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Chapter 2: Problem 16

Present value of a perpetuity What is the present value of a \(\$ 100\) perpetuity if the interest rate is 7 percent? If interest rates doubled to 14 percent, what would its present value be?

Short Answer

Expert verified
At 7% interest, PV is $1428.57; at 14% interest, PV is $714.29.

Step by step solution

01

Understand the Perpetuity Formula

A perpetuity is a type of annuity that provides ongoing, indefinite payments. The present value (PV) of a perpetuity is calculated using the formula: \( PV = \frac{C}{r} \), where \( C \) is the annual payment and \( r \) is the interest rate.
02

Calculate Present Value at 7% Interest

With an annual payment (C) of $100 and an interest rate (r) of 7% or 0.07, plug these values into the formula: \( PV = \frac{100}{0.07} \). Simplifying gives: \( PV = 1428.57 \).
03

Calculate Present Value at 14% Interest

If the interest rate doubles to 14% or 0.14, substitute into the formula: \( PV = \frac{100}{0.14} \). Solving this results in: \( PV = 714.29 \).
04

Comparison of Present Values

Notice how an increase in the interest rate lowers the present value of the perpetuity, showing the inverse relationship between present value and interest rates.

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Key Concepts

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

Perpetuity Formula
Calculating the present value of a perpetuity involves using a simple but powerful formula. A perpetuity is essentially an annuity that provides indefinite, ongoing payments. The central formula to find its present value is given by:\[ PV = \frac{C}{r} \]Here, \( C \) represents the annual payment, and \( r \) is the interest rate expressed as a decimal. This formula helps us understand how much a stream of endless payments is worth today. Whether you're dealing with perpetual bonds or dividends, the perpetuity formula simplifies complex cash flows into a single present value figure. By dividing the annual payment by the interest rate, you can quickly determine how much money you would need today to produce the specified payment stream indefinitely.
Interest Rate Effect
The effect of the interest rate on the present value of a perpetuity is significant and straightforward to grasp. When the interest rate increases, the present value of the perpetuity decreases. This occurs because a higher interest rate implies that money today is worth more compared to money in the future. Conversely, a lower interest rate increases the present value. For example:
  • At a 7% interest rate, the present value of a $100 perpetuity is approximately $1428.57.
  • If the interest rate rises to 14%, the present value drops to approximately $714.29.
This inverse relationship highlights the critical nature of interest rates in financial decision-making. By understanding this effect, one can better navigate financial scenarios and make more informed investment choices.
Annuity Concept
An annuity involves a series of equal payments at regular intervals over time. Though similar to perpetuities, annuities have a finite duration. Annuities are widely used in financial products like retirement plans and insurance to structure payments over a specific period. Key characteristics of annuities include:
  • Fixed Payment Schedule: Payments occur at regular intervals.
  • Defined Period: Unlike perpetuities, annuities end after a certain number of periods.
In contrast, a perpetuity, as examined in the exercise, provides endless payments. Understanding how annuities function helps in grasping more complex financial structures, as they introduce the concept of time-bound financial planning. Calculating annuities involves slightly different formulas that account for the total number of payments, unlike perpetuities, which focus only on the payment amount and interest rate.

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Most popular questions from this chapter

Amortization schedule with a balloon payment You want to buy a house that costs \(\$ 100,000 .\) You have \(\$ 10,000\) for a down payment, but your credit is such that mortgage companies will not lend you the required \(\$ 90,000\). However, the realtor persuades the seller to take a \(\$ 90,000\) mortgage (called a seller take-back mortgage) at a rate of 7 percent, provided the loan is paid off in full in 3 years. You expect to inherit \(\$ 100,000\) in 3 years, but right now all you have is \(\$ 10,000,\) and you can only afford to make payments of no more than \(\$ 7,500\) per year given your salary. (The loan would really call for monthly payments, but assume end-of-year annual payments to simplify things.) a. If the loan were amortized over 3 years, how large would each annual payment be? Could you afford those payments? b. If the loan were amortized over 30 years, what would each payment be, and could you afford those payments? c. \(\quad\) To satisfy the seller, the 30 -year mortgage loan would be written as a "balloon note," which means that at the end of the 3 rd year you would have to make the regular payment plus the remaining balance on the loan. What would the loan balance be at the end of Year \(3,\) and what would the balloon payment be?

Reaching a financial goal Six years from today you need \(\$ 10,000\). You plan to deposit \(\$ 1,500\) annually, with the first payment to be made a year from today, in an account that pays an 8 percent effective annual rate. Your last deposit will be for less than \(\$ 1,500\) if less is needed to have the \(\$ 10,000\) in 6 years. How large will your last payment be?

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.

Nominal interest rate and extending credit As a jewelry store manager, you want to offer credit, with interest on outstanding balances paid monthly. To carry receivables, you must borrow funds from your bank at a nominal 6 percent, monthly compounding. To offset your overhead, you want to charge your customers an EAR (or EFF\%) that is 2 percent more than the bank is charging you. What APR rate should you charge your customers?

Future value for various compounding periods Find the amount to which \(\$ 500\) will grow under each of these conditions: a. 12 percent compounded annually for 5 years. b. 12 percent compounded semiannually for 5 years. c. 12 percent compounded quarterly for 5 years. d. 12 percent compounded monthly for 5 years. e. 12 percent compounded daily for 5 years. f. Why does the observed pattern of FVs occur?
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