with computations about the tvm

1. TIME VALUE OF MONEY
JULY23, 2016

2. INTRODUCTION
 “A Peso Received Today Is Worth More Than a Peso Received in The Future”.
 The time value of money impacts business
finance, consumer finance, and government finance.
 Time value of money results from the concept of interest. - is a
cost of using money over time or the sum of money paid for the use of money

3.  Interest and Time Value of Money are considerations in decisions involving
expenditures for business investments and the acquisition of operating assets.
 Time Value of Money involves 2 major concepts:
 Future Value
 Present Value
 3 factors considered by both of the 2 concepts:
 Principal
 Interest Rate
 Time Period

4. SIMPLE INTEREST
 it is the interest paid or earned on the initial principal only.
 Interest (I) is a sum of money for the use of money.
 Principal (P)
 Rate of interest (r)
 Length of time (t)
 Formula: I = P r t

5. Example:
Venus deposited P5,000 in a bank at 6.5% simple interest
for 2 years. How much will she earns after 2 years,
assuming that no withdrawals were made?
solution:
P = P5,000 r = 6.5% or 0.065 t = 2 years I = ?
I = P r t
= (P5,000) (0.065) (2)
= P650.00 (interest)

6. To find also the rate, r, the time, t, and the principal,
P, and also the future amount, F, we use the following formula:
r = I / P t
t = I / P r
P = I / r t or P = F / 1 + r t
F = P (1 + r t)

7. Compound interest
 It is the interest paid on both the principal and the amount of interest
accumulated in the prior period.
 Compounding is the process called for determining future value when
compound interest is applied.
 Formula:
F = P (1 + i ) n

8. Example:
Venus deposited P5,000 in a bank at 6.5%
compounded annually for 2 years. How much will
she earns after 2 years, assuming that no
withdrawals were made?
solution:
F = P (1 + i ) n
= P5,000 ( 1 + 0.065) 2
= P5,671.125

9. Simple interest compared with compound interest
 Example:
 ABC Corporation leaves its P10,000 on deposits for five years in a bank paying 10%
annual interest.
Simple Interest Compound Interest
Year Beginning
amount
Simple
interest
Ending
amount
Beginnin
g amount
Compound
interest
Ending
amount
1 P10,000 P1,000 P11,000 P10,000 P1,000 P11,000
2 P10,000 P1,000 P12,000 P11,000 P1,100 P12,100
3 P10,000 P1,000 P13,000 P12,100 P1,210 P13,310
4 P10,000 P1,000 P14,000 P13,310 P1,331 P14,641
5 P10,000 P1,000 P15,000 P14,641 P1,464.10 P16,105.10
Total
interest
P5,000 P6,106.10

10. Future value (annual
compounding)
 Formula:
 FVn = PV ( 1 + i)n
 FV
n -- future value
 PV -- initial principal amount
 i -- interest rate (compounded)
 n -- periods

11. Example:
The future value of P10,000 compounded at a 10%
annual interest rate at the end of one year, two
years and five years are computed as follows:
Year Calculation Future Vale
1 FV1 = (P10,000)(1+0.10)1 P11,000
2 FV2 = (P10,000)(1+0.10)2
= (P10,000)(1,21)
P12,100
5 FV5 = (P10,000)(1+0.10)5
=(P10,000)(1.61051)
P16,105.10

12. FUTURE VALUE ( WITH
INTRAPERIOD COMPOUNDING)
 It is compounding that occurs more than once a year.
 Formula:
 FVn = PV (1 + i/m)mn
 FV
n -- future value
 PV -- initial principal amount
 i -- interest rate (compounded)
 n -- periods
 m --number of compounding period per year

13. Example:
Instead of placing P1,000 in Atlanta Bank that pays 10%
interest annually, the financial manager decides to put the
money in National Bank that pays 10% interest
compounded semi-annually. Between the two banks, there
would be a difference in the future value of your
investment after one year.
Atlanta Bank
Annual Compounding
National Bank
Semi-annual Compounding
FV1 = (P1,000)(1 + 0.10)1
= (P1,000)(1.10)
=P1,100
FV1 = (P1,000)(1 +0.10/2)(2)(1)
= (P1,000)(1.1025)
= P 1,102.50

14. Nominal interest rate compared
to effective interest rate
 Nominal Interest Rate is simply the
stated rate while the Effective Interest
Rate or also called Annual Percentage
Rate (APR) is the true interest rate and
may differ from nominal rate depending
on the frequency on compounding.
 Formula;
 APR = (1 + i/m )m - 1

15. Example:
Disney Incorporated deposits money in a bank that pays 10% nominal interest
and compounds interest semi-annually.
APR = (1 + 0.10/2)2 – 1
= (1.05)2 -1
= 1.1025 – 1
= 0.1025 or 10.25%

16. Comparison or results of different compounding periods for
P1,000 invested at a 10% interest rate for one year
Initial
Investment
Compounding
Period
Future Value Annual
Percentage
Rate
P1,000 Annually P1,100 10%
P1,000 Semi-annually P1,102.50 10.25%
P1,000 Quarterly P1,103.81 10.38%
P1,000 Monthly P1,104.71 10.47%

17. Determination of the future
value of a stream of payments
 Future Value Determination Involving a Stream of
Unequal Payments
 In calculating the future value of an unequal stream of
payments, find the future value of each payment at a
specified future date and then sum it up.

18. Example:
A firm plans to deposit P2,000 today and P1,500
one year from now at Mount Carmel Rural Bank.
No future deposits or withdrawals are made and
the bank pays 10% interest compounded annually.
The future value of the account at the end of four
years is computed to be:
FV4 = (P2,000)(1.10)4 + (P1,500)(1.10)3
= (P2,000)(1.464) + (P1,500)(1.331)
= P2,928 + P1,996.50
= P4,924.50

19.  Future Value Determination Involving a Stream of Equal Payments
 A stream of equal payments made at regular time intervals is an annuity
 There are two types of fixed annuities:
 Ordinary Annuity
 Annuity Due

20. 1. Ordinary annuity
 It is one in which payments or receipts occurs at the end of each period.
 Formula:
 FVOA n = A ((( 1 + i ) n – 1) / i )
 FVOA n - future value of ordinary annuity
 A - amount of fixed annuity payment

21. Let's now run through Example 1. Consider the following
annuity cash flow schedule:

22. To calculate the future value of the annuity, we have
to calculate the future value of each cash flow. Let's
assume that you are receiving $1,000 every year for
the next five years, and you invested each payment at
5%. The following diagram shows how much you would
have at the end of the five-year period:

23.  this is formula that serves as a shortcut for finding the accumulated value of
all cash flows received from an ordinary annuity:
FVOA n = A ((( 1 + i ) n – 1) / i )
= $1,000 ((( 1 + .05) 5 -1 ) / .05)
= $5,525.63

24. Example:
Crystal Corporation deposits P1,000 at the end of
each of three years in a bank account paying 10%
interest compounded annually. The value of the
account at the end of the third year is computed
by:
FVOA3 = (P1,000) (((1+0.10)3 -1)/0.10)
= (P1,000)(3.310)
= P3,310

25. 2. Annuity due
 It is one which payments or receipts occur at the
beginning of each period.
 Formula:
 FVAD n = A ((1 + i )n – 1) / i)(1 + i)
 FVAD n - future value of an annuity due
 A - amount of fixed annuity payment

26. When you are receiving or paying cash
flows for an annuity due, your cash
flow schedule would appear as
follows:

27. Example:
$1,000 payment is made at the beginning of the
period rather than at the end (interest rate is still
5%):

28.  The future value of this annuity can use
this formula:
FVAD n = A ((1 + i )n – 1) / i)(1 + i)
= $1000 (( 1 + .05) 5 – 1) / i) ( 1+.05)
=$ 5801.91

29. Example
Instead of depositing P1,000 at the end of the year
for three consecutive years, the firms makes
deposits at the beginning of each year. Interest is
compounded annually at 10%. How much will the
firm have in account after three years?
FVOA3 = (P1,000)(3.310) (1.10)
= (P1,000)(3.641)
= P3,641

30. Present value
 It is the current value of a future amount of money, or
series of payments, evaluated at an appropriate
discount year.
 Discount rate is the rate of the interest that is used to
find present values.

31. Discounting
 It is the process of determining the present value of the future amount.
Formula:
PV = FVn / (1 + i)n
PV = FVn (1 + i)-n

32. Example
Blueberry Company expects to receive P1,100 one year from now. What is the present value of this
amount if the discount rate is 10%?
PV = FVn / (1 + i)n
PV = P1,100 / (1 + 0.10)1
= P1,100 / 1.10
= P1,000

33. Determination of the present
value of a stream of payments
 Present Value Determination Involving a Stream of
Unequal Payments
 To find the present value of an unequal or mixed stream of
payments, simply calculate the present value of each
future amount separately and then add these present
values together.

34. Example:
MNM Company expects to receive payments of
P1,000, P1,500 and P2,000 at the end of one, two
and three years, respectively. The present value of
this stream of payments discounted at 10% is
computed as follows:
PV = (P1,000)(0.909) + (P1,500)(0.826) + (P2,000)(0.751)
= P909 + P1,239 + P1,502
= P3,650

35.  Present Value Determination Involving a Stream of Equal Payments
 The present value of an equal payments is found by using the equation below.
PVOA n = A { ∑ 1 / (1 + i) t }
 This equation, the annuity payment, A, is multiplied by the term in bracket, which
is the sum of the individual present value interest factors.

36. Example:
Summer Corporation expects to receive P1,000 at year’s end for the next 3
years. The present value of this annuity discounted at 10% is computed as
follows:
PVOA n = (P1,000) { ∑ 1 / (1 + i) t }
= (P1,000) (0.909 + 0.826 + 0.751)
= (P1,000) (2.486)
= P2,486.00

37. Determination of the present value of a
perpetuity
 Perpetuity – is an annuity with an infinite life: that is, the payments continue
indefinitely.
 The present value of perpetuity is found by using the equation below:
PV of a perpetuity = annuity / discount rate

38. Example:
Honey Dew Corporation wants to deposit an
amount of money in the bank account that will
allow it to withdraw P1,000 indefinitely at the end
of the year without reducing the amount of initial
deposit. If a bank guarantees to pay the firm 10%
interest on its deposits, the amount of money the
firm has to deposits is computed as follows:
PV of a perpetuity = P1,000 / 0.10 = P10,000

39. BOTTOM LINE
 Time literally is Money - the value of the money you have now is not the same
as it will be in the future and vice versa.
 So, it is important to know how to calculate the time value of money so that
you can distinguish between the worth of investments that offer you returns at
different times.