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Chapter 5: Problem 28

Trust FunDs Carl is the beneficiary of a \(\$ 20,000\) trust fund set up for him by his grandparents. Under the terms of the trust, he is to receive the money over a 5 -yr period in equal installments at the end of each year. If the fund earns interest at the rate of \(9 \% /\) year compounded annually, what amount will he receive each year?

Short Answer

Expert verified
Carl will receive approximately \(\$5142.15\) each year over the 5-year period.

Step by step solution

01

Calculate the annuity factor

An annuity factor is used to determine the equal annual payments required for a given present value, interest rate, and number of years. The annuity factor, denoted as \(AF\), is calculated using the following formula: \(AF = \frac{1-(1+i)^{-n}}{i}\) where \(i\) is the annual interest rate and \(n\) is the number of years. In this problem, the interest rate is \(9 \%\) per year, which can be written as \(i = 0.09\). The number of years is 5, so \(n = 5\). Plugging in the values, we get: \(AF = \frac{1-(1+0.09)^{-5}}{0.09}\)
02

Evaluate the annuity factor

Calculate the value of the annuity factor: \(AF = \frac{1-(1+0.09)^{-5}}{0.09} = \frac{1-(1.09)^{-5}}{0.09} \approx 3.8901\)
03

Calculate the annual payment

Now that we have the annuity factor, we can determine the equal annual payment Carl will receive throughout the 5 years. Use the formula for the present value of an annuity: \(PV = P \times AF\) We are given the present value, \(PV = 20,000\). We want to find the annual payment, or \(P\). Rearrange the formula to solve for \(P\): \(P = \frac{PV}{AF}\) Plugging in the values, we get: \(P = \frac{20,000}{3.8901} \approx \$5142.15\) Therefore, Carl will receive approximately \(\$5142.15\) each year over the 5-year period.

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Key Concepts

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

Annuity
An annuity is a series of equal payments made at regular intervals over a specified period of time. This financial concept is crucial in calculating the distributions from trust funds, like the one Carl receives. In Carl's case, each of these payments is equal, and they happen annually. An annuity can help smoothly distribute funds over time, making it a steady source of income.
In determining the value of these regular payments, you first prepare with parameters like the interest rate and how long the annuity lasts. The trust fund's interest impacts the annual income Carl receives, as it applies each year through the annuity factor equation. It's essential to understand that annuities aim to maintain financial stability over a period and involve calculating present and future values.
Present Value
The Present Value (PV) is the current worth of a sum of money that is set to be received in the future, adjusted for a given interest rate. For Carl's trust fund, the present value is the total starting amount, $20,000. This sum will be divided over several years, considering the increase due to compound interest.
The PV of the annuity takes into account the trust fund's initial value and uses it to compute the equal annual payments Carl receives. It's adjusted using the concept of the time value of money, implying that a sum today is worth more than the same sum in the future due to its earning potential. Understanding present value is essential in transforming accumulated funds from a lump sum to a series of manageable disbursements, like Carl's.
  • The formula rearranges to determine annual payments from PV.
  • Ensures long-term financial planning by realizing current value.
Compound Interest
Compound interest refers to earning interest not only on the initial principal but also on the accumulated interest from previous periods. In Carl's trust fund, the interest compounds annually at 9%. This compounding affects how much money grows in the fund over time.
Every year, the interest gained is reinvested, and the total grows, affecting the annual payout he receives. Compound interest significantly impacts the annuity payments because it increases the total amount available in the future to be divided over the 5-year period.
  • It leads to exponential growth over the years.
  • Essential for understanding the difference between simple and compound interest.
Annual Payments
Annual payments are the regular disbursements Carl will receive from his trust fund. Calculating these payments involves understanding both annuity and present value concepts, mitigated by the annual compounding interest rate.
To find the amount Carl gets each year, we use the annuity formula incorporating the determined annuity factor. The annual payment, approximately $5142.15, reflects equal cash flow, ensuring that Carl evenly utilizes the initial $20,000 over the specified period.
  • These payments are calculated to ensure financial predictability.
  • They reflect careful financial planning, considering interest and future value.

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Most popular questions from this chapter

Suppose payments were made at the end of each quarter into an ordinary annuity earning interest at the rate of \(10 \% /\) year compounded quarterly. If the future value of the annuity after 5 yr is \(\$ 50,000\), what was the size of each payment?

The parents of a 9 -yr-old boy have agreed to deposit \(\$ 10\) in their son's bank account on his 10 th birthday and to double the size of their deposit every year thereafter until his 18 th birthday. a. How much will they have to deposit on his 18 th birthday? b. How much will they have deposited by his 18 th birthday?

An investor purchased a piece of waterfront property. Because of the development of a marina in the vicinity, the market value of the property is expected to increase according to the rule $$ V(t)=80,000 e^{\sqrt{t / 2}} $$ where \(V(t)\) is measured in dollars and \(t\) is the time (in yr) from the present. If the rate of appreciation is expected to be \(9 \%\) compounded continuously for the next 8 yr, find an expression for the present value \(P(t)\) of the property's market price valid for the next 8 yr. What is \(P(t)\) expected to be in 4 yr?

Find the effective rate of interest corresponding to a nominal rate of \(9 \% /\) year compounded annually, semiannually, quarterly, and monthly.

A state lottery commission pays the winner of the "Million Dollar" lottery 20 installments of \(\$ 50,000 /\) year. The commission makes the first payment of \(\$ 50,000\) immediately and the other \(n=19\) payments at the end of each of the next 19 yr. Determine how much money the commission should have in the bank initially to guarantee the payments, assuming that the balance on deposit with the bank earns interest at the rate of \(8 \% /\) year compounded vearly.
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