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Chapter 4: Problem 72

Ordinary Annuities and Annuities Due As discussed in the text, an annuity due is identical to an ordinary annuity except that the periodic payments occur at the beginning of each period and not at the end of the period. Show that the relationship between the value of an ordinary annuity and the value of an otherwise equivalent annuity due is: Annuity due value \(=\) Ordinary annuity value \(\times(1+r)\) Show this for both present and future values.

Short Answer

Expert verified
The relationship between the value of an ordinary annuity and the value of an otherwise equivalent annuity due is given by the equation: Annuity Due Value \(=\) Ordinary Annuity Value \(\times(1+r)\). This holds for both present and future values, as demonstrated by the following formulas: For Present Values: \(PV_{ad} = PV_{oa} \times (1+r)\) For Future Values: \(FV_{ad} = FV_{oa} \times (1+r)\)

Step by step solution

01

Derive the Present Value of Ordinary Annuity Formula

Using the formula for the present value of a series of cash flows, we find that the present value of an ordinary annuity is given by: \[PV_{oa} = P \times \frac{1-\left(1+r\right)^{-n}}{r}\] Where: \(PV_{oa}\) = Present Value of an Ordinary Annuity \(P\) = Periodic Payment \(r\) = Interest Rate per Period \(n\) = Number of Periods Annuity Due:
02

Derive the Present Value of Annuity Due Formula

Since each payment in an annuity due occurs one period earlier than in an ordinary annuity, we can calculate the present value of an annuity due (\(PV_{ad}\)) by multiplying the present value of an ordinary annuity by the interest rate factor \((1+r)\): \[PV_{ad} = PV_{oa} \times (1+r)\] Now let's move on the future value of an ordinary annuity and an annuity due. Ordinary Annuity:
03

Derive the Future Value of Ordinary Annuity Formula

Using the formula for the future value of a series of cash flows, we find that the future value of an ordinary annuity is given by: \[FV_{oa} = P \times \frac{\left(1+r\right)^{n} - 1}{r}\] Where: \(FV_{oa}\) = Future Value of an Ordinary Annuity \(P\) = Periodic Payment \(r\) = Interest Rate per Period \(n\) = Number of Periods Annuity Due:
04

Derive the Future Value of Annuity Due Formula

As in the present value case, we can calculate the future value of an annuity due (\(FV_{ad}\)) by multiplying the future value of an ordinary annuity by the interest rate factor \((1+r)\): \[FV_{ad} = FV_{oa} \times (1+r)\]
05

Demonstrate the Relationship Between Ordinary Annuities and Annuities Due Values

Now, we have shown that both the present and future values of an annuity due can be calculated by multiplying the respective values of an ordinary annuity by the interest rate factor \((1+r)\): For Present Values: \[PV_{ad} = PV_{oa} \times (1+r)\] For Future Values: \[FV_{ad} = FV_{oa} \times (1+r)\] This confirms that the relationship between the value of an ordinary annuity and the value of an otherwise equivalent annuity due is given by the equation: Annuity Due Value \(=\) Ordinary Annuity Value \(\times(1+r)\) and holds for both present and future values.

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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 is a financial product where equal cash flows occur at the end of each specified period. Imagine you have to pay a car loan, monthly, for five years. Each payment you make happens at the month's end, making it an ordinary annuity.
Think of it like a routine where the money leaves your account regularly after experiencing a full earning cycle, such as after receiving your salary.
Ordinary annuities are common in various contexts, including loans, investments, and retirement planning.
  • This steady flow of outflows or inflows helps in planning and mapping out future expenses or savings.
  • Because the payments are due at each period's end, they allow more time for the amount to accrue interest.
Annuity Due
An annuity due shifts the paradigm by making payments happen at the start of each period. Picture paying your rent as soon as the month kicks off. This upfront payment is a prime example of an annuity due.
The key differentiator here is the timing of cash flows, which leads to a distinct difference in valuation compared to ordinary annuities.
Why does this matter?
  • Since payments are made sooner, each payment has an extra period to earn interest in both present and future value computations.
  • The impact is financially significant鈥攊t effectively increases the overall value compared to an ordinary annuity by a factor of \(1+r\).
Practically, annuities due are seen in scenarios like insurance premiums paid in advance, or lease agreements.
Present Value Formula
The Present Value (PV) formula is a financial calculation used to determine the current worth of a series of future payments. For an ordinary annuity, the PV formula is: \[PV_{oa} = P \times \frac{1-(1+r)^{-n}}{r}\] Here, you adjust future payments to understand what they are worth today, considering a specific interest rate.
For an annuity due, this formula slightly tweaks by multiplying the ordinary annuity's PV by \(1+r\), accounting for the time advantage. \[PV_{ad} = PV_{oa} \times (1+r)\]
  • The \(P\) represents periodic payment; \(r\) stands for interest rates; and \(n\) marks the number of periods.
  • Using this formula, individuals can evaluate decisions like investments or loans, where the timing of cash flows is vital.
Future Value Formula
The Future Value (FV) formula enables the calculation of how much a series of payments will grow in the future, assuming a constant interest rate. For an ordinary annuity, it is given by: \[FV_{oa} = P \times \frac{(1+r)^n - 1}{r}\] This formula projects how much money you鈥檒l have from regular investments over time.
To adjust for an annuity due, again, we multiply by \(1+r\), reflecting the payments鈥� extended time to earn interest. \[FV_{ad} = FV_{oa} \times (1+r)\]
  • This approach is crucial for understanding how saving habits contribute to long-term financial goals.
  • It provides clarity on the compounding effect of money over several periods based on initial and ongoing contributions.

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Most popular questions from this chapter

Calculating Annuity Values After deciding to buy a new car, you can either lease the car or purchase it with a three-year loan. The car you wish to buy costs \(\$ 38,000\). The dealer has a special leasing arrangement where you pay \(\$ 1\) today and \(\$ 520\) per month for the next three years. If you purchase the car, you will pay it off in monthly payments over the next three years at an 8 percent APR. You believe that you will be able to sell the car for \(\$ 26,000\) in three years. Should you buy or lease the car? What break-even resale price in three years would make you indifferent between buying and leasing?

Calculating Annuity Present Value An investment offers \(\$ 4,300\) per year for 15 years, with the first payment occurring one year from now. If the required return is 9 percent, what is the value of the investment? What would the value be if the payments occurred for 40 years? For \(\mathbf{7 5}\) years? Forever?

Present Value and Multiple Cash Flows Conoly Co. has identified an investment project with the following cash flows. If the discount rate is 10 percent, what is the present value of these cash flows? What is the present value at 18 percent? At 24 percent?

Simple Interest versus Compound Interest First City Bank pays 9 percent simple interest on its savings account balances, whereas Second City Bank pays 9 percent interest compounded annually. If you made a \(\$ 5,000\) deposit in each bank, how much more money would you earn from your Second City Bank account at the end of 10 years?

Calculating Annuity Values An All-Pro defensive lineman is in contract negotiations. The team has offered the following salary structure: All salaries are to be paid in a lump sum. The player has asked you as his agent to renegotiate the terms. He wants a \(\$ 9\) million signing bonus payable today and a contract value increase of \(\mathbf{\$ 7 5 0 , 0 0 0}\). He also wants an equal salary paid every three months, with the first paycheck three months from now. If the interest rate is 5 percent compounded daily, what is the amount of his quarterly check? Assume \(\mathbf{3 6 5}\) days in a year.
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