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Chapter 10: Problem 5

What is the present value of an annuity that pays \(\$ 20,000\) each year, forever, starting today, from an account that pays \(1 \%\) interest per year, compounded annually?

Short Answer

Expert verified
The present value of the annuity is \( \$2,020,000 \).

Step by step solution

01

Identify the Type of Annuity

This problem involves a perpetuity because the payments continue forever. It is specifically a perpetuity due today since the first payment is made immediately.
02

Determine the Present Value Formula for a Perpetuity

The present value formula for a perpetuity due is given by \( PV = C + \frac{C}{r} \), where \( C \) is the annual payment and \( r \) is the interest rate.
03

Substitute the Given Values into the Formula

Plug in \( C = 20000 \) and \( r = 0.01 \) into the formula. This gives \( PV = 20000 + \frac{20000}{0.01} \).
04

Calculate the Present Value

First, calculate the infinite series part: \( \frac{20000}{0.01} = 2000000 \). Then, add the initial payment to this: \( 20000 + 2000000 = 2020000 \). The present value of the annuity is \( \$2,020,000 \).

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Key Concepts

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

Perpetuity Due Formula
A perpetuity is a type of annuity that provides an infinite series of cash flows. With a perpetuity due, payments are made at the start of each period, beginning immediately.
To find the present value of a perpetuity due, use the formula: \[PV = C + \frac{C}{r}\] where:
  • \(PV\) is the present value you're calculating,
  • \(C\) is the cash flow the annuity provides annually, and
  • \(r\) is the interest rate per period.
The formula slightly adjusts the basic perpetuity formula to account for the payment made at the start of the period, hence the addition of \(C\) to the fraction term.
This results in the immediate payment being included directly in the present value calculation, in addition to the sum of all future payments adjusted to present terms.
Compounded Interest
Compounded interest is when interest is added to the principal, and then in future periods, interest is calculated on this new principal amount that includes the previous interest gained.
This can lead to exponential growth of an investment over time, as each period's interest grows from a progressively larger base.
When the problem mentions an interest rate compounded annually, it means the interest is calculated and added to the principal once every year.
  • This rate impacts the growth of the investment and the calculation of present value, as the rate determines the discount factor \(r\) used in perpetuity formulas.
  • In the original exercise, the compounded rate is 1% annually, which is a fairly low rate, which keeps the present value calculations more modest.
Annuity Payment
An annuity payment is a regular, fixed payment made or received over a specified period. For perpetuities, like in the exercise, these payments continue indefinitely.
In the exercise, the annuity payment is specified as \$20,000 annually. This cash flow recurring every year translates into the \(C\) variable within the perpetuity due formula.
Understanding the difference between types of annuities is key:
  • Ordinary annuities pay at the end of periods.
  • Annuities due make payments at the start of periods, as seen here.
Mastering when and how these payments occur is crucial for plugging the correct values into financial formulas and accurately determining present values or future balances.

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Most popular questions from this chapter

A cigarette puts 1.2 mg of nicotine into the body. Nicotine leaves the body at a continuous rate of \(34.65 \%\) per hour, but more than 60 mg can be lethal. If a person smokes a cigarette with each of the following frequencies, find the long-run quantity of nicotine in the body right after a cigarette. Does the nicotine reach the lethal level? (a) Every hour (b) Every half hour (c) Every 15 minutes (d) Every 6 minutes (e) Every 3 minutes

Every month, \(\$ 500\) is deposited into an account earning \(0.1 \%\) interest a month, compounded monthly. (a) How much is in the account right after the \(6^{\text {th }}\) deposit? Right before the \(6^{\text {th }}\) deposit? (b) How much is in the account right after the \(12^{\text {th }}\) deposit? Right before the \(12^{\text {th }}\) deposit?

(a) An allergy drug with a half-life of 18 weeks is given in 100 -mg doses once a week. At the steady state, find the quantity of the drug in the body right after a dose. (b) The drug does not become effective until the quantity in the body right after a dose reaches 2000 mg. How many weeks after the first dose does the drug become effective?

A dose, \(D,\) of a drug is taken at regular time intervals, and a fraction \(r\) remains after one time interval. Show that at the steady state, the quantity of the drug excreted between doses equals the dose.

Are about bonds, which are issued by a government to raise money. An individual who buys a \(\$ 1000\) bond gives the government \(\$ 1000\) and in return receives a fixed sum of money, called the coupon, every six months or every year for the life of the bond. At the time of the last coupon, the individual also gets back the \(\$ 1000\), or principal. What is the present value of a \(\$ 1000\) bond which pays 550 a year for 10 years, starting one year from now? Assume the interest rate is \(6 \%\) per year, compounded annually.
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