Math 140 - Cooley Business Calculus OCC
Section 5.2 – Applications of Integrating Growth and Decay Models
Definition
If P0 is invested for t years at interest rate k, compounded continuously, then
,
where at t = 0. The value P is called the future value of P0 dollars invested at interest rate k, compounded continuously, for t years.
Definition – Accumulated Future Value of a Continuous Income Stream
Let be a function that represents the rate, per year, of a continuous income stream, let k be the interest rate, compounded continuously, at which the continuous income stream is invested, and let T be the number of years for which the income stream is invested.
Then the accumulated future value of the continuous income stream is given by
.
If is a constant function, it can be factored out of the integral, and the formula becomes, after evaluating and simplifying,
.
If is a nonconstant function, then the second equation above does not apply and the integral in the first equation must be evaluated using some other technique such as integration by parts, tables, a graphing utility, or some other kind of computer software, such as MiniTab.
Definition
The present value, P0, of an amount P due t years later, at interest rate k, compounded continuously, is given
by
.
Definition – Accumulated Present Value of a Continuous Income Stream
Let be a function that represents the rate, per year, of a continuous income stream, let k be the interest rate, compounded continuously, at which the continuous income stream is invested, and let T be the number of years over which income stream is received.
If is a constant function, then B, the accumulated present value of the continuous income stream, is given by
.
If is a nonconstant function, the accumulated present value of the continuous income stream is given by the following integral:
J Exercises:
Find the future value P of each amount P0 invested for time period t at interest rate k, compounded continuously.
1)
2)
Find the present value P0 of each amount P due t years in the future and invested at interest rate k, compounded continuously.
3)
4)
J Exercises:
Find the accumulated future value of each continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously. Round to the nearest $10.
5)
6)
Find the accumulated present value of each continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously.
7)
8)
J Exercises:
9) At age 25, Del earns his CPA and accepts a position in an accounting firm. Del plans to retire at the
age of 65, having received an annual salary of $125,000. Assume an interest rate of 7%, compounded continuously.
a) What is the accumulated present value of his position?
b) What is the accumulated future value of his position?
10) A couple have a new grandchild. They want to create a trust fund for him that will yield $1,000,000
on his 22nd birthday so that he can start his own business when he is out of college.
a) What lump sum would they have to deposit now at 6.2%, compounded continuously, to achieve
$1,000,000?
b) The amount in part (a) is more than they can afford, so they decide to invest a constant money
stream of dollars per year. Find such that the accumulated future value of the
continuous money stream is $1,000,000, assuming an interest rate of 6.2%, compounded
continuously.
J Exercises:
11) Lucky Larry wins $1,000,000 in a state lottery. The standard way in which a state pays such lottery
winnings is at a constant rate of $50,000 per year for 20 yr.
a) If Lucky invests each payment from the state at 7%, compounded continuously, what is the
accumulated future value of the income stream? Round your answer to the nearest $10.
b) What is the accumulated present value of the income stream at 7%, compounded continuously?
This amount represents what the state has to invest at the start of its lottery payments, assuming
the 7% interest rate holds.
c) The risk for Lucky is that he doesn’t know how long he will live or what the future interest rate
will be; it might drop or rise, or it could vary considerably over 20 years. This is the risk he
assumes in accepting payments of $50,000 a year over 20 years. Lucky has taken Mr. Cooley’s,
Math 140 Business Calculus course at Orange Coast College in Costa Mesa, California, so he is
aware of the formulas for accumulated future value and present value. He calculates the
accumulated present value of the income stream for interest rates of 4%, 6%, 8%, and 10%. What
values does he obtain?
d) Lucky thinks “a bird in the hand (present value) is worth two in the bush (future value)” and
decides to negotiate with the state for immediate payment of his lottery winnings. He asks the
state for $600,000. They offer $400,000. Discuss the pros and cons of each amount. Lucky finally
accepts $500,000. Is this a good decision.
- 1 -