fof im chapter 05 - 7th

7/27/2019 Fof Im Chapter 05 - 7th

http://slidepdf.com/reader/full/fof-im-chapter-05-7th 1/79



88 ♦ Keown/Martin/Petty Instructor’s Manual with Solutions

mn

mr

1PV nFV

+=

where m = the number of times compounding occurs during the year

II. Determining the present value, that is, the value in today’s dollars of a sum of money

to be received in the future, involves nothing other than inverse compounding. Thedifferences in these techniques are simply different points of view.

A. Mathematically, the present value of a sum of money to be received in thefuture can be determined by the following equation:

PV = FV n( )

+ nr 1

1

where n = the number of years until payment will be received,

r = the opportunity rate or discount rate

PV = the present value of the future sum of moneyFVn = the future value of the investment at the end of n years

1. The present value of a future sum of money is inversely related to boththe number of years until the payment will be received and theopportunity rate.

III. An annuity is a series of equal dollar payments for a specified number of years.Because annuities occur frequently in finance—for example, bond interest payments— we treat them specially.

A. A compound annuity involves depositing or investing an equal sum of money at

the end of each year for a certain number of years and allowing it to grow.1. This can be done by using our compounding equation and compounding

each one of the individual deposits to the future or by using thefollowing compound annuity equation:

FVn = PMT ( )

∑

−=

+1n

0t

tr 1 = PMT −+r

r n 1)1(

where PMT = the annuity value deposited at the end of eachyear

r = the annual interest (or discount) rate

n = the number of years for which the annuity willlast

FVn = the future value of the annuity at the end of thenth year

©2011 Pearson Education, Inc. Publishing as Prentice Hall



Foundations of Finance, Seventh Edition ♦ 89

B. Pension funds, insurance obligations, and interest received from bonds allinvolve annuities. To compare these financial instruments, we would like toknow the present value of each of these annuities.

1. This can be done by using our present value equation and discountingeach one of the individual cash flows back to the present or by using thefollowing present value of an annuity equation:

PV = PMT( )

∑= +n

1t tr 1

1 = PMT

+−

ir)(1

11 n

where PMT = the annuity withdrawn at the end of each year

r = the annual interest or discount rate

PV = the present value of the future annuity

n = the number of years for which the annuity will last

C. This procedure of solving for PMT, the annuity value when r, n, and PV areknown, is also the procedure used to determine what payments are associatedwith paying off a loan in equal installments. Loans paid off in this way, in

periodic payments, are called amortized loans. Here again, we know three of the four values in the annuity equation and are solving for a value of PMT, theannual annuity.

IV. A perpetuity is an annuity that continues forever; that is, every year from now on, thisinvestment pays the same dollar amount.

A. An example of a perpetuity is preferred stock which yields a constant dollar dividend infinitely.

B. The following equation can be used to determine the present value of a perpetuity:

PV =r

pp

where PV = the present value of the perpetuity

pp = the constant dollar amount provided by the perpetuity

r = the annual interest or discount rate

V. To aid in the calculations of present and future values, tables are provided at the end of the text.

A. To aid in determining the value of FV n in the compounding formula

FVn = PV (l + r) n

The value of (l + r) n is referred to as FUTURE VALUE FACTOR r,n calculated





as (1 + r) n.

B. To aid in the computation of present values

PV = FV n ( )nr 1

1

+

With ( )nr 1

1

+ , equal to the Present Value Factor

C. Because of the time-consuming nature of compounding an annuity,

FVn = PMT ( )∑−

=+

1n

0t

tr 1 where

( )∑−

=

+1n

0t

tr 1 =−+

r

r n 1)1(

and is referred to as the ANNUITY PRESENT VALUE FACTOR r,n

D. To simplify the process of determining the present value of an annuity

PV = PMT( )

+∑=

n

1ttr 1

1 where

( )∑= +

n

1tt

r 1

1 is referred to as the ANNUITY PRESENT

VALUE FACTOR r,n= +−

ii)(1

11 n





ANSWERS TOEND-OF-CHAPTER QUESTIONS

5-1. The concept of time value of money is a recognition that a dollar received today isworth more than a dollar received a year from now, or at any future date. It exists because there are investment opportunities on money; that is, we can place our dollar received today in a savings account and one year from now have more than a dollar.

5-2. Compounding and discounting are inverse processes of each other. In compounding,money is moved forward in time, while in discounting, money is moved back in time.This can be shown mathematically in the compounding equation:

FVn = PV (1 + r) n

We can derive the discounting equation by multiplying each side of this equation

by ( )n

1

1 r + and we get:

PV = FV n ( )nr 1

1

+

5-3. We know that FV n = PV(1 + r) n

Thus, an increase in r will increase FV n, and a decrease in n will decrease FV n.

5-4. Bank C, which compounds continuously, pays the highest interest. This occurs because, while all banks pay the same interest, 5%, bank C compounds the 5%

continuously. Continuous compounding allows interest to be earned more frequentlythan any other compounding period.

5-5. An annuity is a series of equal dollar payments for a specified number of years.Examples of annuities include mortgage payments, interest payments on bonds, fixedlease payments, and any fixed contractual payment. A perpetuity is an annuity thatcontinues forever; that is, every year from now on this investment pays the same dollar amount. The difference between an annuity and a perpetuity is that a perpetuity has notermination date, whereas an annuity does.

5-6. This problem involves a comparison of three websites, all of which are very good.





c. FV12 = PV (1 + r) n

FV12 = $775 (1 + 0.12) 12

FV12 = $775 (3.896)

FV12 = $3,019.40Or:

N = 12I/Y = 12PV = –775PMT = 0CPT FV = $3,019

d. N = 5I/Y = 5PV = –21,000PMT = 0CPT FV = $26,802.

5-2. I/Y = 5PV = –500,000PMT = 0

FV = 1,039.50CPT N = 15

b. I/Y = 9PV = –35.0PMT = 0FV = 53.87CPT N = 5

c. I/Y = 20PV = –100.0PMT = 0FV = 298.60CPT N = 6





d. I/Y = 2PV = –53.0PMT = 0FV = 78.76

CPT N = 20

5-3. a. N = 12CPT I/Y = 12PV = –500PMT = 0FV = 1,948

b. N = 7

CPT I/Y = 4.999PV = –300PMT = 0FV = 422.10

c. N = 20CPT I/Y = 9PV = –50PMT = 0FV = 280.20

d. N = 5CPT I/Y = 20PV = –200PMT = 0FV = 497.60

5-4. a. N = 10I/Y = 10CPT PV = –308.43PMT = 0FV = 800





b. N = 5I/Y = 5CPT PV = –235.06PMT = 0

FV = 300

c. N = 8I/Y = 3CPT PV = –789PMT = 0FV = 1,000

d. N = 8I/Y = 20CPT PV = –233PMT = 0FV = 1,000

5-5. a. N = 10I/Y = 5PV = 0PMT = –500CPT FV = 6,289

b. N = 5I/Y = 10PV = 0PMT = –100CPT FV = 610.51

c. N = 7I/Y = 7PV = 0PMT = –35CPT FV = 302.89





d. N = 3I/Y = 2PV = 0PMT = –25

CPT FV = 76.51

5-6. a. N = 10I/Y = 7CPT PV = 17,559PMT = –2,500FV = 0

b. N = 3

I/Y = 3CPT PV = –198PMT = 70FV = 0

c. N = 7I/Y = 6CPT PV = –1,563.06PMT = 280FV = 0

d. N = 10I/Y = 10CPT PV = –3,072.28PMT = 500FV = 0

5-7. a. FV n = PV (1 + r) n

compounded for 1 year FV1 = $10,000 (1 + 0.06) 1

FV1 = $10,000 (1.06)

FV1 = $10,600

Or:





N = 1I/Y = 6PV = –10,000PMT = 0

CPT FV = $10,600

compounded for 5 years

FV5 = $10,000 (1 + 0.06) 5

FV5 = $10,000 (1.338)

FV5 = $13,380

Or: N = 5I/Y = 6PV = –10,000PMT = 0CPT FV = $13,382


FV15 = $10,000 (1 + 0.06) 15

FV15 = $10,000 (2.397)

FV15 = $23,970


b. FV n = PV (1 + r) n

compounded for 1 year at 8%

FV1 = $10,000 (1 + 0.08) 1

FV1 = $10,000 (1.080)

FV1 = $10,800





Or: N = 1I/Y = 8PV = –10,000

PMT = 0CPT FV = $10,800

compounded for 5 years at 8%

FV5 = $10,000 (1 + 0.08) 5

FV5 = $10,000 (1.469)

FV5 = $14,690



FV15 = $10,000 (1 + 0.08) 15

FV15 = $10,000 (3.172)

FV15 = $31,720Or:

N = 15I/Y = 8PV = –10,000PMT = 0CPT FV = $31,722


FV1 = $10,000 (1 + 0.1) 1FV1 = $10,000 (1 + 1.100)

FV1 = $11,000





Or: N = 1I/Y = 10PV = –10,000

PMT = 0CPT FV = $11,000


FV5 = $10,000 (1 + 0.1) 5

FV5 = $10,000 (1.611)

FV5 = $16,110



FV15 = $10,000 (1 + 0.1) 15

FV15 = $10,000 (4.177)

FV15 = $41,770


c. There is a positive relationship between both the interest rate used to compounda present sum and the number of years for which the compounding continuesand the future value of that sum.

5-8. a. N = 35I/Y = 4PMT = 0FV = 2,000,000





CPT PV = –$506,831





Now solving for bimonthly compounding (six times per year):

FVn = PVmn

mr

1

+

FV5 = $5,000 (1 + ) 6(5)

FV5 = $5,000 (1 + 0.01) 30

FV5 = $5,000 (1.348)FV5 = $6,740

Using a financial calculator where you can change the number of times compoundingoccurs per year, you can solve this problem in one of two ways.

One way to solve this problem if you’re using a Texas Instruments BAII-Pluscalculator is to first make P/Y = 6

N = 5 × 6 = 30

I/Y = 6

PV = –5,000PMT = 0

CPT FV = $6,739

OR if you don’t want to use P/Y button (that is, set P/Y=1)

N = 5 × 6 = 30

I/Y = 6/6

PV = –5,000

PMT = 0

CPT FV = $6,739

c. FV n = PV (1 + r) n

FV5 = $5,000 (1 + 0.12) 5

FV5 = $5,000 (1.762)

FV5 = $8,810

Or:

N = 5I/Y = 12PV = –5,000PMT = 0CPT FV = $8,812





FV5 = PVmn

mr

1

+

FV5 = $5,000 +

212.0

1 2×5

FV5 = $5,000 (1 + 0.06) 10

FV5 = $5,000 (1.791)

FV5 = $8,955

Or:



N = 5 × 2 = 10I/Y = 12PV = –5,000PMT = 0CPT FV = $8,954

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 5 × 2 = 10I/Y = 12/2PV = –24PMT = 0CPT FV = $8,954

Now solving for bimonthly compounding (six times per year):

FV5 = PVmn

mr

1

+

FV5 = $5,0006(5)

0.121

6 + ÷

FV5 = $5,000 (1 + 0.02) 30

FV5 = $5,000 (1.811)

FV5 = $9,055







N = 5 × 6 = 30I/Y = 12PV = –5,000PMT = 0CPT FV = $9,057

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 5 × 6 = 30I/Y = 12/6PV = –5,000

PMT = 0CPT FV = $9,057

d. FV n = PV (1 + r) n

FV12 = $5,000 (1 + 0.06) 12

FV12 = $5,000 (2.012)

FV12 = $10,060

Or: N = 12I/Y = 6PV = –5,000PMT = 0CPT FV =$10,061

e. An increase in the stated interest rate will increase the future value of a givensum. Likewise, an increase in the length of the holding period will increase thefuture value of a given sum.





5-10. Annuity A:

N = 12

I/Y = 11

CPT PV = –$55,185

PMT = –8,500

FV = 0

Since the cost of this annuity is $50,000 and its present value is $55,182, givenan 11% opportunity cost, this annuity has value and should be accepted.

Annuity B:

N = 25

I/Y = 11

CPT PV = –$58,952

PMT = –7,000

FV = 0

Since the cost of this annuity is $60,000 and its present value is only $58,954,given an 11% opportunity cost, this annuity should not be accepted.

Annuity C:

N = 20

I/Y = 11

CPT PV = –$63,707

PMT = –8,000

FV = 0

Since the cost of this annuity is $70,000 and its present value is only $63,704,given an 11% opportunity cost, this annuity should not be accepted.

5-11. Year 1: FV n = PV (1 + r) n

FV1 = 15,000 (1 + 0.2) 1FV1 = 15,000 (1.200)

FV1 = 18,000 books





OR: N = 1I/Y = 20PV = –15,000

PMT = 0CPT FV = 18,000 books

Year 2: FV n = PV (1 + r) n

FV 2 = 15,000 (1 + 0.2) 2

FV 2 = 15,000 (1.440)

FV2 = 21,600 books

OR: N = 2I/Y = 20PV = –15,000PMT = 0CPT FV = 21,600 books


FV3 = 15,000 (1.20) 3

FV3 = 15,000 (1.728)

FV3 = 25,920 booksOR:

N = 3I/Y = 20PV = –15,000PMT = 0CPT FV = 25,920 books





Book sales

25,000

20,000

15,000

1 2 3

years

The sales trend graph is not linear, because this is a compound growth trend. Just ascompound interest occurs when interest paid on the investment during the first period

is added to the principal of the second period, interest is earned on the new sum. Book sales growth was compounded; thus, the first year the growth was 20% of 15,000 books, the second year 20% of 18,000 books, and the third year 20% of 21,600 books.

5-12. FV n = PV (1 + r) n

FV1 = 47(1 + 0.12) 1

FV1 = 47(1.12)

FV1 = 52.6 Home Runs in 2010

OR: N = 1I/Y = 12PV = –47PMT = 0CPT FV = 52.6 home runs

FV2 = 47(1 + 0.12) 2

FV2 = 47(1.21)

FV2 = 58.96 Home Runs in 2011

OR:





N = 2I/Y = 12PV = –47PMT = 0CPT FV = 58.96 home runs

FV3 = 47(1 + 0.12) 3

FV3 = 47(1.331)

FV3 = 66.03 Home Runs in 2012.

OR: N = 3I/Y = 12PV = –47PMT = 0CPT FV = 66.03 home runs

FV3 = 47(1 + 0.12) 4

FV4 = 47(1.464)


OR: N = 4I/Y = 12PV = –47

PMT = 0CPT FV = 73.96 home runs

FV5 = 47(1 + 0.12) 5

FV5 = 47(1.611)


OR: N = 5I/Y = 12

PV = –47PMT = 0CPT FV = 82.83 home runs





5-17. N = 10I/Y = 9PV = 0CPT PMT= –$658,200.90

FV = 10,000,000

5-18. One dollar at 12.0% compounded monthly for one year

FVn = PV (1 + r) n

FV12 = $1(1 + .01) 12

= $1(1.127)

= $1.127



N = 1 × 12 = 12I/Y = 12PV = –1PMT = 0CPT FV = $1.1268

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 1 × 12 = 12

I/Y = 12/12

PV = –1

PMT = 0

CPT FV = $1.1268

One dollar at 13.0% compounded annually for one year

FVn = PV (1 + r) n

FV1 = $1(1 + .13) 1

= $1(1.13)

= $1.13

OR





N = 1

I/Y = 13

PV = –1

PMT = 0

CPT FV = $1.13

The loan at 12% compounded monthly is more attractive.

5-19. Investment A

N = 5I/Y = 20CPT PV = –29,906PMT = 10,000FV = 0

Investment B

Step 1: First, discount the annuity back to the beginning of year 5, which is the end of year 4. Then, discount this equivalent sum to present.

Step 1: N = 6I/Y = 20CPT PV = –33,255PMT = 10,000FV = 0

Step 2:

N = 4I/Y = 20CPT PV = 16,037

PMT = 0FV = –33,255





Investment C

PV = FV n

+ nr) (11

= $10,000

+1

)20. 1(

1+ $50,000

+6

)20. 1(

1

+ $10,000

+ 10)20. 1(

1

= $10,000(.833) + $50,000(.335) + $10,000(.162)

= $8,330 + $16,750 + $1,620

= $26,700

OR: Simply calculate the present value of all three single cash flows and then add themtogether:

N = 1I/Y = 20CPT PV = 8,333

PMT = 0FV = 10,000

N = 6I/Y = 20

CPT PV = –16,745PMT = 0

FV = 50,000 N = 10I/Y = 20

CPT PV = –1,615PMT = 0FV = 10.000

Then add all the present values together: $8,333 + $16,745 + $1,615 = $26,693





5-22. Using a financial calculator where you can change the number of timescompounding occurs per year, you can solve this problem in one of two ways.


CPT N = 18 and since P/Y = 2, there are two periods per year, so N = 9 yearsI/Y = 16PV = –1PMT = 0FV = 4


CPT N = 18 and since the interest rate is expressed in semi-annualterms, there are two periods per year, so N=9 years

I/Y = 16/2

PV = –1

PMT = 0

FV = 4

5-23. Step 1 (First, discount the annuity back to the beginning of year 11, which is the end of year 10.):

N = 5I/Y = 6CPT PV = –42,124PMT = 10,000FV = 0

Step 2 (Then, discount this equivalent sum to present.): N = 10I/Y = 6CPT PV = 23,473PMT = 0FV = –42,124





Step 3 (Then, determine the present value of the $20,000 withdrawal at the end of year 15):

N = 15I/Y = 6

CPT PV = 8,345PMT = 0FV = –20,000

Step 4: (Add the present values together):

Thus, you would have to deposit $23,473 + $8,345 or $31,818 today.

5-24. N = 10I/Y = 10PV = –40,000CPT PMT= $6,510FV = 0

5-25. N = 5

CPT I/Y = 19.86%

PV = –30,000

PMT = 10,000

FV = 0

5-26. N = 5

CPT I/Y = 22.0%PV = –10,000

PMT = 0

FV = 27,027

5-27. N = 5I/Y = 12PV = –25,000CPT PMT = $6,935FV = 0





5-28. The present value of $10,000 in 12 years at 11 percent is:

N = 12I/Y = 11

CPT PV = 2,858PMT = 0FV = –10,000

The present value of $25,000 in 25 years at 11% is:

N = 25I/Y = 11CPT PV = 8,345PMT = 0FV = –1,840

5-29. N = 5I/Y = 12PV = 0CPT PMT = $3,148FV = –20,000

5-30. a. N = 15

I/Y = 7

PV = 0

CPT PMT= $1,990

FV = –50,000

b. N = 15I/Y = 7CPT PV = 18,122PMT = 0FV = –50,000

c. The contribution of the $10,000 deposit toward the $50,000 goal is N = 10I/Y = 7PV = –10,000PMT = 0CPT FV = 19,672





Thus, only $30,328 need be accumulated by annual deposit. N = 15I/Y = 7PV = 0

CPT PMT= $1,207FV = –30,328

5-31. This problem can be subdivided into (1) the compound value of the $100,000 in thesavings account, (2) the compound value of the $300,000 in stocks, (3) the additionalsavings due to depositing $10,000 per year in the savings account for 10 years, and (4)the additional savings due to depositing $10,000 per year in the savings account at theend of years 6-10. (Note the $20,000 deposited in years 6-10 is covered in parts 3 and4.)

(1) N = 10I/Y = 7PV = –100,000PMT = 0CPT FV = 196,715

(2) N = 10I/Y = 12PV = –300,000PMT = 0CPT FV = 931,754

(3) Compound annuity of $10,000, 10 years N = 10I/Y = 7PV = 0PMT = –10,000CPT FV = 138,164

(4) Compound annuity of $10,000 (years 6-10) N = 5I/Y = 7

PV = 0PMT = –10,000CPT FV = 57,507





At the end of ten years, you will have $196,715 + $931,754 + $138,164 + $57,507 =$1,324,140.

N = 20I/Y = 10

PV = –1,324,140CPT PMT = 155,533FV = 0

5-32. N = 20I/Y = 15PV = –100,000CPT PMT = 15,976FV = 0

5-33. N = 30I/Y = 10PV = –150,000CPT PMT = 15,912FV = 0

5-34. This is an annuity due problem:

At 10%: N = 20I/Y = 10

CPT PV = 425,678 x 1.1 = 468,246PMT = –50,000FV = 0

At 20%: N = 20I/Y = 20CPT PV = 243,479 x 1.2 = 292,175PMT = –50,000FV = 0

5-35. N = 46CPT I/Y = 30.14%PV = –0.12PMT = 0FV = 22,000





number of periods (n) = 20

payment (PMT) = $0

present value (PV) = $30,000

future value (FV) = $250,000

type (0 = at end of period) = 0

guess =

r = 11.18%

Excel formula: = RATE(number of periods,payment,present value,futurevalue,type,guess)

Notice that present value ($30,000) took on a negative value.

5-39. Two things to keep in mind when you're working this problem: first, you'll have toconvert the annual rate of 8 percent into a monthly rate by dividing it by 12, and second,you'll have to convert the number of periods into months by multiplying 25 times 12 for a total of 300 months.

Excel formula: = PMT(rate, number of periods, present value, future value, type)

rate (r) = 8%/12number of periods (n) = 300

present value (PV) = $300,000future value (FV) = $0type (0 = at end of period) = 0

monthly mortgage payment = ($2,315.45)

Notice that monthly payments take on a negative value because you pay them.

You can also use Excel to calculate the interest and principal portion of any loanamortization payment. You can do this using the following Excel functions:

Calculation: Formula:

Interest portion of payment = IPMT(rate, period, number of periods, present value, future value, type)

Principal portion of payment = PPMT(rate, period, number of periods, present value, future value, type)

where period refers to the number of an individual periodic payment.Thus, if you would like to determine how much of the 48th monthly payment wenttoward interest and principal you would solve as follows:

Interest portion of payment 48: ($1,884.37)

The principal portion of payment 48: ($431.08)

5-40. a. N = 384





I/Y = 6

PV = –24

PMT = 0

CPT FV = 125.217 billion dollars

b. One way to solve this problem if you’re using a Texas Instruments BAII-Pluscalculator is to first make P/Y = 12

N = 384 x 12 = 4,608

I/Y = 6

PV = –24

PMT = 0



N = 384 x 12 = 4,608

I/Y = 6/12 = .5

PV = –24

PMT = 0


c. N = 10

I/Y = 10

CPT PV = –23.13 billion dollars

PMT = 0

FV = 60. billion

d. N = 10

CPT I/Y = 14.87%

PV = –15 billion

PMT = 0

FV = 60. billion

e. N = 40I/Y = 7

PV = –28. billion

CPT PMT = 2.10 billion dollars

FV = 0





5-41. What will the car cost in the future?

N = 6

I/Y = 3

PV = –15,000

PMT = 0

CPT FV = 17,910.78 dollars

How much must Bart put in an account today in order to have $17,910.78 in 6years?

N = 6

I/Y = 7.5

CPT PV = –11,605.50 dollars

PMT = 0

FV = 17,910.78

5-42. N = 45

I/Y = 8.75

PV = 0

CPT PMT= –2,054.81 dollars

FV = 1,000,000

5-43. First, we must calculate what Mr. Burns will need in 20 years. Once we know howmuch he will need, we can then calculate how much he needs to deposit each year inorder to come up with that amount (note: once you calculate the present value, youmust multiply your answer, in this case –$4.192 billion times (1 + r) because this is anannuity due):

N = 10

I/Y = 20

CPT PV = –4.1925 billion × 1.20 = -5.031 billion dollars

PMT = 1 billion dollars

FV = 0

Next, we will determine how much Mr. Burns needs to deposit each year for 20 yearsto reach this goal of accumulating $5.031 billion at the end of the 20 years:

N = 20I/Y = 20PV = 0CPT PMT= –26.9 million dollarsFV = 5.031 billion





5-44. What’s the $100,000 worth in 25 years (keep in mind that Homer invested the money 5years ago and we want to know what it will be worth in 20 years)?

N = 25

I/Y = 7.5

PV = –100,000PMT = 0


Now we determine what the additional $1,500 per year will grow to (note that sinceHomer will be making these investments at the beginning of each year for 20 years wehave an annuity due, thus, once you calculate the present value, you must multiply your answer, in this case $64,957.02 times (1 + r)):

N = 20

I/Y = 7.5

PV = 0

PMT = –1,500

CPT FV = 64,957.02 × 1.075 = 69,828.80 dollars

Finally, we must add the two values together:

$609,833.96 + $69,828.80 = $679,662.76

5-45. Since this problem involves monthly payments we must first divide the annual interestrate by 12 to convert it to monthly terms. Then, N becomes the number of months or compounding periods,

N = 60I/Y = 6.2/12

PV = –25,000

CPT PMT= 485.65 dollars

FV = 0

OR: If you’re using a Texas Instruments BAII-Plus calculator you can solve this problem by using the P/Y function and make P/Y = 12:

N = 5 × 12 = 60

I/Y = 6.2PV = –25,000

CPT PMT = 485.65

FV = 0





Now, calculate how much the monthly payments would be if Suzie took the $1,000cash back and reduced the amount owed from $25,000 to $24,000. Again, since this

problem involves monthly payments there are two ways to solve it:

One way to solve this problem, if you’re using a Texas Instruments BAII-Plus

calculator, is to first make P/Y = 12 N = 60I/Y = 6.9PV = -24,000CPT PMT = $474.10FV = 0

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 60I/Y = 6.9/12

PV = –24,000CPT PMT = $470.10FV = 0

5-48. Since this problem involves quarterly payments there are two ways to solve it:

One way to solve this problem, if you’re using a Texas Instruments BAII-Pluscalculator, is to first make P/Y = 4

N = 16I/Y = 6.4%PV = 0

PMT = –1,000CPT FV = $18,071.11

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 16I/Y = 6.4/4PV = 0PMT = –1,000CPT FV = $18,071.11

5-49. Since this problem involves monthly payments there are two ways to solve it:


CPT N = 41.49 (rounded up to 42 months)I/Y = 12.9PV = –5000PMT = 150FV = 0





OR if you don’t want to use P/Y button (that is, set P/Y = 1)CPT N = 41.49 (rounded up to 42 months)I/Y = 12.9/12PV = –5000

PMT = 150FV = 0

5-50. a. Since this problem begins using annual payments, make sure your calculator isset to P/Y=1.

N = 12CPT I/Y = 8.37%PV = –160,000PMT = 0FV = 420,000

b. Again, since this problem begins using annual payments, make sure your calculator is set to P/Y = 1 N = 10CPT I/Y = 11.6123%PV = –140,000PMT = 0FV = 420,000

c. Since this problem involves monthly payments there are two ways to solve it.One way to solve this problem, if you’re using a Texas Instruments BAII-Pluscalculator, is to use the P/Y = function and make P/Y = 12. Then, N becomesthe number of months or compounding periods:

N = 120

I/Y = 6

PV = –140,000

CPT PMT = –$1,008.57

FV = 420,000


N = 120

I/Y = 6 /12

PV = –140,000

CPT PMT = $1,008.57

FV = 420,000





5-52. N = 20CPT I/Y = 13.00PV = –21,074.25PMT = 3000

FV = 0

5-53. At retirement, Milhouse wants: $300,000 for a boat and an annuity of $80,000 per year for 15 years at 6%. First, calculate the PV of the annuity (which is how muchhe will need to withdraw $80,000 each year), then add $300,000:

N = 50I/Y = 6%CPT PV = –$1,260,948PMT = $80,000

FV = 0 Now add the $300,000 he wants to buy a boat to the $1,260,948, and at the end of 43years he will need to have $1,560,948

Next, determine how much he needs to deposit at the end of each year for 43 years at9% in order to accumulate $1,560,948:

N = 43I/Y = 9PV = 0

CPT PMT= –3,540.80FV = 1,560,948

5-54. N = 200I/Y = 3.98PV = –12,345PMT = 0CPT FV = $30,300,773

5-55. N = 7CPT I/Y = 15%PV = –4,510PMT = 0FV = 12,000





5-56. CPT N = 45I/Y = 4.5%PV = –45,530PMT = 0

FV = 330,000

5-57. N = 28I/Y = 7%CPT PV = –60,000.PMT = 0FV = 398,930

SOLUTION TO MINI CASEa. Discounting is the inverse of compounding. We really have only one formula to move

a single cash flow through time. In some instances, we are interested in bringing thatcash flow back to the present (finding its present value) when we already know thefuture value. In other cases, we are merely solving for the future value where we knowthe present value.

b. The present value of an annuity factor is actually derived from the present value factor.This can be seen by examining the value of each. The present value factor gives valuesof

nr) (11+

for various values of i and n, while the present value of an annuity factor gives valuesof

( )

+∑=

n

1ttr 1

1

for various values of r and n. Thus, the value in the present value of annuity for an n-year annuity for any discount rate r is merely the sum of the first n values of the present

value factor.c. (1) N = 10

I/Y = 8PV = –5,000PMT = 0CPT FV = $10,795





(2) CPT N = 15I/Y = 10PV = –400PMT = 0

FV = $1,671(3) N = 10

CPT I/Y = 15PV = –1,000PMT = 0FV = $4,046

d. One way to solve this problem if you’re using a Texas Instruments BAII-Pluscalculator is to use the P/Y = function and make P/Y = 2. Then, N becomes thenumber of months or compounding periods:

N = 10I/Y = 10

PV = –1,000

PMT = 0

CPT FV = 1,629


N = 10

I/Y = 10/2

PV = –1,000

PMT = 0

CPT FV = 1,629

e. An annuity due is an annuity in which the payments occur at the beginning of each period as opposed to occurring at the end of each period, which is when the paymentoccurs in an ordinary annuity.

f. N = 7

I/Y = 10

CPT PV = –4,868PMT = 1,000

FV = 0





PV(annuity due)

N = 7

I/Y = 10

CPT PV = –4,868 x 1.1 = 5,355

PMT = 1,000

FV = 0

g. N = 7

I/Y = 10

PV = 0

PMT = –1,000

CPT FV = 9,487

FVn(annuity due) N = 7

I/Y = 10

PV = 0

PMT = –1,000

CPT FV = 9,487 x 1.1 = 10,436

h. N = 25

I/Y = 10

PV = –100,000

CPT PMT= 11,017

FV = 0

i. PV =r

PP

=08.000,1$

= $12,500





j. Step 1 (First, discount the annuity back to the beginning of year 10, which is the end of year 9.):

N = 10I/Y = 10

CPT PV = –6,144.57PMT = 1,000FV = 0

Step 2 (Then, discount this equivalent sum to present.): N = 9I/Y = 10CPT PV = 2,605PMT = 0FV = –6,144.57

k. Step 1 (First, discount the perpetuity back to the beginning of year 10, which is the endof year 9.):

=10.000,1$

= $10,000

Step 2 (Then, discount this equivalent sum to present.):

N = 9

I/Y = 10

CPT PV = 4,241

PMT = 0FV = –10,000





5-6A. (Present Value of an Annuity) What is the present value of the following annuities?

a. $3,000 a year for 10 years discounted back to the present at 8%

b. $50 a year for 3 years discounted back to the present at 3%

c. $280 a year for 8 years discounted back to the present at 7%

d. $600 a year for 10 years discounted back to the present at 10%

5-7A. (Compound Value) Trish Nealon, who recently sold her Porsche, placed $20,000 in asavings account paying annual compound interest of 7%.

a. Calculate the amount of money that will have accrued if she leaves the moneyin the bank for 1, 5, and 15 years.

b. If she moves her money into an account that pays 9% or one that pays 11%,

rework part a using these new interest rates.c. What conclusions can you draw about the relationships between interest rates,

time, and future sums from the calculations you have done above?

5-8A. (Compound Interest with Nonannual Periods) Calculate the amount of money that will be in each of the following accounts at the end of the given deposit period:

Compounding Period

Annual (Compounded Deposit Amount Interest Every Period

Account Deposited Rate Month) (Years)Korey Stringer $2,000 12% 2 2Eric Moss 50,000 12 1 1Ty Howard 7,000 18 2 2Rob Kelly 130,000 12 3 2Matt Christopher 20,000 14 6 4Juan Porter 15,000 15 4 3

5-9A. (Compound Interest with Nonannual Periods)

a. Calculate the future sum of $6,000, given that it will be held in the bank 5 yearsat an annual interest rate of 6%.

b. Recalculate part a using a compounding period that is (1) semiannual and (2) bimonthly.

c. Recalculate parts a and b for a 12% annual interest rate.

d. Recalculate part a using a time horizon of 12 years (annual interest rate is still6%).





e. With respect to the effect of changes in the stated interest rate, and holding periods on future sums in parts c and d, what conclusions do you draw whenyou compare these figures with the answers found in parts a and b?

5-10A. (Solving for r in Annuities) Ellen Denis, a sophomore mechanical engineering student,receives a call from an insurance agent, who believes that Ellen is an older womanready to retire from teaching. He talks to her about several annuities that she could buythat would guarantee her an annual fixed income. The annuities are as follows:

Initial Payment into Duration

Annuity Amount of Money of Annuity Annuity (at t = 0) Received per year (Years)A $50,000 $8500 12B $60,000 $7000 25C $70,000 $8000 20

If Ellen could earn 12% on her money by placing it in a savings account, should she place it instead in any of the annuities? Which ones, if any? Why?

5-11A. (Future Value) Sales of a new marketing book were 10,000 copies this year and wereexpected to increase by 15% per year. What are expected sales during each of the nextthree years? Graph this sales trend and explain.

5-12A. (Future Value) Reggie Jackson, formerly of the New York Yankees, hit 41 home runsin 1980. If his home-run output grew at a rate of 12% per year, what would it have

been over the following 5 years?

5-13A. (Loan Amortization) Stefani Moore purchased a new house for $150,000. She paid$30,000 down and agreed to pay the rest over the next 25 years in 25 equal annual

payments that included principal payments plus 10% compound interest on the unpaid balance. What will these equal payments be?

5-14A. (Solving for PMT in an Annuity) To pay for your child’s education, you wish to haveaccumulated $25,000 at the end of 15 years. To do this, you plan on depositing anequal amount in the bank at the end of each year. If the bank is willing to pay 7%compounded annually, how much must you deposit each year to obtain your goal?

5-15A. (Solving for r in Compound Interest) If you were offered $2,376.50 ten years from nowin return for an investment of $700 currently, what annual rate of interest would youearn if you took the offer?

5-16A. (Present Value and Future Value of an Annuity) In 10 years, you plan to retire and buya house in Marco Island, Florida. The house you are looking at currently costs$125,000 and is expected to increase in value each year at a rate of 5%. Assuming youcan earn 10% annually on your investment, how much must you invest at the end of each of the next 10 years to be able to buy your dream home when you retire?





5-17A. (Compound Value) The Knutson Corporation needs to save $15 million to retire a $15million mortgage that matures on December 31, 2002. To retire this mortgage, thecompany plans to put a fixed amount into an account at the end of each year for 10years, with the first payment occurring on December 31, 1993. The Knutson

Corporation expects to earn 10% annually on the money in this account. What annualcontribution must it make to this account to accumulate the $15 million by December 31, 2002?

5-18A. (Compound Interest with Nonannual Periods) After examining the various personalloan rates available to you, you find that you can borrow funds from a financecompany at 24% compounded monthly or 26% compounded annually. Whichalternative is the most attractive?

5-19A. (Present Value of an Uneven Stream of Payments) You are given three investmentalternatives to analyze. The cash flows from these three investments are as follows:

Investment End of Year A B C

1 $15,000 $20,0002 15,0003 15,0004 15,0005 15,000 $15,0006 15,000 60,0007 15,0008 15,000

9 15,00010 15,000 20,000

Assuming a 20% discount rate, find the present value of each investment.

5-20A. (Present Value) The Shin Corporation is planning to issue bonds that pay no interest but can be converted into $1,000 at maturity, 8 years from their purchase. To pricethese bonds competitively with other bonds of equal risk, it is determined that theyshould yield 9%, compounded annually. At what price should the Shin Corporation sellthese bonds?

5-21A. (Perpetuities) What is the present value of the following?

a. A $400 perpetuity discounted back to the present at 9%

b. A $1,500 perpetuity discounted back to the present at 13%

c. A $150 perpetuity discounted back to the present at 10%

d. A $100 perpetuity discounted back to the present at 6%





5-22A. (Solving for n with Nonannual Periods) About how many years would it take for your investment to grow sevenfold if it were invested at 10% compounded annually?

5-23A. (Complex Present Values) How much do you have to deposit today so that beginning11 years from now you can withdraw $10,000 a year for the next 5 years (periods 11through 15) plus an additional amount of $15,000 in that last year (period 15)? Assumean interest rate of 7%.

5-24A. (Loan Amortization) On December 31, Loren Billingsley bought a yacht for $60,000, paying $15,000 down and agreeing to pay the balance in 10 equal annual installmentsthat include both principal and 9% interest on the declining balance. How big wouldthe annual payments be?

5-25A. (Solving for r in an Annuity) You lend a friend $45,000, which your friend will repayin 5 equal annual payments of $9,000 with the first payment to be received one year from now. What rate of return does your loan receive?

5-26A. (Solving for r in Compound Interest) You lend a friend $15,000, for which your friendwill repay you $37,313 at the end of 5 years. What interest rate are you charging your “friend”?

5-27A. (Loan Amortization) To purchase some new machinery, a firm borrows $30,000 fromthe bank at 13% compounded annually. This loan is to be repaid in equal annualinstallments at the end of each year over the next 4 years. How much will each annual

payment be?

5-28A. (Present Value Comparison) You are offered $1,000 today, $10,000 in 12 years, or $25,000 in 25 years. Assuming that you can earn 11% on your money, which shouldyou choose?

5-29A. (Compound Annuity) You plan to buy some property in Florida five years from today.To do this, you estimate that you will need $30,000 at that time for the purchase. Youwould like to accumulate these funds by making equal annual deposits in your savingsaccount, which pays 10% annually. If you make your first deposit at the end of thisyear and you would like your account to reach $30,000 when the final deposit is made,what will be the amount of your deposit?

5-30A. (Complex Present Value) You would like to have $75,000 in 15 years. To accumulatethis amount, you plan to deposit an equal sum in the bank each year, which will earn8% interest compounded annually. Your first payment will be made at the end of theyear.

a. How much must you deposit annually to accumulate this amount? b. If you decide to make a lump-sum deposit today instead of the annual deposits,

how large should this lump-sum deposit be? (Assume you can earn 8% on thisdeposit.)

c. At the end of 5 years you will receive $20,000 and deposit this in the bank toward your goal of saving $75,000 at the end of 15 years. In addition to this





deposit, how much must you deposit in equal annual deposits to reach your goal? (Again, assume that you can earn 8% on this deposit.)

5-31A. (Comprehensive Present Value) You are trying to plan for retirement in 10 years, andcurrently you have $150,000 in a savings account and $250,000 in stocks. In addition,

you plan to add to your savings by depositing $8,000 per year in your savings account at the end of each of the next 5 years and then $10,000 per year at the end of each year for the final 5 years until retirement.

a. Assuming your savings account returns 8% compounded annually while your investments in stocks will return 12% compounded annually, how much willyou have at the end of 10 years? (Ignore taxes.)

b. If you expect to live for 20 years after you retire, and at retirement you depositall of your savings in a bank account paying 11 percent, how much can youwithdraw each year after retirement (20 equal withdrawals beginning one year after you retire) to end up with a zero balance at death?

5-32A. (Loan Amortization) On December 31, Eugene Chung borrowed $200,000, agreeing torepay this sum in 20 equal annual installments that included both the principal and 10%interest on the declining balance. How large will the annual payments be?

5-33A. (Loan Amortization) To buy a new house, you must borrow $250,000. To do this, youtake out a $250,000, 30-year, 9% mortgage. Your mortgage payments, which are madeat the end of each year (one payment each year), include both principal and 9% intereston the declining balance. How large will your annual payments be?

5-34A. (Present Value) The state lottery’s million-dollar payout provides for one milliondollars to be paid over 24 years in $40,000 amounts. The first $40,000 payment ismade immediately with the remaining 24 payments occurring at the end of each of thenext 24 years. If 10% is the appropriate discount rate, what is the present value of thisstream of cash flows? If 20% is the appropriate discount rate, what is the present valueof the cash flows?

5-35A. (Solving for i in Compound Interest—Financial Calculator Needed) In March 1963,issue number 39 of Tales of Suspense was issued. The original price for that issue was12 cents. By March of 1997, 34 years later, the value of this comic book had risen to$2,000. What annual rate of interest would you have earned if you had bought thecomic in 1963 and sold it in 1997?

5-36A. (Comprehensive Present Value) You have just inherited a large sum of money and youare trying to determine how much you should save for retirement and how much youcan spend now. For retirement, you will deposit today (January 1, 1997) a lump sum ina bank account paying 10% compounded annually. You do not plan to touch thisdeposit until you retire in 5 years (January 1, 2002), and you plan to live for 20additional years and then to drop dead on December 31, 2021. During your retirement,you would like to receive income of $60,000 per year to be received on the first day of each year, with the first payment on January 1, 2002, and the last payment on January1, 2021. Complicating this objective is your desire to have one final 3-year fling during





which time you’d like to track down all the original members of the “Mr. Ed Show”and “The Monkeys” and get their autographs. To finance this you want to receive$300,000 on January 1, 2017 and nothing on January 1, 2018, and January 1, 2019, asyou will be on the road. In addition, after you pass on (January 1, 2022), you wouldlike to have a total of $100,000 to leave to your children.

a. How much must you deposit in the bank at 10% on January 1, 1997 in order toachieve your goal? (Use a time line in order to answer this question.)

b. What kinds of problems are associated with this analysis and its assumptions?

SOLUTIONS TO ALTERNATIVE PROBLEMS

5-1A. a. FV n = PV (1 + r) n

FV11

= $4,000(1 + 0.09) 11

FV11 = $4,000 (2.580)

FV11 = $10,320

b. FV n = PV (1 + r) n

FV10 = $8,000 (1 + 0.08) 10

FV10 = $8,000 (2.159)

FV10 = $17,272

c. FVn = PV (1 + r) n

FV12 = $800 (1 + 0.12) 12

FV12 = $800 (3.896)

FV12 = $3,117

d. FV n = PV (1 + r) n

FV6 = $21,000 (1 + 0.05) 6

FV6 = $21,000 (1.340)FV6 = $28,140





5-2A. a. FV n = PV (1 + r) n

$1,043.90 = $550 (1 + 0.06) n

1.898 = FUTURE VALUE FACTOR 6%, n yr.

Thus, n = 11 years (because the value of 1.898 occurs in the 11-year row of the 6% column of Appendix B).

b. FV n = PV (1 + r) n

$88.44 = $40 (1 + .12) n


Thus, n = 7 years

c. FVn = PV (1 + r) n

$614.79 = $110 (1 + 0.24) n


Thus, n = 8 years

d. FVn = PV (1 + r) n

$78.30 = $60 (1 + 0.03) n


Thus, n = 9 years

5-3A a. FVn

= PV (1 + r) n

$1,898.60 = $550 (1 + r) 13

3.452 = FUTURE VALUE FACTOR i%, 13 yr.

Thus, i = 10% (because the Appendix B value of 3.452 occurs in the12-year row in the 10% column)

b. FV n = PV (1 + r) n

$406.18 = $275 (1 + r) 8


Thus, i = 5%

c. FVn = PV (1 + r) n

$279.66 = $60 (1 + r) 20


Thus, i = 8%





d. FVn = PV (1 + r) n

$486.00 = $180 (1 + r) 6


Thus, i = 18%

5-4A. a. PV = FV n

+ nr) (11

PV = $800

+ 10)1.0 1(

1

PV = $800 (0.386)

PV = $308.80

b. PV = FV n

+n

r) (1

1

PV = $400

+ 6)06.0 1(

1

PV = $400 (0.705)

PV = $282.00

c. PV = FV n

+ n)1 1(

1

PV = $1,000

+ 8)05.0 1(1

PV = $1,000 (0.677)

PV = $677

d. PV = FV n

+ nr) (11

PV = $900

+ 9)2.0 1(

1

PV = $900 (0.194)

PV = $174.60





5-5A. a. FV n = PMT ( )

+∑

−

=

1n

0t

tr 1

FV = $500 ( )

+∑

−

=

110

0t

t0.06 1

FV10 = $500 (13.181)

FV10 = $6,590.50

b. FV n = PMT ( )

+∑

−

=

1n

0t

tr 1

FV5 = $150 ( )

+∑

−

=

15

0t

t11.0 1

FV5 = $150 (6.228)FV5 = $934.20

c. FVn = PMT ( )

+∑

−

=

1n

0t

tr 1

FV7 = $35 ( )

+∑−

=

18

0t

t0.07 1

FV7 = $35 (10.260)

FV7 = $359.10

d. FVn = PMT ( )

+∑

−

=

1n

0t

tr 1

FV3 = $25 ( )

+∑

−

=

13

0t

t02.01

FV3 = $25 (3.060)

FV3 = $76.50





5-6A. a. PV = PMT( )

+∑=

n

1ttr 1

1

PV = $3,000( )

+∑=

10

1tt

08.0 1

1

PV = $3,000 (6.710)

PV = $20,130

b. PV = PMT( )

+∑=

n

1ttr 1

1

PV = $50( )

∑= +3

1t t0.03 1

1

PV = $50 (2.829)PV = $141.45

c. PV = PMT( )

+∑=

n

1ttr 1

1

PV = $280( )

+

∑=

8

1tt07.0 1

1

PV = $280 (5.971)

PV = $1,671.88

d. PV = PMT( )

+∑=

n

1ttr 1

1

PV = $600( )

+

∑=

10

1tt0.1 1

1

PV = $600 (6.145)

PV = $3,687.00





5-7A. a. FV n = PV (1 + r) n

compounded for 1 year

FV 1 = $20,000 (1 + 0.07) 1

FV 1 = $20,000 (1.07)FV 1 = $21,400


FV 5 = $20,000 (1 + 0.07) 5

FV 5 = $20,000 (1.403)

FV 5 = $28,060


FV 15 = $20,000 (1 + 0.07) 15

FV 15 = $20,000 (2.759)

FV 15 = $55,180

b. FV n = PV (1 + i) n


FV 1 = $20,000 (1 + 0.09) 1

FV1 = $20,000 (1.090)

FV 1 = $21,800

compounded for 5 years at 9%FV 5 = $20,000 (1 + 0.09) 5

FV 5 = $20,000 (1.539)

FV 5 = $30,780


FV 15 = $20,000 (1 + 0.09) 15

FV 15 = $20,000 (3.642)

FV 15 = $72,840


FV 1 = $20,000 (1 + 0.11) 1

FV 1 = $20,000 (1.11)

FV 1 = $22,200






FV5 = $20,000 (1 + 0.11) 5

FV5 = $20,000 (1.685)

FV5 = $33,700compounded for 15 years at 11%

FV15 = $20,000 (1 + 0.11) 15

FV15 = $20,000 (4.785)

FV15 = $95,700

c. There is a positive relationship between both the interest rate used to compounda present sum and the number of years for which the compounding continues,and the future value of that sum.

5-8A. FVn

= PVmn

r 1 m

+ ÷

Account PV r m nmn

r 1

m + ÷ PV

mnr

1m

+ ÷ Korey Stringer 2,000 12% 6 2 1.268 $2,536Eric Moss 50,000 12 12 1 1.127 56,350Ty Howard 7,000 18 6 2 1.426 9,982Rob Kelly 130,000 12 4 2 1.267 164,710Matt Christopher 20,000 14 2 4 1.718 34,360Juan Porter 15,000 15 3 3 1.551 23,265

5-9A. a. FV n = PV (1 + r) n

FV5 = $6,000 (1 + 0.06) 5

FV5 = $6,000 (1.338)

FV5 = $8,028

b. FV n = PVmn

r 1

m + ÷

FV5 = $6,0002(5)

0.061

2 + ÷

FV5 = $6,000 (1 + 0.03) 10

FV5 = $6,000 (1.344)






FV 2 = 10,000(1 + 0.15) 2

FV 2 = 10,000(1.322)

FV 2 = 13,220 booksYear 3: FV n = PV (1 + r) n

FV 3 = 10,000(1 + 0.15) 3

FV 3 = 10,000(1.521)

FV 3 = 15,210 books

Book sales

20,000

15,000

10,000

1 2 3years

The sales trend graph is not linear because this is a compound growth trend.Just as compound interest occurs when interest paid on the investment duringthe first period is added to the principal of the second period, interest is earnedon the new sum. Book sales growth was compounded; thus, the first year thegrowth was 15% of 10,000 books, the second year 15% of 11,500 books, andthe third year 15% of 13,220 books.

5-12A. FV n = PV (1 + r) n

FV 1 = 41(1 + 0.12) 1

FV 1 = 41(1.12)

FV 1 = 45.92 Home Runs in 1981 (in spite of the baseball strike).FV 2 = 41(1 + 0.12) 2

FV 2 = 41(1.254)

FV 2 = 51.414 Home Runs in 1982

FV 3 = 41(1 + 0.12) 3

FV 3 = 41(1.405)





FV3 = 57.605 Home Runs in 1983.

FV3 = 41(1 + 0.12) 4

FV4 = 41(1.574)

FV4 = 64.534 Home Runs in 1984 (for what at that time would have been a new major league record).

FV5 = 41(1 + 0.12) 5

FV5 = 41(1.762)

FV5 = 72.242 Home Runs in 1985 (again for a new major league record, but not up to Barry Bonds’ 73 homers in 2001).

Actually, Reggie never hit more than 41 home runs in a year. In 1982, he hit 15 only; in1983 he hit 39; in 1984, he hit 14; in 1985, 25; and 26 in 1986. He retired at the end of 1987 with 563 career home runs.

5-13A. PV = PMT( )

+∑=

n

1ttr 1

1

$120,000 = PMT ( )

+∑

=

25

1t 1.0 11

t

$120,000 = PMT(9.077)

Thus, PMT = $13.220.23 per year for 25 years

5-14A. FV n = PMT ( )

+∑

−

=

1n

0t

tr 1

$25,000 = PMT ( )

+∑

−

=

115

0t

t07.0 1

$25,000 = PMT(25.129)

Thus, PMT = $994.87





5-15A. FV n = PV (1 + r) n

$2,376.50 = $700 (FUTURE VALUE FACTOR r%, 10 yr. )

3.395 = FUTURE VALUE FACTOR i%, 10 yr

Thus, i = 13%5-16A. The value of the home in 10 years

FV 10 = PV (1 + .05) 10

= $125,000(1.629)

= $203,625

How much must be invested annually to accumulate $203.625?

$203,625 = PMT ( )

+∑

−

=

110

0t

t10. 1

$203,625 = PMT(15.937)

PMT = $12,776.87

5-17A. FV n = PMT ( )

+∑

−

=

1n

0t

tr 1

$15,000,000 = PMT ( )

+∑

−

=

110

0t

t.10 1

$15,000,000 = PMT(15.937)

Thus, PMT = $941,2065-18A. One dollar at 24.0% compounded monthly for one year

FVn = PV (1 + r) n

FV 12 = $1(1 + .02) 12

= $1(1.268)

= $1.268

One dollar at 26.0% compounded annually for one year

FVn = PV (1 + r) n

FV 1 = $1(1 + .26) 1

= $1(1.26)

= $1.26

The loan at 26% compounded monthly is more attractive.





5-19A. Investment A

PV = PMT( )

+∑=

n

1ttr 1

i

= $15,000( )

+∑=

5

1tt20 . 1

1

= $15,000(2.991)

= $44,865

Investment B

First, discount the annuity back to the beginning of year 5, which is the end of year

4. Then, discount this equivalent sum to present.

PV = PMT( )

+∑=

n

1ttr 1

1

= $15,000( )

+

∑=

6

1tt20. 1

1

= $15,000(3.326)

= $49,890—then discount the equivalent sum back to present.

PV = FV n

+ nr) (11

= $49,890

+ 4)20.1(

1

= $49,890(.482)

= $24,046.98





Investment C

PV = FV n

+ nr)(11

= $20,000

+ 1)20.1(1 + $60,000

+ 6)20.1(

1

+ $20,000

+ 10)20.1(

1

= $20,000(.833) + $60,000(.335) + $20,000(.162)

= $16,660 + $20,100 + $3,240

= $40,000

5-20A. PV = FV n

+n

r)(1

1

PV = $1,000

+ 8)09.1(

1

= $1,000(.502)

= $502

5-21A. a. PV =r

PP

PV =

PV = $4,444

b. PV =r

PP

PV =

PV = $11,538

c. PV =r

PP

PV =

PV = $1,500d. PV =

r PP

PV =

PV = $1,667





5-22A. FV n = PVmn

mr

1

+

7 = 12n

0.101

2 + ÷

7 = (1 + 0.05) 2n

7 = FUTURE VALUE FACTOR 5%, 2n yr.

A value of 7.040 occurs in the 5% column and 40-year row of the table in Appendix B.Therefore, 2n = 40 years, and n = approximately 20 years.

5-23A. The Present value of the $10,000 annuity over years 11-15.

PV = PMT( ) ( )

+

+

∑∑==

10

1tt

15

1tt .07 1

1 -

07. 1

1

= $10,000(9.108 - 7.024)

= $10,000(2.084)

= $20,840

The present value of the $15,000 withdrawal at the end of year 15:

PV = FV 15

+ 15)07.1(

1

= $15,000(.362)

= $5,430

Thus, you would have to deposit $20,840 + $5,430 or $26,270 today.

5-24A. PV = PMT( )

+

∑=

10

1tt09. 1

1

$45,000 = PMT(6.418)

PMT = $7,012





5-25A. PV = PMT( )

+∑=

5

1tti 1

1

$45,000 = $9,000 (ANNUITY PRESENT VALUE FACTOR r%, 5 yr. )

5.0 = ANNUITY PRESENT VALUE FACTOR i%, 5 yr.

i = 0%

5-26A. PV = FV n

+ nr)(11

$15,000 = $37,313 (PRESENT VALUE FACTOR r%, 5 yr. )

.402 = PRESENT VALUE FACTOR 20%, 5 yr.

Thus, i = 20%

5-27A. PV = PMT( )

+∑=

n

1ttr 1

1

$30,000 = PMT( )

+

∑=

4

1tt13. 1

1

$30,000 = PMT(2.974)

PMT = $10,087

5-28A. The present value of $10,000 in 12 years at 11% is:

PV = FV n t1

(1 r) ÷+

PV = $10,000 12

1

(1 .11)

÷+

PV = $10,000 (.286)

PV = $2,860





The present value of $25,000 in 25 years at 11% is:

PV = $25,000 25

1

(1 .11)

÷+

= $25,000 (.074)= $1,850

Thus, take the $10,000 in 12 years.

5-29A. FV n = PMT ( )

+∑

−

=

1n

0t

tr 1

$30,000 = PMT ( )

+∑

−

=

15

0t

t10. 1

$30,000 = PMT(6.105)

PMT = $4,914

5-30A. a. FV = ( )

+∑

−

=

1n

0t

tr 1

$75,000 = ( )

+∑

−

=

115

0t

t08. 1

$75,000 = PMT (ANNUITY FUTURE VALUE FACTOR 8%, 15 yr. )

$75,000 = PMT(27.152)

PMT = $2,762.23 per year

b. PV = FV n

+ nr)(11

PV = $75,000 (PRESENT VALUE FACTOR 8%, 15 yr. )

PV = $75,000(.315)

PV = $23,625 deposited today





d. Compound annuity of $2,000 (years 6-10)

FV 5 = $2,000 ( )

+∑

−

=

15

0t

t08. 1

= $2,000 (5.867)= $11,734

At the end of 10 years, you will have $323,850 + $776,500 + $115,896+ $11,734 = $1,227,980.

PV = PMT( )

+∑=

20

1tt11. 1

1

$1,227,980 = PMT (7.963)

PMT = $154,210.72

5-32A. PV = PMT (ANNUITY PRESENT VALUE FACTOR r%, n yr. )

$200,000 = PMT (ANNUITY PRESENT VALUE FACTOR 10%, 20 yr. )

$200,000 = PMT(8.514)

PMT = $23,491

5-33A. PV = PMT (ANNUITY PRESENT VALUE FACTOR r%, n yr .)

$250,000 = PMT (ANNUITY PRESENT VALUE FACTOR 9%, 30 yr. )

$250,000 = PMT(10.274)

PMT = $24,333

5-34A. At 10%:

PV = $40,000 + $40,000 (ANNUITY PRESENT VALUEFACTOR 10%, 24 yr. )

PV = $40,000 + $40,000 (8.985)

PV = $40,000 + $359,400

PV = $399,400





At 20%:

PV = $40,000 + $40,000 (ANNUITY PRESENT VALUEFACTOR 20%, 24 yr. )

PV = $40,000 + $40,000 (4.938)

PV = $40,000 + $197,520

PV = $237,520

5-35A. FV = PMT (FUTURE VALUE FACTOR r%, n yr. )

$2,000 = .12(FUTURE VALUE FACTOR i%, 35 yr. )

Solving using a financial calculator:

r = 32.70%

5-36A. a.

$60,000 per year $300,000

$60,000

$100,000

1/06 1/11

1/16 1/21 1/26 1/31

There are a number of equivalent ways to discount these cash flows back to present, one of which is as follows (in equation form):

PV = $60,000 (ANNUITY PRESENT VALUE FACTOR 10%, 19 yr.

– ANNUITY PRESENT VALUE FACTOR 10%, 4 yr. )

+ $300,000 (PRESENT VALUE FACTOR 10%, 20 yr. )

+ $60,000 (PRESENT VALUE FACTOR 10%, 23 yr. + PRESENT VALU

+ $100,000 (PRESENT VALUE FACTOR 10%, 25 yr. )

= $60,000 (8.365-3.170) + $300,000 (.149)

+ $60,000 (0.112 + .102) + $100,000 (.092)

= $311,700 + $44,700 + $12,840 + $9,200

= $378,440

b. If you live longer than expected, you could end up with no money later on in life.5-37A. rate (r) = 8%





number of periods (n) = 7 payment (PMT) = $0

present value (PV) = $900type (0 = at end of period) = 0

Future value = $1,542.44Excel formula: =FV(rate, number of periods, payment, present value, type)

Notice that present value ($900) took on a negative value.

5-38A. In 20 years you’d like to have $250,000 to buy a home, but you only have $30,000. Atwhat rate must your $30,000 be compounded annually for it to grow to $250,000 in 20years?

number of periods (n) = 20 payment (PMT) = $0

present value (PV) = $30,000future value (FV) = $250,000

type (0 = at end of period) = 0guess =

r = 11.18%

Excel formula: =RATE(number of periods, payment, present value, future value, type,guess)

Notice that present value ($30,000) took on a negative value.

5-39A. To buy a new house you take out a 25 year mortgage for $300,000. What will your monthly interest rate payments be if the interest rate on your mortgage is 8 percent?

Two things to keep in mind when you're working this problem: first, you'll have toconvert the annual rate of 8 percent into a monthly rate by dividing it by 12, andsecond, you'll have to convert the number of periods into months by multiplying 25times 12 for a total of 300 months.

Excel formula: =PMT(rate, number of periods, present value, future value, type)

rate (r) = 8%/12

number of periods (n) = 300 present value (PV) = $300,000future value (FV) = $0type (0 = at end of period) = 0

monthly mortgage payment = ($2,315.45)

Notice that monthly payments take on a negative value because you pay them.





d. N = 10

CPT I/Y = 14.87%

PV = –15 billion

PMT = 0

FV = 60. billion

e. N = 40

I/Y = 7

PV = –30. billion

CPT PMT = 2.25 billion dollars

FV = 0

5-41A. What will the car cost in the future?

N = 6

I/Y = 3

PV = –15,000

PMT = 0


How much must Bart put in an account today in order to have $17,910.78 in 6years?

N = 6

I/Y = 7.5

CPT PV = –11,605.50 dollars

PMT = 0

FV = 17,910.78

5-42A. N = 45

I/Y = 8.75

PV = 0

CPT PMT = –2,054.81 dollarsFV = 1,000,000





5-43A. First, we must calculate what Mr. Burns will need in 20 years. Once we know howmuch he will need, we can calculate how much he needs to deposit each year in order to come up with that amount (note: once you calculate the present value, you mustmultiply your answer, in this case -$4.192 billion times (1 + i) because this is anannuity due):

N = 10I/Y = 20CPT PV = –4.1925 billion × 1.20 = -5.031 billion dollars

PMT = 1 billionFV = 0

Next, we will determine how much Mr. Burns needs to deposit each year for 20 yearsto reach this goal of accumulating $5.031 billion at the end of the 20 years:

N = 20I/Y = 20PV = 0CPT PMT= -26.9 million dollarsFV = 5.031 billion

5-44A. What’s the $100,000 worth in 25 years (keep in mind that Homer invested the money 5years ago and we want to know what it will be worth in 20 years)?

N = 25

I/Y = 7.5

PV = –100,000

PMT = 0CPT FV = 609,833.96 dollars

Now we determine what the additional $1,500 per year will grow to (note that sinceHomer will be making these investments at the beginning of each year for 20 years wehave an annuity due, thus, once you calculate the present value, you must multiply your answer, in this case $64,957.02 times (1 + r)):

N = 20

I/Y = 7.5

PV = 0

PMT = –1,500

CPT FV = 64,957.02 × 1.075 = 69,828.80 dollars

Finally, we must add the two values together:

$609,833.96 + $69,828.80 = $679,662.76





5-45A. Since this problem involves monthly payments we must first, make P/Y = 12 . Then, N becomes the number of months or compounding periods,

N = 60

I/Y = 6.2

PV = –25,000

CPT PMT = 485.65 dollars

FV = 0

5-46A. Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,

N = 36

CPT I/Y = 11.62%

PV = –999

PMT = 33

FV = 0

5-47A. First, what will be the monthly payments if Suzie goes for the 4.9 percent financing?Since this problem involves monthly payments we must first, make P/Y = 12. Then, N

becomes the number of months or compounding periods,

N = 60

I/Y = 4.9

PV = –25,000

CPT PMT = 470.64 dollarsFV = 0

Now, calculate how much the monthly payments would be if Suzie took the $1,000cash back and reduced the amount owed from $25,000 to $24,000. Again, since this

problem involves monthly payments we must first, make P/Y = 12.

N = 60

I/Y = 6.9

PV = -24,000

CPT PMT = 474.10 dollars

FV = 0





5-53A. Since this problem involves quarterly compounding we must first, make P/Y = 4. Then, N becomes the number of quarters or compounding periods,

N = 16

I/Y = 6.4%

PV = 0

PMT = –1000


5-49A. Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,

CPT N = 41.49 (rounded up to 42 months)

I/Y = 12.9

PV = -5000

PMT = 150

FV = 0

5-50A. a. Since this problem begins using annual payments, make sure your calculator isset to P/Y=1.

N = 12

CPT I/Y = 8.37%

PV = –160,000

PMT = 0FV = 420,000

b. Again, since this problem begins using annual payments, make sure your calculator is set to P/Y=1

N = 10

CPT I/Y = 11.6123%

PV = –140,000

PMT = 0

FV = 420,000





c. Since this problem now involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,

N = 120

I/Y = 6

PV = –140,000

CPT PMT = –1,008.57 dollars

FV = 420,000

d. Since this problem now involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods. Also,since Professor ME will be depositing both the $140,000 (immediately) and$500 (monthly), they must have the same sign,

N = 120

CPT I/Y = 8.48%PV = –140,000

PMT = –500

FV = 420,000

fof im chapter 05 - 7th

Documents