Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, an auction house sold a sculpture at auction for a price of $10,341,500. Unfortunately for the previous owner, he had purchased it in 1999 at a price of $12,437,500.

Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, an auction house sold a sculpture at auction for a price of $10,341,500. Unfortunately for the previous owner, he had purchased it in 1999 at a price of $12,437,500.

  

What was his annual rate of return on this sculpture? (Negative amount should be indicated by a minus sign. Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Annual rate of return

%

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

  

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t – 1

r = ($10,341,500 / $12,437,500)1/4 – 1 = – 4.51%

Notice that the interest rate is negative. This occurs when the FV is less than the PV.

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

4

±$12,437,500

$10,341,500

N

I/Y

PV

PMT

FV

Solve for

– 4.51%

Assume that in 2010, a gold dollar minted in 1893 sold for $127,000. For this to have been true, what rate of return did this coin return for the lucky numismatist? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

Assume that in 2010, a gold dollar minted in 1893 sold for $127,000. For this to have been true, what rate of return did this coin return for the lucky numismatist? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Rate of return

%

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t – 1

r = ($127,000 / $1)1/117 – 1 = 0.1057, or 10.57%

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

117

$1

±$127,000

N

I/Y

PV

PMT

FV

Solve for

10.57%

In 1895, the first Putting Green Championship was held. The winner’s prize money was $180. In 2010, the winner’s check was $1,380,000.

In 1895, the first Putting Green Championship was held. The winner’s prize money was $180. In 2010, the winner’s check was $1,380,000.

 

What was the percentage increase per year in the winner’s check over this period? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

Interest rate

%

If the winner’s prize increases at the same rate, what will it be in 2037? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

Future value

$

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

   

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t – 1

r = ($1,380,000 / $180)1/115 – 1 = 0.0809, or 8.09%

To find the FV of the first prize in 2037, we use:

FV = PV(1 + r)t

FV = $1,380,000(1.0809)27 = $11,269,824.51

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

115

±$180

$1,380,000

N

I/Y

PV

PMT

FV

Solve for

8.09%

Enter

27

8.09%

$1,380,000

N

I/Y

PV

PMT

FV

Solve for

$11,269,824.51

Your coin collection contains 58 1952 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2068, assuming they appreciate at a 4.9 percent annual rate? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Future value $ Explanation: To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $58(1.049)116 = $14,906.81 Calculator Solution: Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation. Enter 116 4.9% $58 N I/Y PV PMT FV Solve for $14,906.81

Your coin collection contains 58 1952 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2068, assuming they appreciate at a 4.9 percent annual rate? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Future value

$

Explanation:

To find the FV of a lump sum, we use:

FV = PV(1 + r)t

FV = $58(1.049)116 = $14,906.81

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

116

4.9%

$58

N

I/Y

PV

PMT

FV

Solve for

$14,906.81

You have just received notification that you have won the $1 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you’re around to collect), 65 years from now. What is the present value of your windfall if the appropriate discount rate is 9 percent? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Present value $ Explanation: To find the PV of a lump sum, we use: PV = FV / (1 + r)t PV = $1,000,000 / (1.09)65 = $3,692.14 Calculator Solution: Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation. Enter 65 9% $1,000,000 N I/Y PV PMT FV Solve for $3,692.14

You have just received notification that you have won the $1 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you’re around to collect), 65 years from now.

 

What is the present value of your windfall if the appropriate discount rate is 9 percent? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Present value

$

Explanation:

To find the PV of a lump sum, we use:

PV = FV / (1 + r)t

PV = $1,000,000 / (1.09)65 = $3,692.14

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

65

9%

$1,000,000

N

I/Y

PV

PMT

FV

Solve for

$3,692.14

You're trying to save to buy a new $191,000 Ferrari. You have $41,000 today that can be invested at your bank. The bank pays 4.9 percent annual interest on its accounts. How long will it be before you have enough to buy the car?

You're trying to save to buy a new $191,000 Ferrari. You have $41,000 today that can be invested at your bank. The bank pays 4.9 percent annual interest on its accounts.

 

How long will it be before you have enough to buy the car? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Number of years

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for t, we get:

t = ln(FV / PV) / ln(1 + r)

t = ln($191,000 / $41,000) / ln(1.049) = 32.17 years

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

4.9%

$41,000

±$191,000

N

I/Y

PV

PMT

FV

Solve for

32.17

Assume that in January 2010, the average house price in a particular area was $276,400. In January 2002, the average price was $193,300. What was the annual increase in selling price?

Assume that in January 2010, the average house price in a particular area was $276,400. In January 2002, the average price was $193,300.

  

What was the annual increase in selling price? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Annual increase in selling price

%

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t – 1

r = ($276,400 / $193,300)1/8 – 1 = 0.0457, or 4.57%

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

8

$193,300

±$276,400

N

I/Y

PV

PMT

FV

Solve for

4.57%

At 6.10 percent interest, how long does it take to double your money? (Round your answer to 2 decimal places. (e.g., 32.16))

At 6.10 percent interest, how long does it take to double your money? (Round your answer to 2 decimal places. (e.g., 32.16))

 

Length of time

years

At 6.10 percent interest, how long does it take to quadruple it? (Round your answer to 2 decimal places. (e.g., 32.16))

Length of time

years

Explanation:

To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for t, we get:

t = ln(FV / PV) / ln(1 + r)

The length of time to double your money is:

FV = $2 = $1(1.061)t

t = ln(2) / ln(1.061) = 11.71 years

The length of time to quadruple your money is:

FV = $4 = $1(1.061)t

t = ln(4) / ln(1.061) = 23.41 years

  

Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (the difference in these answers is due to rounding). This is an important concept of time value of money.

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

6.10%

$1

±$2

N

I/Y

PV

PMT

FV

Solve for

11.71

Enter

6.10%

$1

±$4

N

I/Y

PV

PMT

FV

Solve for

23.41

Solve for the unknown number of years in each of the following (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)):

Solve for the unknown number of years in each of the following (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)):

 

	Present Value			Years	Interest Rate			Future Value
	$	510			9	%		$	1,212
		760			10				1,629
		17,900			17				260,563
		21,000			15				391,887

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for t, we get:

t = ln(FV / PV) / ln(1 + r)

  

FV = $1,212 = $510(1.09)t;		t = ln($1,212/ $510) / ln(1.09)	=	10.04 years
FV = $1,629 = $760(1.10)t;		t = ln($1,629/ $760) / ln(1.10)	=	8.00 years
FV = $260,563 = $17,900(1.17)t;		t = ln($260,563 / $17,900) / ln(1.17)	=	17.06 years
FV = $391,887 = $21,000(1.15)t;		t = ln($391,887 / $21,000) / ln(1.15)	=	20.94 years

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

9%

$510

±$1,212

N

I/Y

PV

PMT

FV

Solve for

10.04

Enter

10%

$760

±$1,629

N

I/Y

PV

PMT

FV

Solve for

8.00

Enter

17%

$17,900

±$260,563

N

I/Y

PV

PMT

FV

Solve for

17.06

Enter

15%

$21,000

±$391,887

N

I/Y

PV

PMT

FV

Solve for

20.94

Solve for the unknown interest rate in each of the following (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)):

Solve for the unknown interest rate in each of the following (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)):

 

Present Value				Years		Interest Rate		Future Value
	$	330		4		%		$	422
		450		18					1,571
		48,000		19					266,917
		47,261		25					803,425

Explanation:

We can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:

 

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t – 1

  

FV = $422 = $330(1 + r)4;		r = ($422 / $330)1/4 – 1	=	0.0634, or 6.34%
FV = $1,571 = $450(1 + r)18;		r = ($1,571 / $450)1/18 – 1	=	0.0719, or 7.19%
FV = $266,917 = $48,000(1 + r)19;		r = ($266,917 / $48,000)1/19 – 1	=	0.0945, or 9.45%
FV = $803,425 = $47,261(1 + r)25;		r = ($803,425 / $47,261)1/25 – 1	=	0.1200, or 12.00%

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

4

$330

±$422

N

I/Y

PV

PMT

FV

Solve for

6.34%

Enter

18

$450

±$1,571

N

I/Y

PV

PMT

FV

Solve for

7.19%

Enter

19

$48,000

±$266,917

N

I/Y

PV

PMT

FV

Solve for

9.45%

Enter

25

$47,261

±$803,425

N

I/Y

PV

PMT

FV

Solve for

12.00%

For each of the following, compute the present value (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)):

For each of the following, compute the present value (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)):

 

Present Value	Years		Interest Rate			Future value
$	14		8	%		$	16,451
	5		14				61,557
	30		15				896,073
	35		8				560,164

Explanation:

To find the PV of a lump sum, we use:

 

PV = FV / (1 + r)t

PV = $16,451 / (1.08)14	=	$	5,600.92
PV = $61,557 / (1.14)5	=	$	31,970.78
PV = $896,073 / (1.15)30	=	$	13,533.44
PV = $560,164 / (1.08)35	=	$	37,886.44

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

14

8%

$16,451

N

I/Y

PV

PMT

FV

Solve for

$5,600.92

Enter

5

14%

$61,557

N

I/Y

PV

PMT

FV

Solve for

$31,970.78

Enter

30

15%

$896,073

N

I/Y

PV

PMT

FV

Solve for

$13,533.44

Enter

35

8%

$560,164

N

I/Y

PV

PMT

FV

Solve for

$37,886.44