈ \$400.98
So, each payment should be approximately \$400.98 to meet the given present value of the annuity.
Key Concepts
Present ValueCompounded InterestAnnuity Payment Formula
Present Value
Present value is a financial concept that helps determine how much a series of future payments is worth today.
In our scenario, the present value is $42,000. This is the current worth of the annuity.
The annuity will distribute payments over a future period — in this case, monthly over 9.25 years.
The power of present value lies in its ability to encapsulate these future payments into a single, intuitive figure.
The calculation accounts for interest rates, commonly used to reflect investment growth or inflation reduction.
When we say an annuity has a present value of $42,000, we mean that if you had $42,000 today and invested it at the given interest rate, you would receive exactly equal to all future payments planned through the annuity's time span.
This concept allows investors and planners to compare different financial products easily.
In our scenario, the present value is $42,000. This is the current worth of the annuity.
The annuity will distribute payments over a future period — in this case, monthly over 9.25 years.
The power of present value lies in its ability to encapsulate these future payments into a single, intuitive figure.
The calculation accounts for interest rates, commonly used to reflect investment growth or inflation reduction.
When we say an annuity has a present value of $42,000, we mean that if you had $42,000 today and invested it at the given interest rate, you would receive exactly equal to all future payments planned through the annuity's time span.
This concept allows investors and planners to compare different financial products easily.
Compounded Interest
Compounded interest is a method of calculating interest on both the original amount invested (principal) and any accumulated interest from previous periods.
This means that interest isn't just earned on the initial $42,000, but each period’s newly accumulated total.
In the case of our ordinary annuity, interest is compounded monthly.
To figure this out, we take the annual rate of 6.25% and split it into 12 months, giving us a monthly rate of around 0.520833%.
With compounded interest, your investment can grow more quickly since the interest periodically adds to the principal and itself earns interest over the subsequent periods.
It’s a powerful financial concept used in various long-term investment decisions.
This means that interest isn't just earned on the initial $42,000, but each period’s newly accumulated total.
In the case of our ordinary annuity, interest is compounded monthly.
To figure this out, we take the annual rate of 6.25% and split it into 12 months, giving us a monthly rate of around 0.520833%.
With compounded interest, your investment can grow more quickly since the interest periodically adds to the principal and itself earns interest over the subsequent periods.
It’s a powerful financial concept used in various long-term investment decisions.
- Compounding accelerates growth: As interest accumulates, it leads to a faster increase in the investment balance.
- A critical factor is the number of compounding periods: More frequent periods equate to faster growth.
Annuity Payment Formula
The Annuity Payment Formula is vital in determining how much each payment in an annuity must be.
It helps us reverse-engineer the present value or future value into actual periodic payment amounts, like monthly or annual disbursements.
The formula used in our example is: \[\text{PMT} = \text{PV} \times \frac{i}{1 - (1+i)^{-n}}\]where:
This formula is central to financial calculations, particularly in planning savings, loans, and understanding costs in relation to time.
It helps us reverse-engineer the present value or future value into actual periodic payment amounts, like monthly or annual disbursements.
The formula used in our example is: \[\text{PMT} = \text{PV} \times \frac{i}{1 - (1+i)^{-n}}\]where:
- PMT is the payment per period
- PV is the present value
- i is the interest rate per period
- n is the total number of payments
This formula is central to financial calculations, particularly in planning savings, loans, and understanding costs in relation to time.
Other exercises in this chapter
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