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Exponentials, Simple and Compound Interest, Annuities, NPV and Amortization

1. Rational Functions
Graph the following function when a=3 and b=2. Develop a generic expression (i.e., as a function of "a" and "b") to find the "x" and "y" intercepts for this function (see attachment)

2. Exponential Functions
Once a new automobile enters the market, the manufacturers try to estimate their residual values after a period of time based on predicted performance of the new model and past performance of similar models. Ultimately this residual value is not only determined by the expected performance and reliability of a vehicle, but by consumer perception of its value after a number of years "on the street". The residual value after 3 years is a common indicator of annual depreciation for a vehicle. The following table (see attachment) shows 3-yr depreciations for three 2004 car models and their respective MSRP's or Manufacturer's Suggested Retail Price.
a. Use these values to estimate the annual depreciation rate for each of the models listed (Hint: Use the MSRP and the value after 3 years to estimate the exponential depreciation rate, k). Assume the depreciation model (see attachment)
b. Calculate the value for each car after 5 years. Which one should have a higher market value after 5 years?
c. Draw a graph showing all depreciation curves for each of the cars. Make sure you identify each one and all the intercepts are clearly labeled. What can you say about Acura TSX and Audi A4 during the first five years?

3. Logarithms
Use "Sastre's" long-term stock price model and the table provided (see attachment) to predict the stock price per share of Home Depot and Microsoft at the end of 2004. Find the current stock price for these companies and compare them with the values obtained with the model. For what company does the model work best? Any thoughts why?

4. Simple Interest
A person invests $10,000 at 8% per year for 10 years.
d. Find the simple interest earned
e. Find the future value
f. At the end of the 10-years, this person decides to re-invest the interest earned into the same account for another 10 years. What is the future value of this new investment?
g. Instead of re-investing the interest earned after 10 years, what would be the future value of the original $10,000 investment if the account was kept for 20 years? Which one is a better investment strategy?

5. Compound Interest
A person deposits $10,000 into a bank account that earns interest at 6% compounded semiannually for 3 years. At the end of the 3-year time interval, the person reinvests the original $10,000 plus its interest and deposits an additional $5000. At this time the interest rate changes to 7% compounded continuously.
h. How much is in the account 5 years after the original $10,000 deposit?
i. What is the effective interest rate during the first 3 years?
j. How long does this person have to wait for the account to reach $40,000?

6. Ordinary Annuities
Stacey deposits $600 at the end of each quarter for 11 years. Each deposit earns interest at 12% compounded quarterly.
k. Find the total amount in this account at the end of 11 years
l. Charles wants the same amount at the end of 11 years, but instead of putting $600 every quarter, he wants to deposit one lump-sum payment now. How much should this amount be if it is to be compounded at 12% quarterly for 11 years

7. Net Present Value
A business manager decided to take a capital loan for $375,000 at 9% interest rate compounded annually to invest in a new project that will generate the following cash inflows over a 5-year period, realized at year-end: $80,000; $90,000; $110,000; $130,000 and $140,000.
m. Construct a 5-year time line to represent cash inflows and outflows
n. Compute NPV. Is this a sound investment?

8. Amortization
Develop an amortization schedule for a $10,500 loan that must be repaid in monthly installments for 18 months at a nominal interest rate of 12%.

Attachments

Solution Preview

1. Rational Functions

Graph the following function when a=3 and b=2. Develop a generic expression (i.e., as a function of "a" and "b") to find the "x" and "y" intercepts for this function.

a) Write g(x) = y = 1 / (x+a) + b
then when a=3 and b=2, we have:
y = 1 / (x+3) + 2
y - 2 = 1 / (x+3)
(y - 2) (x+3) = 1

b) find intercepts of y = 1 / (x+a) + b:
when x = 0, y = 1 / (0+a) + b = 1/a +b
when y = 0,
1 / (x+a) + b = 0
1 / (x+a) = - b
1 = - b (x+a)
x+a = -1 / b
x = -a - 1/b
thus, the two intercepts are:
(-a -1/b , 0) and (0 , 1/(x+a) +b)

2. Exponential Functions

Once a new automobile enters the market, the manufacturers try to estimate their residual values after a period of time based on predicted performance of the new model and past performance of similar models. Ultimately this residual value is not only determined by the expected performance and reliability of a vehicle, but by consumer perception of its value after a number of years "on the street". The residual value after 3 years is a common indicator of annual depreciation for a vehicle. The following table shows 3-yr depreciations for three 2004 car models and their respective MSRP's or Manufacturer's Suggested Retail Price.
Car Make MSRP 3yr Depreciation
Acura TSX $27,060 47%
Audi A4 $31,070 54%
BMW 330 $36,915 49%
Data Source: http://www.acura.com/models/

a. Use these values to estimate the annual depreciation rate for each of the models listed (Hint: Use the MSRP and the value after 3 years to estimate the exponential depreciation rate, k.). Assume the depreciation model .

For Acura TSX:
C = $27,060
V = (1- 3yr depreciation)*C = (1-47%)C = 53%C
t = 3
We substitute the numbers into the formula: V = C * EXP(-kt):
53%C = C * EXP(-3k)
0.53 = EXP(-3k)
-3k = ln(0.53) = -0.635
k = 0.212
Then the residual value after 1 year (t = 1) is:
V = C * EXP(-kt) = C * EXP(-0.212) = 0.81 C
So the depreciation is D = C - V = C - 0.81 C = 0.19 C
Thus, the annual depreciation rate for Acura TSX is
d1 = D / C = 0.19 C / C = 19%

For Audi A4:
C = $31,070
V = (1- 3yr depreciation)*C = (1-54%)C = 46%C
t = 3
We substitute the numbers into the formula: V = C * EXP(-kt):
46%C = C * EXP(-3k)
0.46 = EXP(-3k)
-3k = ln(0.53) = -0.777
k = 0.259
Then the residual value after 1 year (t = 1) is:
V = C * EXP(-kt) = C * EXP(-0.259) = 0.77 C
So the depreciation is D = C - V = C - 0.77 C = 0.23 C
Thus, the annual depreciation rate for Audi A4 is
d2 = ...

Solution Summary

Rational and Exponential Functions, Logarithms, Simple Interest, Compoud Interest, Ordinary Annuities, Net Present Value, Amortization are investigated. The solution is detailed and well explained. The solution was given a rating of "5" by the student who originally posted the question.

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