Problem 23 - Precalculus: Mathematical for Calculus

Question

An Annuity That Lasts Forever 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

Invest $50000 at 10% annually and $36407.77 at 8% quarterly for perpetuities.

Step 1: Draw the Time Line

To represent an annuity in perpetuity, draw a timeline marking each period at which an amount \( R \) is received forever. Label each point on the timeline starting from year 1 onwards, and indicate the cash flows as recurring indefinitely.

Step 2: Setup the Infinite Series

The amount needed to set up an annuity in perpetuity can be represented as an infinite geometric series: \[ A_p = \frac{R}{1+i} + \frac{R}{(1+i)^2} + \frac{R}{(1+i)^3} + \dots \] where \( i \) is the interest rate per period.

Step 3: Derive the Sum of the Infinite Series

Since the series is geometric, we use the formula for the sum of an infinite geometric series, \( S = \frac{a}{1-r} \), where \( a \) is the first term (\( \frac{R}{1+i} \)) and \( r \) is the common ratio (\( \frac{1}{1+i} \)). The series sum becomes:\[ A_p = \frac{\frac{R}{1+i}}{1 - \frac{1}{1+i}} = \frac{R}{i} \]

Step 4: Calculate Investment for 10% Annually Compounded Interest

Using \( i = 0.10 \) and \( R = 5000 \), substitute into \( A_p = \frac{R}{i} \) to get:\[ A_p = \frac{5000}{0.10} = 50000 \] Thus, \$50000 must be invested now.

Step 5: Adjust the Interest Rate for Quarterly Compounding

For \( i = 8\% \) compounded quarterly, find the effective annual interest rate:\( i_{eff} = (1 + \frac{0.08}{4})^4 - 1 \). Calculate to get \( i_{eff} \approx 0.0824 \).

Step 6: Calculate Investment for 8% Quarterly Compounded Interest

With \( i_{eff} \approx 0.0824 \) and \( R = 3000 \), substitute into \( A_p = \frac{R}{i} \):\[ A_p = \frac{3000}{0.0824} \approx 36407.77 \] Thus, approximately \$36407.77 must be invested now.

Key Concepts

Infinite Series

An infinite series is a sum of infinite terms that when added together approach a specific value. In the context of annuities, this infinite series arises from the requirement for a perpetual payout. Each term in the series represents the present value of a future payment. Since these payments continue indefinitely, we have an infinite number of terms in the series. This means the series forms an ongoing sum that represents the present value needed for continuous payments of a certain amount, R, for each period.
Such series are crucial in calculating how much you need to invest now to receive a specific payout forever. The sum allows us to quantify a never-ending sequence of cash flows in a manageable, finite value.

Geometric Series

A geometric series is a series with a constant ratio between successive terms. In financial contexts, such as annuities, the geometric series helps determine the series of payments' present value across time. Each term in this series is a future payment discounted to its present value using the formula \( \frac{R}{(1+i)^n} \), where \( R \) is the recurring payment, \( i \) is the interest rate, and \( n \) is the period number.
The key property of geometric series is convergence, which means as we sum an infinite number of terms with a fixed common ratio, they sum to a finite number. This is especially helpful in perpetual annuities where payments continue infinitely but their present value is represented by a definitive sum: \( A_p = \frac{R}{i} \).
With this sum formula, investors can determine how much to invest to ensure a consistent future cash flow.

Compound Interest

Compound interest refers to earning interest on both the original amount of money invested (principal) and on the interest that accumulates over time. This process is essential to annuities, as it indicates how much the investment grows each period. The formula for compound interest can be expressed as \((1 + i)^n\), where \( i \) is the interest rate, and \( n \) is the number of compounding periods.
In the context of annuities, understanding the effective interest rate, especially when compounding occurs multiple times a year, such as quarterly, is crucial. For instance, an 8% annual interest rate compounded quarterly results in an effective annual interest rate of approximately 8.24%. This effective rate needs to be used when calculating the present value of payments in perpetuity.
Mastering compound interest is vital for determining the accurate present value of an infinite series of future payments.

Investment Strategy

Developing an investment strategy involves determining how much needs to be invested today to achieve future financial goals. When it comes to annuities in perpetuity, the strategy revolves around calculating the precise amount necessary to secure endless future payments, given an interest rate.
The investment equation, \( A_p = \frac{R}{i} \), becomes the linchpin of the strategy, where \( R \) is the desired recurring payment and \( i \) is the interest rate. An accurate calculation ensures that the fund grows adequately to support ongoing payments without depleting the principal.
This approach requires a thorough understanding of interest calculations, effective rates, and future values to craft a plan that captures both cash flow needs and risk management. It's not just about deploying capital but doing so wisely to serve long-term financial objectives sustainably.