This document discusses the time value of money concept. It defines key terms like present value, future value, interest rates, and annuities. It provides formulas to calculate future value, present value, and annuities. Examples are given to demonstrate calculating simple and compound interest, present and future value of cash flows over single and multiple periods, and ordinary and due annuities. The document also covers topics like finding interest rates, time periods, and loan amortization.

1. Time value of Money
By: Sajad Nazari

2. Learning Objectives
• Understand how the time value of money works and
why it is important in finance
• Learn how to calculate the present value and future
value of single and lump sums ;
• To know and identify the different types of annuities for
both present value and future value of both an ordinary
annuity and an annuity due;
• Calculate the present value and future value of an
uneven cash flow stream;
• Explain the difference between nominal, periodic and effective
interest rates
• Discuss the basics loan amortization

3. Time Value of Money
• It indicates the relationship between time and
money.
• A peso received today is worth more than a
peso to be received tomorrow."
• This is because the peso amount you received
today can be invested to earn interest.
• The single most important in all financial
concepts.

4. Uses of TMV
• Valuation of Stocks and Bonds
• Setting up loan payment schedules
• Making Corporate Decisions regarding
investments.

5. Time Lines
• An important tool used in time value
analysis;
• It is a graphical representation used to
show the timing of cash flows.
Periods 0 1 2 3
10%

6. Definitions:
• Interest is money paid for the use of money.
Symbol: I
• Principal is the amount of money borrowed or
invested. Symbol P
• Interest Rate is the rate, or percent. Symbol: r
• Maturity Value / Final Value is the increased
amount resulting from the increase process.
Symbol: F
• Time a period of time. Days, monthly, quarterly,
semi-annually, annually.
Symbol: t

7. Simple Interest
• Mr. Lopez invest $10,000 at 5% annual rate for 2 years.
How much will be the interest?
• I = Prt
• P = Principal R= interest Rate t = time
• 0 1 2
Periods 5%
PV =$10,000 I =?
• I=Prt
• = $10,000 x 5% x 2
• =$ 1,000

8. Derived Formulas:
• Interest (I) = Prt
• Principal (P) = I / rt
• Principal (P) = F- i
• Rate (r) = I / Pt
• Time (t) = I / Pr
• Future Value (Fv) = P(1+rt)
• Present Value (Pv) = Fv
1+rt

9. Time Value Concepts
• Future Value
• Present Value
• Future Value of Ordinary Annuity
• Future Value of Annuity Due
• Present Value of Ordinary Annuity
• Present Value of Annuity Due

10. FUTURE VALUE
• The amount to which a cash flow or
series of cash flows will grow over a
given period when compounded at a
given interest rate. – Brigham, 2011
• It is the amount of money that will
grow to at some point in the future. –
Cabrera, 2011
• FV = PV(1+i)n
• PV = Present Value / Principal
• I = interest rate
• n = Number of periods / term

11. Compounding, Simple Interest
and Compound Interest
• Compounding is the arithmetic
process of determining the final
value of cash flow or series of cash
flows when compounded interest is
applied. – Brigham, 2011
• Simple Interest occurs when interest
is not earned on interest. – Brigham,
2011
• Compound Interest occurs when
interest is earned on prior periods’
interest. –Brigham, 2011

12. Simple interest
• Mr. Lopez invest $10,000 at 5%
annual rate for 2 years. How much
will be the Maturity value?
• 0 1 2
Periods
PV = 10,000 FV = ?
• F = P(1+rt)n
• = $10,000(1.1)
• = $11,000

13. Future Value (compounded)
• Mr. Lopez invest $10,000 compounded
at 5% annual rate for 2 years. How
much will be the Maturity value?
• 0 1 2
PV = $10,000 FV= ?
• FV= PV(1+i)n
• PV = Present Value
• i = interest rate
• N= number of years / time

14. Future Value (compounded)
• 0 1 2
PV=$10,000 $10,500 $11,025
• PV = 10,000
• i = .05
• N= 2
• FV= PV(1+i)n FV = PV(1+i)n
• =$10,000(1+.05) = $10,500(1.05)
• = $10,500 = $11,025
• FV = PV(1+i)n
• = $10,000(1.0250)
• =$ 11,025 the maturity value after 2 years

15. Mr. Lopez invest $10,000 compounded at
5% annual rate for 2 years. How much
will be the Maturity value? (table 1)
• FV= PV(1+i)n
• PV = $10,000
• i = .05
• N= 2
• FV = $10,000(1.10250)
= $11,025

16. Calculating FV using
Spread Sheet

17. Procedure:

18. Procedure:

19. Procedure:

20. Procedure:

21. Procedure:

22. Present Value
• The value today of a future cash
flow or series of cash flows. –
Brigham, 2011
• It is the amount of money today
that is equivalent to a given
amount to be received or paid in
the future. –Cabrera, 2011
• it is just a reverse of the future
value, in a way that instead of
compounding the money forward
into the future, we discount it back
to the present.

23. Formula:
• PV = F V
(1+i)n
• FV = Future Value
• i = interest rate
• n = number of years

24. Single-Period Case
• Suppose you need $50,000.00 to buy laptop
next year. You can earn 10% on your money by
putting in on the bank. How much do you have
to put up today?
• 0 1
PV =? FV = $50,000
• PV= FV
(1+i)n
• = $50,000
(1+.10)1
• = $45,454.545 the amount needed to invest
today

25. Single-Period Case
• Suppose you need $50,000.00 to buy
laptop next year. You can earn 10%
on your money by putting in on the
bank. How much do you have to put
up today? (table 2)
• PV= FV
(1+i)n
• FV = $50,000 i=.10 n=1
• PV = $50,000 (.090909)
=$45,454.5

26. Using Spread Sheets

27. Multiple Period Case
• Angelo would like to buy a new automobile. He
has $600,000, but the car costs $800,000. If he
can earn 12%, how much does he need to
invest today in order to buy the car in two
years? Does he have enough money, assuming
the price will still the same?
• 0 1 2
PV = ? $800,000
• FV = Future Value
• i = Interest Rate
• N = Number of years / Time

28. Multiple Period Case
• PV= FV
(1+i)n
• = $800,000
(1+.12) 2
• =$637,755.102 the amount
Angelo must invest today.
• = $600,000 - $637,755.102
• = $37,755.102
• Angelo is still short of $37,755.102

29. Angelo would like to buy a new automobile. He
has $600,000, but the car costs $800,000. If he
can earn 12%, how much does he need to
invest today in order to buy the car in two
years? Does he have enough money,
assuming the price will still the same? (table 2)
• PV= FV
(1+i)n
• FV = $800,000 i=.12 n=2
• PV = $800,000 ( 0.79719)
= $637,752

30. ANNUITIES
• It is a series of equal sized cash
flows occurring over equal
intervals of time. –Cabrera,
2011
• A series of equal payment at
fixed intervals for a specified
number of periods – Brigham,
2011

31. Two types of Annuity
 Ordinary Annuity – exists when the
cash flows occur at the end of each
period.
• 0 1 2 3
Periods
Payments -100 -100 -100
 Annuity Due – exists when the cash
flows occur at the beginning of each
period.
 0 1 2 3
Periods
Payments -100 -100 -100

32. Future Value of Ordinary Annuity
• It is the future value of a series of equal
sized cash flows with the first payment
taking place at the end of the first
compounding period. The last payment
will not earn any interest since it is
made at the end of the annuity period. –
Cabrera, 2011
• FVA = R (1+i)n-1
i
• R = Periodic Payment
• i = Interest Rate
• n = time / term

33. Future Value of Ordinary Annuity (FVA)
• Suppose you deposit $2,000 at the end
of year 1, another $2,000 at the end of
year 2, how much will you have in 5
years, if you deposit ₱2,000 at the end
of each year? Assume a 10% interest
rate throughout. (table 3)
• FVA = R (1+i)n-1
I
• R = $2,000 i=.10 n=5
• FVA = $2,000(6.10510)
• = $12,210.2

34. Using Spread Sheet

35. FUTURE VALUE OF AN ANNUITY DUE
• The future value of a series of equal
sized cash flows with the first
payment taking place at the
beginning of the annuity period.
– Cabrera, 2011
• FVAD = R [(1+i)n-1] x (1+i)
I
• R= Periodic Payments
• i= interest rate
• N= number of years / periods

36. FUTURE VALUE OF AN ANNUITY DUE
• Jose deposits $3,500 every beginning of the
month at his bank that credits 3% monthly for
a year. How much he will have at the end of
the term? (table 5)
• FVAD = R [(1+i)n-1] x (1+i)
I
• R = $3,500 i=.03 n=12
• FVAD = $3,500 (14.61779)
= $51,162.265

37. Using spread sheet:

38. Future Value of Annuity Due (FVAD) (uneven cash flows)
• Suppose you deposit today $100 in an account paying
8%. In one year, you will deposit another $200 and
₱300 at the beginning of the third year, how much will
you have in three years?
• FVAD = R [(1+i)n-1] x (1+i)
• FVAD = $100 (1.0800)
= $108.00
• FVAD = $308 ( 1.0800)
= $332.64
• FVAD = $632.64(1.0800)
= $683.25
• FVAD = $683.25

39. Present Value of Ordinary Annuity
• The Present Value of a series of
equal sized cash flows with the
first payment taking place at the
end of the first compounding
period. –Cabrera, 2011
• PVA = R 1-/(1+i)n
i
• R = Periodic Payment
• i = Interest Rate
• n = time / term

40. Present Value of Ordinary Annuity
• How much is the cash equivalent of the IPAD
that can be purchased by giving a down
payment of $3,000 and $2,500 payable at the
end of each period for 5 months at 5%?
(table 4)
• PVA = R 1-/(1+i)n
I
• R = $2,500 i=.05 n=5
• PVA = $2,500 (4.32948)
= $10,823.7
• Cash equivalent = PVA + down payment
• = $10,823.7 + $3,000
• = $13,823.7

41. Using spread sheet (PVA)

42. Present Value of Ordinary Annuity (uneven cash flows)
• You are offered an investment that will pay you $200
in one year, $400 the next year, $600 the next year
and $800 at the end of the next year. You can earn
12% on very similar investments. What is the most
you should pay for this one?
• PVA = R 1-/(1+i)n
i
• PVA = $200 ( 0.89286)
= $178.572
• PVA = $578.572 (0.89286)
= $559.4398
• PVA = $1,159.44 (0.89286)
= $1,035.217
• PVA = $1,835.217 (0.89286)
= $1,638.592
• PVA = $ 1,638.592

43. Present Value of Annuity Due
• It is the present value of a series of
equal sized cash flows with the first
payment taking place at the
beginning of the annuity period. –
Cabrera,2011
• PVAD = R [1- 1/(1+I)n] x (1+i)
i
• R = Periodic Payment
• i = Interest Rate
• n = time / term

44. Present Value of Annuity Due
• A machine can be bought for $4,000 down payment
and 8 equal monthly payments of $1,200 payable
every beginning of the month. If money is worth 6%
compounded monthly, what is the cash equivalent of
the machine? (table 6)
• PVAD = R [1- 1/(1+I)n] x (1+i)
I
• R = $1,200 i=.06 n=8 dp = $4,000
• PVAD = $1,200 (6.58238)
= 7,898.856
• Down payment + PVAD
• = $4,000+ $7898.856
• = $11,898.856

45. Present Value of Annuity Due

46. FINDING ANNUITY PAYMENTS
• What is the annuity payment if the present value is
$100,000, the interest rate is 10% and there are 8 periods?
• Present value = $18,744.40176

47. Finding the Interest Rate, r
• Suppose your parents will retire in 18 years. They
currently have $250,000, and they think they will need
$1,000,000 at retirement. What annual interest rate
must they earn to reach their goal, assuming they
don't save any additional funds?
• R = I / Pt
• Interest rate = 8%

48. Finding the number of years, n
• Sometimes, we need to know how long it will
take to accumulate a certain some of money
• Ex. How long will it take $10,000 to double if it
was invested in the bank that paid 6% per
year?
• T = I /Pr

49. Types of Interest Rates
• Nominal Interest Rate (Quoted or
stated) – The contracted, or quoted or
stated interest rate. It is also called
annual percentage rate (APR); the
periodic rate times the number of
periods per year.
• Effective Annual Rate – The annual rate
of interest actually being earned, as
opposed to the quoted rate. This is the
rate that would produce the same future
value under annual compounding as
would more frequent compounding at a
given nominal rate.

50. Amortization
• A method of repaying an interest bearing debt by a
series of equal payments at expected time interval.
• A = R [1-(1+i)-n]
i
• R = Periodic Payment
• i = Interest Rate
• n = time / term
• Amortized loan – a loan that is to be repaid in
equal amounts on a monthly, quarterly, or annual
basis.
• Amortization Schedule - a table showing how a
loan will be repaid.

51. Sample Problem:
• A homeowner borrows $100,000 on a mortgage loan.
The loan is to be repaid in five equal payments at the
end of each of the next 5 years. The lender charges 6%
on the balance at the beginning of each year.
• Annuity Payment = $23,739.64

52. Amortization Schedule
Year Beginning
Amount (1)
Payment
(2)
Interest
(3)
Repayment
Of Principal (4)
Ending
Balance (5)
(1x.06) (2-3) (1-4)
1 $100,000.00 $23,379.64 $6,000.00 $17,739.64 $76,620.36
2 76,620.36 23,379.64 4,597.22 18,872.42 53,240.72
3 53,240.72 23,379.64 3,194.44 20,185.20 29,861.08
4 29,861.08 23,379.64 1,791.665 21,587.98 6,481.44
5 6,481.44 23,379.64 388.886 22,990.75 0.00

53. References:
• Brigham, Eugene F., Houston, Joel F.,
Financial Management Fundamentals
(12th Edition), Manila Philippines, 2011.
• Ma. Elenita Balatbat Cabrera,
Management Accounting Concepts and
Applications 2011 Edition
• Arce, Aquino, CoPo, Gabuyo, Laddaran,
Mananquil, Mathematics of Investment
2010 Edition.