Time Value of Money Analysis

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5-42 Integrated Case
Time Value of Money Analysis. You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money analysis covering the following questions: a. Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2; (2) an ordinary annuity of $100 per year for 3 years; and (3) an uneven cash flow stream of -$50, $100, $75 and $50 at the end of Years 0 through 3. (1)

100
0
1
2
100
0
1
2

(2)
I%I%
I%I%

(3)
100
50
75
0
1
2
3
-50
100
50
75
0
1
2
3
-50

b.
1. What’s the future value of $100 after 3 years if it earns 10%, annual compounding? FV = PV (1 + I)N = $100 (1.10)3 = $133.10
2. What’s the present value of $100 to be received in 3 years if the interest rate is 10%, annual compounding? PV = FV / (1 + I)N = $100 / 1.103 = $75.13
c. What annual interest rate would cause $100 to grow to $125.97 in 3 years?

FV = PV (1+I)N

$125.97 = $100 (1 + I)3

Using a financial calculator, I = 8.0%

d. If a company’s sales are growing at a rate of 20% annually, how long will it take sales to double? FV = PV (1+I)N
$100,000 = $50,000 (1.02)N
Using a financial calculator, N = 3.80 Years
e. What’s the difference between an ordinary annuity and an annuity due? What type of annuity is shown here? How would you change it to the other type of annuity?

100100
100100
100100
00
11
22
33
I%
100100
100100
100100
00
11
22
33
I%

* An ordinary annuity is an annuity whose payments occur at the end of each period while an annuity due is an annuity whose payments occur at the beginning of each period. * Shown here is an ordinary annuity because it shows payments at the end of each period. * To change it to an annuity due you would move the payments on period left, with no payment under the 3, as shown below: 100

100
0
1
2
3
I%
100
100
100
0
1
2
3
I%
100

f.
3. What is the future value of a 3-year, $100 ordinary annuity if the annual interest rate is 10%? FVA = $100 (1.10)3-1 + $100 (1.10)3-2 + $100 (1.10)3-3 = $331.00 4. What is its present value?

PVA = $100 / 1.103-1 + $100 / 1.103-2 + $100 / 1.103-3 = $248.68 5. What would the future and present values be if it was an annuity due? FVADUE = FVAORDINARY(1+I) = $331.00 (1.10) = $364.10

PVADUE = PVAORDINARY(1+I) = $248.68 (1.10) = $273.55
g. A 5-year $100 ordinary annuity has an annual interest rate of 10%. 6. What is its present value?
PVA = $100 [(1 – 1/(1.10)5) / 0.10] = $379.08
7. What would the present value be if it was a 10-year annuity? Using a financial calculator, PV10 Year = $614.46
8. What would the present value be if it was a 25-year annuity? Using a financial calculator, PV25 Year = $907.70
9. What would the present value be if it was a perpetuity? PVPERPETUITY = $100 / 0.10 = $1,000
h. A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of each year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%. 10. If she keeps saving in this manner, how much will she have accumulated at age 65? FVA = $1,095 [(1.1245 – 1) / 0.12] = $1,487,261.89

11. If a 40-year-old investor began saving in this manner, how much would he have at age 65?

FVA = $1,095 [(1.1225 – 1) / 0.12] = $146,000.59

12. How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the 20-year-old investor?

$1,487,261.89 = PMT [(1.1225 – 1) / 0.12]
PMT = $11,154.42
i. What is the present value of the following uneven cash flow stream? The annual interest rate is 10%.

0
100
1
300
2
300
3
10%
-50I%
4
0
100
1
300
2
300
3
10%
-50I%
4

PV = $100 / 1.11 + $100 / 1.12 + $100 / 1.13 + $100 / 1.14 = $530.08 j.
13. Will the future value be...
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