Problem 30 - Precalculus Mathematics for Calculus

Question

An annuity in perpetuity is one that continues forever. Such annuities are useful in setting up scholarship funds to ensure that the award continues. (a) Draw a time line (as in Example 1) to show that to set up an annuity in perpetuity of amount \(R\) per time period, the amount that must be invested now is $$A_{p}=\frac{R}{1+i}+\frac{R}{(1+i)^{2}}+\frac{R}{(1+i)^{3}}+\dots+\frac{R}{(1+i)^{n}}+\cdots$$ where \(i\) is the interest rate per time period. (b) Find the sum of the infinite series in part (a) to show that $$A_{p}=\frac{R}{i}$$ (c) How much money must be invested now at \(10 \%\) per year, compounded annually, to provide an annuity in perpetuity of \(\$ 5000\) per year? The first payment is due in one year. (d) How much money must be invested now at \(8 \%\) per year, compounded quarterly, to provide an annuity in perpetuity of \(\$ 3000\) per year? The first payment is due in one year.

Step-by-Step Solution

Answer

(a) Timeline uses discounted payments. (b) Sum is \( A_p = \frac{R}{i} \). (c) Invest \$50,000 at 10%. (d) Invest \$37,500 at 8% quarterly.

Step 1: Drawing the Time Line

The time line of the annuity in perpetuity involves a sequence of payments extending forever. Each payment amount is \( R \). On the time line, these payments are denoted as occurring at times \( t=1, 2, 3, \ldots \). Each payment is discounted back to the present value using the factor \( \frac{1}{(1+i)^n} \), where \( i \) is the interest rate per period, giving us the present value of each payment: \( \frac{R}{(1+i)}, \frac{R}{(1+i)^2}, \ldots \). Summing these present values for all time periods gives the total present value \( A_p \).

Step 2: Finding the Sum of the Infinite Series

The present value \( A_p \) of the annuity can be expressed as an infinite geometric series: \[ A_p = \frac{R}{1+i} + \frac{R}{(1+i)^2} + \frac{R}{(1+i)^3} + \cdots \] This series has a first term \( a = \frac{R}{1+i} \) and a common ratio \( r = \frac{1}{1+i} \). The sum \( S \) of an infinite geometric series is given by \( S = \frac{a}{1-r} \). Substituting the values of \( a \) and \( r \), we get: \[ A_p = \frac{\frac{R}{1+i}}{1 - \frac{1}{1+i}} = \frac{R}{i} \] Thus, the present value \( A_p \) of an annuity in perpetuity is \( \frac{R}{i} \).

Step 3: Calculating Present Value at 10% Annual Interest

To find the present value needed for an annual perpetuity with \( R = \\(5000 \) at an interest rate of \( i = 0.10 \) per year: \[ A_p = \frac{R}{i} = \frac{5000}{0.10} = 50000 \] Thus, \\)50,000 must be invested now to provide a perpetual annuity of \$5000 per year at 10% annual interest.

Step 4: Calculating Present Value at 8% Compounded Quarterly

Given \( R = \\(3000 \) and the nominal annual interest rate \( i_{nom} = 0.08 \), where interest is compounded quarterly. The effective quarterly interest rate is \( i = \frac{0.08}{4} = 0.02 \). For the annual payment, it's cumulative over the four quarters: \[ A_p = \frac{R}{i_{annual}} = \frac{3000}{0.02 \times 4} = \frac{3000}{0.08} = 37500 \] Therefore, \\)37,500 must be invested now to provide a perpetual annuity of \$3000 per year at 8% per annum, compounded quarterly.

Key Concepts

Infinite Series

In the context of annuities in perpetuity, an infinite series represents the ongoing sequence of payments that happen forever. Imagine you start receiving an amount, say \(R, at regular intervals. Now, suppose these payments continue indefinitely. To make mathematical sense of this, we use the concept of a geometric infinite series to find the present value of these never-ending payments.
Consider the series \[ A_p = \frac{R}{1+i} + \frac{R}{(1+i)^2} + \frac{R}{(1+i)^3} + \cdots \]Here, each term of the series represents the present value of a future payment. The first payment of \)R is discounted by the interest rate i, hence the first term is \( \frac{R}{1+i} \). The second payment is discounted for an additional period, giving \( \frac{R}{(1+i)^2} \), and so on. This is called an infinite geometric series because it extends without end and has a constant ratio between terms.
The key takeaway is that although the series appears to be infinite, we can still find a sum—provided that the common ratio is less than one. This condition allows the calculation of a finite present value for the seemingly endless stream of payments. This leads us to the next concept: present value.

Present Value

The present value (PV) of an annuity is an important concept that lets us calculate how much a series of future payments is worth today. To determine this, every future payment is adjusted or 'discounted' according to the interest rate. The formula for the present value of an annuity in perpetuity is\[ A_p = \frac{R}{i} \]where \( A_p \) is the present value, R is the payment amount, and i is the interest rate.
In simpler terms, if you want to receive a perpetual annual payment of $R every year, you need to invest \( \frac{R}{i} \) today. This formula arises from summing the infinite series discussed earlier and using the properties of geometric series. This calculation assumes a constant interest rate and regular payment intervals.
Understanding present value helps you make informed financial decisions, such as determining how much to invest to yield a specific regular payout forever. It's handy for creating endowments or trust funds, where perpetual payment is essential.

Compound Interest

Compound interest is the process by which interest earned on an investment is reinvested to earn additional interest over time. In terms of an annuity in perpetuity, it plays a critical role in determining how much needs to be invested now for a perpetual payment.
The interest rate applied to the investment is typically assumed to compound at regular intervals (annually, quarterly, monthly, etc.). Compounding means that we calculate interest not only on the initial principal but also on any interest accumulated from previous periods. This compounding effect can significantly grow the investment over time.
For example, if the interest is compounded quarterly, the nominal annual interest rate will be divided by four, leading to a quarterly rate. This rate affects the calculation of present value because it changes how much interest is applied in each compounding period, thus adjusting the effective annual rate. Knowing whether the interest is compounded and at what frequency helps to accurately compute how much needs to be invested to receive specific annuity payments indefinitely.
In essence, compound interest leverages the time value of money, making each dollar of interest earned generate additional earnings on its own, further enhancing the sustainability of perpetual annuities.