Applications of Exponential Functions in Finance by Tqurg9

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Exponential Applications -Finance (Notes)

There are two important functions that financial institutions use to calculate the balance of their clients’

                                              F rI
                                         A  P 1
                                                       nt

accounts. The first of these is:            o
                                              H nK          where A is the amount when Po dollars is deposited at an

interest rate of r compounded n times per year for t years.

Example 1                                                         Example 2

How much does a person have in their account                      How much does a person need to deposit in their
after 5 years if they deposited $2000 at an                       account so that after 6 years they have $5000 if
interest rate of 2.3% compounded monthly?                         the account pays an interest rate of 4%
                                                                  compounded daily?
Po=2000, t=5, r=0.023, n=12 (monthly). Find A.
                                                                  A=5000, t=6, r=0.04, n=365 (daily) Find Po.
        F .023I
A  2000 1 
                           ( 12 )( 5 )


        H 12 K                                                             F .04 I
                                                                  5000  P 1.
                                                                                             ( 365 )( 6 )




A  2000a .00192f
        1
                             ( 60 )                                        H 365K
                                                                              o



                                                                  5000  P a .0010959f
                                                                           1
                                                                                                       ( 2190 )

A  2000a
        100192f
                         ( 60 )                                               o
         .
                                                                  5000  P a
                                                                           10010959f
                                                                                                   ( 2190 )
                                                                            .
A  2000(112175)
          .                                                                   o


A  224350
        .                                                         5000  Po (127123)
                                                                              .
                                                                     1                         1
                                                                         5000  Po (127123)
                                                                                     .
                                                                   .
                                                                  127123                     .
                                                                                            127123
                                                                  393319  Po
                                                                       .

The second common function is the one financial institutions use to compound interest continuously. It
          rt
is: A  Pe

Example 3                                                         Example 4
How much does a person have in their account                      How much will a person need to deposit in their
after 5 years if they deposited $2000 at an                       account so that after 6 years they have $5000 if
interest rate of 2.3% compounded continuously?                    the account pays an interest rate of 4%
P=2000, t=5, r=0.023, Find A.                                     compounded continuously?
                                                                  A=5000, t=6, r=0.04, Find Po.
A  2000e 0.023( 5)
                                                                  5000  Pe.24
A  2000e.115
                                                                  5000  P(127125)
                                                                            .
A  2000(112187)
          .
                                                                     1                       1
A  2243.75                                                              5000  P(127125)
                                                                                   .
                                                                   .
                                                                  127125                   .
                                                                                          127125
                                                                  393314  P
                                                                       .

SATEC/Algebra II/Exponential Functions/6b525301-4602-4ed3-843b-bf4b90150343.doc/Rev. 07-01                        Page 1/3
A businesswoman wishes to invest $25,000 at 6% interest at your bank which compounds
interest monthly. She wants to know how many years it will take to double her money at
this interest rate. The function for the amount in an account after “t” years is:
         F r I
A(t )  PG J where
                   nt


         H nK
         1

        A(t) is the amount in the account after t years
        P is the principal invested
        r is the interest rate expressed as a decimal
        n is the number of compounding periods in a year

Find the length of time in years it will take for the account to reach $50,000.
(Hint: use logarithms and their properties to solve.) SHOW ALL WORK!




The NCAA holds a championship basketball tournament each spring. The nation’s top 64
teams in Division 1 are invited to play. When a team loses, it is out of the tournament.
Determine the number of teams left in the tournament after round 5. Write an equation
in the form y=abx to solve this. (Hint: a is the initial number of teams, and b is the rate
at which they are being eliminated.) Be sure to show all work.




A biologist is observing a strain of bacteria growing in a petri dish. The population of the
bacteria increases exponentially over time. If the 1st bacteria took 1 hour to divide into
2 bacteria, how long will it take for 30 bacteria in the disk to grow to at least 1000?
Make sure your write down your equation and show all work.




SATEC/Algebra II/Exponential Functions/6b525301-4602-4ed3-843b-bf4b90150343.doc/Rev. 07-01   Page 2/3
Exponential Applications -Finance

Answer the following, showing all your work on a separate piece of paper.

1. Jackie is buying a house. The amount                           5. Elsa has her eye on a $14,450 car.
   of their loan for the house is                                    She plans to take out a loan for the
   $108,000. The loan Jackie will take                               car. The Dealer offers her 8.4%
   out charges 5.55% annual interest. If                             interest rate compounded monthly.
   the loan is for 15 years, how much will                           How much will she save if she takes
   the monthly payments be?                                          out a 4-year loan as opposed to a 5-
                                                                     year loan?
2. Maria Gonzalez needed to decide into
   which bank to deposit her $3000                                6. Which would be the better deal for
   savings. Bank One offered her 4.7%                                Lonnie a loan for $2500 at 2.5%
   interest compounded daily but she had                             interest compounded annually for four
   to keep her money in the bank for at                              years, or a loan for $2500 at 3.5%
   least 4 years. Frost Bank offered                                 interest compounded continuously for
   4.8% interest compounded                                          three years? Explain your reasoning.
   semiannually with the same condition
   that she had to keep her money in the                          7. Karen has $5000 to invest in two
   bank for at least 4 years. Which bank                             stocks. She decides to invest $3200
   should she go with and why?                                       in LIQD stock that pays 11.3%
                                                                     interest and $1800 in DISNEY stock
3. John Matterson wishes to save $7000                               that pays 10.5% compounded daily.
   as a down payment for a car that he                               How much money would she make
   plans to buy in 6 years. How much                                 after investing for two years?
   money should he deposit into a Money
   Market account that pays 6.5%                                  8. Estimate the time it will take for an
   interest compounded continuously to                               investment of $2500 to double in
   earn his down payment?                                            value if it is invested at 8% interest.

4. Jason and Lucia are buying a house.                            9. Estimate the time it will take for an
   The amount of their loan for the                                  investment of $2500 to triple in value
   house is $98,000. They secured a                                  if it is invested at 8% interest.
   loan, which charges 6.75% annual
   interest. If the loan is for 30years,
   how much will the monthly payments
   be?




SATEC/Algebra II/Exponential Functions/6b525301-4602-4ed3-843b-bf4b90150343.doc/Rev. 07-01     Page 3/3

								
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