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The Time Value of 
Money 
Chapter 6
Learning Objectives 
• Explain why a dollar today is worth more than a dollar in the 
future 
• Define the terms future value 
• Calculate the future value of an amount and an annuity
The Time Value of Money 
• Consume Today or Tomorrow? 
• TVM is based on the belief that people prefer 
to consume goods today rather than wait to 
consume the same goods tomorrow 
•An apple we can have today is more 
valuable to us than an apple we can have 
in one year. 
•Money has a time value because buying 
an apple today is more important than 
buying an apple in one year.
The Time Value of Money 
• Consume Today or Tomorrow? 
• A dollar someone has today can be spent for 
consumption or loaned to earn interest 
• A dollar loaned earns interest that increases 
wealth and the ability to consume 
• The rate of interest determines the trade-off 
between consumption today and saving 
(investing)
The Time Value of Money 
• Future Value versus Present Value 
• Cash-flows are evaluated based on future value or present value 
• Future value measures what cash-flows are worth after a certain 
amount of time has passed 
• Present value measures what future cash-flows are worth 
before a certain amount of time has passed
Comparison of Future Value and 
Present Value
The Time Value of Money 
• Future Value versus Present Value 
• Compounding is the process of increasing cash-flows to a future 
value 
• Discounting is the process of reducing future cash-flows to a 
present value
Future Value 
• What a dollar invested today will be worth in the future depends on 
 Length of the investment period 
Method to calculate interest 
 Interest rate 
 2 types of methods to calculate interest 
Simple method-calculated only on the original principal each year 
Compound interest-calculated on both the original principal and on any 
accumulated interest earned up to that point. Future value implies the 
compound method 
Interest rate 
Simple interest is paid on the original principal amount only 
Compound interest consists of both simple interest and interest-on-interest
Chapter 6: The Time Value of Money
Chapter 6: The Time Value of Money
Present Value of an Amount to be Received in the 
Future 
• Taking future values back to the present is called discounting 
Present Value=Future Value x Present Value Factor 
PV = Present value (initial investment amount) 
i = the interest rate 
n= no. of time periods of the investment
Example 1 
• Suppose we place $10000 in a savings account that pays 10% 
interest compounded annually. How will our savings grow? 
Value at the end of year 1 = present value X (1+i) 
i = interest rate 
• = $10000 ퟏ + ퟎ. ퟏ ퟒ 
• = $10000 (1.4641) 
• = $14,641
Future Value: Example 2 
• If we place $1,000 in a savings account paying 5% interest 
compounded annually, how much will our account accrue to in 10 
years? 
• Future value = present value X (ퟏ + 풓)풏 
• 푭푽풏 = $1,000 (ퟏ + ퟎ. ퟎퟓ)ퟏퟎ 
• = $1,000 (1.62889) 
• = $1,628.89 
13 
Refer Table B-1
Example 3: 
• What is the future value of $10,000 with an interest rate of 16 
percent and one annual period of compounding? With an 
annual interest rate of 16 percent and two semiannual periods 
of compounding? With an annual interest rate of 16 percent 
and four quarterly periods of compounding? 
Annually: 
푭푽 = 푷푽 ∗ (ퟏ + 풊)풏 
푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ+. ퟏퟔ)ퟏ 
푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ. ퟏퟔ)ퟏ 
푭푽 = $ퟏퟏ, ퟔퟎퟎ
Semi-annually: 
푭푽 = 푷푽 푿 ퟏ + 
풊 
풎 
풏푿풎 
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ + 
.ퟏퟔ 
ퟐ 
ퟏ푿ퟐ 
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟖ ퟐ 
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟔퟒ 
• 푭푽 = $ퟏퟏ, ퟔퟔퟒ 
• Quarterly: 
• 푭푽 = 푷푽 푿 ퟏ + 
풊 
풎 
풏푿풎 
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ + 
.ퟏퟔ 
ퟒ 
ퟏ푿ퟒ 
푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟒ ퟒ 
푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟗퟗ 
푭푽 = $ퟏퟏ, ퟔퟗퟗ
Present Value and Discounting 
• Present Value concepts 
• A present value calculation takes end-of-the-period cash flows 
and reverses the effect of compounding to determine the 
equivalent beginning-of-the-period cash flows 
•Present Value Equation 
•P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 푿 푷푽푭풊,풏 
•P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [ 
ퟏ 
(ퟏ+풓)풏] 
•This is discounting and the interest rate i is called the discount rate. 
•Present value (PV) is often referred to as the discounted value of 
future cash-flows.
Present Value and Discounting 
• Time and the discount rate affect present value 
•The greater the amount of time before a cash flow is to occur, 
the smaller the present value of the cash-flow. 
•The higher the discount rate, the smaller the present value of a 
future cash-flow.
Example 4 
What is the value of $500 to be received 10 years from today if our 
discount rate is 6%. 
P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [ 
ퟏ 
(ퟏ+풓)풏] 
FV = $500, n = 10, r = 6% or 0.06 
= $500 [ 
ퟏ 
(ퟏ+ퟎ.ퟎퟔ)ퟏퟎ] 
= $500 (0.558) 
= $279.20 
18 
Refer Table B -3
Example 5 
What is the present value of an investment that yields $1,000 to be 
received in 7 years and $1,000 to be received in 10 years if the discount 
rate is 6 percent? 
ퟏ 
풑풓풆풔풆풏풕 풗풂풍풖풆 = 푭푽풏 [ 
(ퟏ+풓)풏] + 푭푽풏 [ 
ퟏ 
(ퟏ+풓)풏] 
PV= $ퟏ, ퟎퟎퟎ [ 
ퟏ 
(ퟏ+ퟎ.ퟎퟔ)ퟕ] + $1,000 [ 
ퟏ 
(ퟏ+ퟎ.ퟎퟔ)ퟏퟎ] 
• = $1,000 (0.665) + $1,000 (0.558) 
• = $665 + $558 
• = $1223
Future and Present Values of Annuities 
• Annuity- A series of equal payment made or received at 
regular time intervals 
• Ordinary Annuity- A series of equal annuity payments made 
or received at the end of each period 
• Future value of an annuity-What an equal series of payments 
will be worth at some future date 
• Future Value Factor of an Annuity (FVFA)-A factor that when 
multiplied by a stream of equal payments equals the future 
value of that stream 
• Future Value of an Annuity Table- Table of factors that shows 
the future value of equal flows at the end of each period, 
given a particular interest rate
Example 6 
Carlos Menendez is planning to invest $3,500 every year for the next six years in 
an investment paying 12 percent annually. What will be the amount he will have at 
the end of the six years? (Round to the nearest dollar.) 
A) $21,000 
B) $28,403 
C) $24,670 
D) $26,124
Chapter 6: The Time Value of Money
Example 7 
You plan to save $1,250 at the end of each of the next three years to pay for a 
vacation. If you can invest it at 7 percent, how much will you have at the end of 
three years? (Round to the nearest dollar.) 
A) $3,750 
B) $3,918 
C) $4,019 
D) $4,589
Chapter 6: The Time Value of Money
Example 8: 
Maricela Sanchez needs to have $25,000 in five years. If she can earn 8 percent on 
any investment, what is the amount that she will have to invest every year at the 
end of each year for the next five years? (Round to the nearest dollar.) 
A) $5,000 
B) $4,261 
C) $4,640 
D) $4,445
Chapter 6: The Time Value of Money
Example 9 
Jane Ogden wants to save for a trip to Australia. She will need $12,000 at the end of 
four years. She can invest a certain amount at the beginning of each of the next four 
years in a bank account that will pay her 6.8 percent annually. How much will she 
have to invest annually to reach her target? (Round to the nearest dollar.) 
A) $3,000 
B) $2,980 
C) $2,538 
D) $2,711
Chapter 6: The Time Value of Money
Annuities Continued 
• Present Value of an Annuity- What the series of payments in 
the future is worth today 
• Present Value Factor of an Annuity (PVFA)- A factor that 
when multiplied by a stream of equal payments equals the 
present value of that stream 
• Present Value of an Annuity Table- Table of factors that 
shows the value today of equal flows at the end of each 
future period, given a particular interest rate
How to calculate present value of an 
annuity 
(6.1) 30 
o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 
PVA CF PVFA 
PVFA 
1 
  
(1 ) 
1 
  
 
1 
0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
  
i 
i 
CF 
i 
CF 
n 
ퟏ−풑풓풆풔풆풏풕 풗풂풍풖풆 풇풂풄풕풐풓 
풓 
o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 
ퟏ−(ퟏ+풓)−풏 
풓
Level Cash Flows: Annuities and 
• A contract will pay $2,000 at the end of each year for three 
years and the appropriate discount rate is 8%. What is a fair 
price for the contract? 
$5,154.19 
Perpetuities 
• Present Value of an Annuity Example 
(1 1/(1 0.08) 
   
0.08 
PVA $2000 
3 
 
3  
 
 
  
Refer Table B-4 31
Example 1 
What is the value of 10-year $1,000 annuity discounted back to 
the present at 5 percent?
푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 
ퟏ − (ퟏ + 풓)−풏 
풓 
푷푽 = $ퟏ, ퟎퟎퟎ 
ퟏ − (ퟏ + ퟎ. ퟎퟓ)−ퟏퟎ 
ퟎ. ퟎퟓ 
• = $1,000 (7.722) 
• = $7,722
Example 2 
• Transit Insurance Company has made an investment in another company that will 
guarantee it a cash flow of $37,250 each year for the next five years. If the company uses a 
discount rate of 15 percent on its investments, what is the present value of this 
investment? (Round to the nearest dollar.) 
A) $101,766 
B) $124,868 
C) $251,154 
D) $186,250 
Annual payment = PMT = $37,250 
No. of payments = n = 5 
Required rate of return = 15% 
Present value of investment = PVA5
  
    
   
5 
1 
1 
(1 ) 
n 
1 
1 
(1.15) 
$37,250 $37,250 3.3522 
0.15 
n 
i 
PVA PMT 
i 
  
  
  
   
      
  
  
 $124,867.78
Example 3 
Herm Mueller has invested in a fund that will provide him a cash flow of $11,700 
for the next 20 years. If his opportunity cost is 8.5 percent, what is the present value 
of this cash flow stream? (Round to the nearest dollar.) 
A) $234,000 
B) $132,455 
C) $110,721 
D) $167,884
  
    
   
20 
1 
1 
(1 ) 
1 
1 
n 
(1.085) 
$11,700 $11,700 9.4633 
0.085 
n 
i 
PVA PMT 
i 
  
  
  
   
      
  
  
 $110,721.04
Example 4 
Myers, Inc., will be making lease payments of $3,895.50 for a 10-year period, 
starting at the end of this year. If the firm uses a 9 percent discount rate, what is 
the present value of this annuity? (Round to the nearest dollar.) 
A) $23,250 
B) $29,000 
C) $25,000 
D) $20,000
  
    
   
10 
1 
1 
(1 ) 
1 
1 
n 
(1.09) 
$3,895.50 $3,895.50 6.4177 
0.09 
n 
i 
PVA PMT 
i 
  
  
  
   
      
  
  
 $24,999.99
Ordinary Annuity versus Annuity 
Due 
• Present Value of Annuity Due 
•Cash flows are discounted for one period less than in an ordinary 
annuity. 
• Future Value of Annuity Due 
•Cash flows are earn compound interest for one period more than 
in an ordinary annuity. 
40 
PVA PVA (1 ) (6.4) 
1 
Due 
1 
Due 
i 
   
FVA FVA (1 i 
) 
  
Ordinary Annuity versus Annuity Due 
41
Perpetuities 
• A stream of equal cash flows that goes on forever 
• Preferred stock and some bonds are perpetuities 
• Equation for the present value of a perpetuity can be derived 
from the present value of an annuity equation 
42 
PVP CF esent value factor for an annuity 
1 
 
 
1 1 0 
( i) 
( . ) 
  
Pr 0 
CF 
i 
( ) 
i 
CF 
i 
CF 
6 3 
 
1 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
  

Valuing Perpetuity 
Example 1 
• Suppose you decide to endow a chair in finance. The goal of the 
endowment is to provide $100,000 of financial support per year 
forever. If the endowment earns a rate of 8%, how much money 
will you have to donate to provide the desired level of support? 
43 
$1,250,000 
$100,000 
CF 
0    
0.08 
i 
PVP
Example 2 
• What is the value of a $500 perpetuity discounted back to 
the present at 8 percent?
푷푽 = 
푪푭 
풊 
푷푽 = 
$ퟓퟎퟎ 
ퟎ.ퟎퟖ 
= $6250
Example 3 
Your father is 60 years old and wants to set up a cash flow stream that would be 
forever. He would like to receive $20,000 every year, beginning at the end of this 
year. If he could invest in account earning 9 percent, how much would he have to 
invest today to receive his perpetual cash flow? (Round to the nearest dollar.) 
A) $222,222 
B) $200,000 
C) $189,000 
D) $235,200
Annual payment needed = PMT = $20,000 
Investment rate of return = i = 9% 
Term of payment = Perpetuity 
Present value of investment needed = PV 
PMT $20,000 
PV of Perpetuity 
  
i 0.09 
 $222,222.22
Example 4 
A lottery winner was given a perpetual payment of $11, 444. She could invest the 
cash flows at 7 percent. What is the present value of this perpetuity? (Round to 
the nearest dollar.) 
A) $112,344 
B) $163,486 
C) $191,708 
D) $201,356
Annual payment needed = PMT = 
$11,444 
Investment rate of return = i = 7% 
Term of payment = Perpetuity 
Present value of investment needed = PV 
PMT $11,444 
PV of Perpetuity 
  
i 0.07 
 $163,485.71
Cash Flows That Grow at a Constant 
Rate 
• Growing Annuity 
• equally-spaced cash flows that increase in size at a constant rate 
for a finite number of periods 
• Multiyear product or service contract with periodic cash flows 
that increase at a constant rate for a finite number of years 
• Growing Perpetuity 
• equally-spaced cash flows that increase in size at a constant rate 
forever 
• Common stock whose dividend is expected to increase at a 
constant rate forever
Cash Flows That Grow at a Constant 
Rate 
• Growing Annuity 
• Calculate the present value of growing annuity (only) when the 
growth rate is less than the discount rate. 
51 
CF 
  
(6.5) 
1 g 
 
1 i 
 
1 
i - g 
PVA 
n 
1 
 
 
 
n  
 
 
 
 
 
 
 
 
  
Growing Annuity: Example 
• A coffee shop will operate for fifty more years. Cash flow was 
$300,000 last year and increases by 2.5% each year. The discount 
rate for similar firms is 15%. Estimate the value of the firm. 
1 $300,000 (1 0.025) $307,500 
    
$307,500 
CF 
$2,460,000 0.9968 
  
$2,452,128 
1.025 
1.15 
 
1 
0.15 0.025 
50 
0 
 
 
 
 
 
 
 
 
  
 
 
PVA
Cash Flows That Grow at a Constant Rate 
• Growing Perpetuity 
• Use Equation 6.6 to calculate the present value of growing 
perpetuity (only) when the growth rate is less than discount 
rate. 
• It is derived from equation 6.5 when the number of periods 
approaches infinity 
53 
CF 
PVP 1 
  
(6.6) 
i - g 
0 
Cash Flows That Grow at a Constant Rate 
• Growing perpetuity example 
• A firm’s cash flow was $450,000 last year. You expect the cash 
flow to increase by 5% per year forever. If you use a discount 
rate of 18%, what is the value of the firm? 
54 
1 $450,000 (1 0.05) $307,500 
    
$472,500 
$3,634,615 
$472,500 
0.13 
0.18 0.05 
0 
 
 
 
 
CF 
PVP
The Effective Annual Interest Rate 
• Describing interest rates 
• The most common way to quote interest rates is in terms of 
annual percentage rate (APR). It does not incorporate the 
effects of compounding. 
• The most appropriate way to quote interest rates is in terms of 
effective annual rate (EAR). It incorporates the effects of 
compounding. 
55
The Effective Annual Interest Rate 
• Calculate Annual Percentage Rate (APR) 
APR = (periodic rate) x m 
m is the # of periods in a year 
• APR does not account for the number of compounding periods 
or adjust the annualized interest rate for the time value of 
money 
• APR is not a precise measure of the rates involved in borrowing 
and investing 
56
Future and Present Value Calculations and 
Excel Functions for Special Situations 
• The more frequent the compounding for any given interest 
level and time period, the higher the future value. 
• In the Excel RATE and NPER functions, the Payment box or 
loan payment box must be a negative value to represent cash 
outflows
Summary 
• Future values determine the value of dollar payments in the 
future 
• Present value indicates the current value of future dollars 
• Formulas are used to calculate both future and present 
values 
• All calculations can be made using tables or spreadsheets

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Time Value of MoneySajad Nazari
 

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Chapter 6: The Time Value of Money

  • 1. The Time Value of Money Chapter 6
  • 2. Learning Objectives • Explain why a dollar today is worth more than a dollar in the future • Define the terms future value • Calculate the future value of an amount and an annuity
  • 3. The Time Value of Money • Consume Today or Tomorrow? • TVM is based on the belief that people prefer to consume goods today rather than wait to consume the same goods tomorrow •An apple we can have today is more valuable to us than an apple we can have in one year. •Money has a time value because buying an apple today is more important than buying an apple in one year.
  • 4. The Time Value of Money • Consume Today or Tomorrow? • A dollar someone has today can be spent for consumption or loaned to earn interest • A dollar loaned earns interest that increases wealth and the ability to consume • The rate of interest determines the trade-off between consumption today and saving (investing)
  • 5. The Time Value of Money • Future Value versus Present Value • Cash-flows are evaluated based on future value or present value • Future value measures what cash-flows are worth after a certain amount of time has passed • Present value measures what future cash-flows are worth before a certain amount of time has passed
  • 6. Comparison of Future Value and Present Value
  • 7. The Time Value of Money • Future Value versus Present Value • Compounding is the process of increasing cash-flows to a future value • Discounting is the process of reducing future cash-flows to a present value
  • 8. Future Value • What a dollar invested today will be worth in the future depends on  Length of the investment period Method to calculate interest  Interest rate  2 types of methods to calculate interest Simple method-calculated only on the original principal each year Compound interest-calculated on both the original principal and on any accumulated interest earned up to that point. Future value implies the compound method Interest rate Simple interest is paid on the original principal amount only Compound interest consists of both simple interest and interest-on-interest
  • 11. Present Value of an Amount to be Received in the Future • Taking future values back to the present is called discounting Present Value=Future Value x Present Value Factor PV = Present value (initial investment amount) i = the interest rate n= no. of time periods of the investment
  • 12. Example 1 • Suppose we place $10000 in a savings account that pays 10% interest compounded annually. How will our savings grow? Value at the end of year 1 = present value X (1+i) i = interest rate • = $10000 ퟏ + ퟎ. ퟏ ퟒ • = $10000 (1.4641) • = $14,641
  • 13. Future Value: Example 2 • If we place $1,000 in a savings account paying 5% interest compounded annually, how much will our account accrue to in 10 years? • Future value = present value X (ퟏ + 풓)풏 • 푭푽풏 = $1,000 (ퟏ + ퟎ. ퟎퟓ)ퟏퟎ • = $1,000 (1.62889) • = $1,628.89 13 Refer Table B-1
  • 14. Example 3: • What is the future value of $10,000 with an interest rate of 16 percent and one annual period of compounding? With an annual interest rate of 16 percent and two semiannual periods of compounding? With an annual interest rate of 16 percent and four quarterly periods of compounding? Annually: 푭푽 = 푷푽 ∗ (ퟏ + 풊)풏 푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ+. ퟏퟔ)ퟏ 푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ. ퟏퟔ)ퟏ 푭푽 = $ퟏퟏ, ퟔퟎퟎ
  • 15. Semi-annually: 푭푽 = 푷푽 푿 ퟏ + 풊 풎 풏푿풎 • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ + .ퟏퟔ ퟐ ퟏ푿ퟐ • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟖ ퟐ • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟔퟒ • 푭푽 = $ퟏퟏ, ퟔퟔퟒ • Quarterly: • 푭푽 = 푷푽 푿 ퟏ + 풊 풎 풏푿풎 • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ + .ퟏퟔ ퟒ ퟏ푿ퟒ 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟒ ퟒ 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟗퟗ 푭푽 = $ퟏퟏ, ퟔퟗퟗ
  • 16. Present Value and Discounting • Present Value concepts • A present value calculation takes end-of-the-period cash flows and reverses the effect of compounding to determine the equivalent beginning-of-the-period cash flows •Present Value Equation •P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 푿 푷푽푭풊,풏 •P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [ ퟏ (ퟏ+풓)풏] •This is discounting and the interest rate i is called the discount rate. •Present value (PV) is often referred to as the discounted value of future cash-flows.
  • 17. Present Value and Discounting • Time and the discount rate affect present value •The greater the amount of time before a cash flow is to occur, the smaller the present value of the cash-flow. •The higher the discount rate, the smaller the present value of a future cash-flow.
  • 18. Example 4 What is the value of $500 to be received 10 years from today if our discount rate is 6%. P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [ ퟏ (ퟏ+풓)풏] FV = $500, n = 10, r = 6% or 0.06 = $500 [ ퟏ (ퟏ+ퟎ.ퟎퟔ)ퟏퟎ] = $500 (0.558) = $279.20 18 Refer Table B -3
  • 19. Example 5 What is the present value of an investment that yields $1,000 to be received in 7 years and $1,000 to be received in 10 years if the discount rate is 6 percent? ퟏ 풑풓풆풔풆풏풕 풗풂풍풖풆 = 푭푽풏 [ (ퟏ+풓)풏] + 푭푽풏 [ ퟏ (ퟏ+풓)풏] PV= $ퟏ, ퟎퟎퟎ [ ퟏ (ퟏ+ퟎ.ퟎퟔ)ퟕ] + $1,000 [ ퟏ (ퟏ+ퟎ.ퟎퟔ)ퟏퟎ] • = $1,000 (0.665) + $1,000 (0.558) • = $665 + $558 • = $1223
  • 20. Future and Present Values of Annuities • Annuity- A series of equal payment made or received at regular time intervals • Ordinary Annuity- A series of equal annuity payments made or received at the end of each period • Future value of an annuity-What an equal series of payments will be worth at some future date • Future Value Factor of an Annuity (FVFA)-A factor that when multiplied by a stream of equal payments equals the future value of that stream • Future Value of an Annuity Table- Table of factors that shows the future value of equal flows at the end of each period, given a particular interest rate
  • 21. Example 6 Carlos Menendez is planning to invest $3,500 every year for the next six years in an investment paying 12 percent annually. What will be the amount he will have at the end of the six years? (Round to the nearest dollar.) A) $21,000 B) $28,403 C) $24,670 D) $26,124
  • 23. Example 7 You plan to save $1,250 at the end of each of the next three years to pay for a vacation. If you can invest it at 7 percent, how much will you have at the end of three years? (Round to the nearest dollar.) A) $3,750 B) $3,918 C) $4,019 D) $4,589
  • 25. Example 8: Maricela Sanchez needs to have $25,000 in five years. If she can earn 8 percent on any investment, what is the amount that she will have to invest every year at the end of each year for the next five years? (Round to the nearest dollar.) A) $5,000 B) $4,261 C) $4,640 D) $4,445
  • 27. Example 9 Jane Ogden wants to save for a trip to Australia. She will need $12,000 at the end of four years. She can invest a certain amount at the beginning of each of the next four years in a bank account that will pay her 6.8 percent annually. How much will she have to invest annually to reach her target? (Round to the nearest dollar.) A) $3,000 B) $2,980 C) $2,538 D) $2,711
  • 29. Annuities Continued • Present Value of an Annuity- What the series of payments in the future is worth today • Present Value Factor of an Annuity (PVFA)- A factor that when multiplied by a stream of equal payments equals the present value of that stream • Present Value of an Annuity Table- Table of factors that shows the value today of equal flows at the end of each future period, given a particular interest rate
  • 30. How to calculate present value of an annuity (6.1) 30 o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 PVA CF PVFA PVFA 1   (1 ) 1    1 0                     i i CF i CF n ퟏ−풑풓풆풔풆풏풕 풗풂풍풖풆 풇풂풄풕풐풓 풓 o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 ퟏ−(ퟏ+풓)−풏 풓
  • 31. Level Cash Flows: Annuities and • A contract will pay $2,000 at the end of each year for three years and the appropriate discount rate is 8%. What is a fair price for the contract? $5,154.19 Perpetuities • Present Value of an Annuity Example (1 1/(1 0.08)    0.08 PVA $2000 3  3      Refer Table B-4 31
  • 32. Example 1 What is the value of 10-year $1,000 annuity discounted back to the present at 5 percent?
  • 33. 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 ퟏ − (ퟏ + 풓)−풏 풓 푷푽 = $ퟏ, ퟎퟎퟎ ퟏ − (ퟏ + ퟎ. ퟎퟓ)−ퟏퟎ ퟎ. ퟎퟓ • = $1,000 (7.722) • = $7,722
  • 34. Example 2 • Transit Insurance Company has made an investment in another company that will guarantee it a cash flow of $37,250 each year for the next five years. If the company uses a discount rate of 15 percent on its investments, what is the present value of this investment? (Round to the nearest dollar.) A) $101,766 B) $124,868 C) $251,154 D) $186,250 Annual payment = PMT = $37,250 No. of payments = n = 5 Required rate of return = 15% Present value of investment = PVA5
  • 35.          5 1 1 (1 ) n 1 1 (1.15) $37,250 $37,250 3.3522 0.15 n i PVA PMT i                     $124,867.78
  • 36. Example 3 Herm Mueller has invested in a fund that will provide him a cash flow of $11,700 for the next 20 years. If his opportunity cost is 8.5 percent, what is the present value of this cash flow stream? (Round to the nearest dollar.) A) $234,000 B) $132,455 C) $110,721 D) $167,884
  • 37.          20 1 1 (1 ) 1 1 n (1.085) $11,700 $11,700 9.4633 0.085 n i PVA PMT i                     $110,721.04
  • 38. Example 4 Myers, Inc., will be making lease payments of $3,895.50 for a 10-year period, starting at the end of this year. If the firm uses a 9 percent discount rate, what is the present value of this annuity? (Round to the nearest dollar.) A) $23,250 B) $29,000 C) $25,000 D) $20,000
  • 39.          10 1 1 (1 ) 1 1 n (1.09) $3,895.50 $3,895.50 6.4177 0.09 n i PVA PMT i                     $24,999.99
  • 40. Ordinary Annuity versus Annuity Due • Present Value of Annuity Due •Cash flows are discounted for one period less than in an ordinary annuity. • Future Value of Annuity Due •Cash flows are earn compound interest for one period more than in an ordinary annuity. 40 PVA PVA (1 ) (6.4) 1 Due 1 Due i    FVA FVA (1 i )   
  • 41. Ordinary Annuity versus Annuity Due 41
  • 42. Perpetuities • A stream of equal cash flows that goes on forever • Preferred stock and some bonds are perpetuities • Equation for the present value of a perpetuity can be derived from the present value of an annuity equation 42 PVP CF esent value factor for an annuity 1   1 1 0 ( i) ( . )   Pr 0 CF i ( ) i CF i CF 6 3  1                  
  • 43. Valuing Perpetuity Example 1 • Suppose you decide to endow a chair in finance. The goal of the endowment is to provide $100,000 of financial support per year forever. If the endowment earns a rate of 8%, how much money will you have to donate to provide the desired level of support? 43 $1,250,000 $100,000 CF 0    0.08 i PVP
  • 44. Example 2 • What is the value of a $500 perpetuity discounted back to the present at 8 percent?
  • 45. 푷푽 = 푪푭 풊 푷푽 = $ퟓퟎퟎ ퟎ.ퟎퟖ = $6250
  • 46. Example 3 Your father is 60 years old and wants to set up a cash flow stream that would be forever. He would like to receive $20,000 every year, beginning at the end of this year. If he could invest in account earning 9 percent, how much would he have to invest today to receive his perpetual cash flow? (Round to the nearest dollar.) A) $222,222 B) $200,000 C) $189,000 D) $235,200
  • 47. Annual payment needed = PMT = $20,000 Investment rate of return = i = 9% Term of payment = Perpetuity Present value of investment needed = PV PMT $20,000 PV of Perpetuity   i 0.09  $222,222.22
  • 48. Example 4 A lottery winner was given a perpetual payment of $11, 444. She could invest the cash flows at 7 percent. What is the present value of this perpetuity? (Round to the nearest dollar.) A) $112,344 B) $163,486 C) $191,708 D) $201,356
  • 49. Annual payment needed = PMT = $11,444 Investment rate of return = i = 7% Term of payment = Perpetuity Present value of investment needed = PV PMT $11,444 PV of Perpetuity   i 0.07  $163,485.71
  • 50. Cash Flows That Grow at a Constant Rate • Growing Annuity • equally-spaced cash flows that increase in size at a constant rate for a finite number of periods • Multiyear product or service contract with periodic cash flows that increase at a constant rate for a finite number of years • Growing Perpetuity • equally-spaced cash flows that increase in size at a constant rate forever • Common stock whose dividend is expected to increase at a constant rate forever
  • 51. Cash Flows That Grow at a Constant Rate • Growing Annuity • Calculate the present value of growing annuity (only) when the growth rate is less than the discount rate. 51 CF   (6.5) 1 g  1 i  1 i - g PVA n 1    n            
  • 52. Growing Annuity: Example • A coffee shop will operate for fifty more years. Cash flow was $300,000 last year and increases by 2.5% each year. The discount rate for similar firms is 15%. Estimate the value of the firm. 1 $300,000 (1 0.025) $307,500     $307,500 CF $2,460,000 0.9968   $2,452,128 1.025 1.15  1 0.15 0.025 50 0             PVA
  • 53. Cash Flows That Grow at a Constant Rate • Growing Perpetuity • Use Equation 6.6 to calculate the present value of growing perpetuity (only) when the growth rate is less than discount rate. • It is derived from equation 6.5 when the number of periods approaches infinity 53 CF PVP 1   (6.6) i - g 0 
  • 54. Cash Flows That Grow at a Constant Rate • Growing perpetuity example • A firm’s cash flow was $450,000 last year. You expect the cash flow to increase by 5% per year forever. If you use a discount rate of 18%, what is the value of the firm? 54 1 $450,000 (1 0.05) $307,500     $472,500 $3,634,615 $472,500 0.13 0.18 0.05 0     CF PVP
  • 55. The Effective Annual Interest Rate • Describing interest rates • The most common way to quote interest rates is in terms of annual percentage rate (APR). It does not incorporate the effects of compounding. • The most appropriate way to quote interest rates is in terms of effective annual rate (EAR). It incorporates the effects of compounding. 55
  • 56. The Effective Annual Interest Rate • Calculate Annual Percentage Rate (APR) APR = (periodic rate) x m m is the # of periods in a year • APR does not account for the number of compounding periods or adjust the annualized interest rate for the time value of money • APR is not a precise measure of the rates involved in borrowing and investing 56
  • 57. Future and Present Value Calculations and Excel Functions for Special Situations • The more frequent the compounding for any given interest level and time period, the higher the future value. • In the Excel RATE and NPER functions, the Payment box or loan payment box must be a negative value to represent cash outflows
  • 58. Summary • Future values determine the value of dollar payments in the future • Present value indicates the current value of future dollars • Formulas are used to calculate both future and present values • All calculations can be made using tables or spreadsheets