Time Value of Money
INTEREST
 term used in business
 cost of using money over time
 interest expense = borrower / debtor
 interest income = lender / creditor
Time Value of Money – term used by economist
3 Factors:
1. Principal
2. Interest Rate
3. Time Period
2 Concepts:
a. Future Value
b. Present Value
FUTURE VALUE:
 compounds money forward in time to determine its
worth in the future
Compounding is the process of determining future value when
compound interest is applied.
Can Be Computed Using:
A. Simple Interest (interest paid or earned on the initial
principal only)
B. Compound Interest (interest paid on both the principal
and the amount of interest accumulated in prior
periods)
Simple Interest:
A. ABC Corporation deposits P 10,000 in a bank at 10%
interest in a year. How much is the future value of the
principal at the end of year 1?
Interest (I) = Principal (P) x Rate (R) x Time (T)
Future Value (FV) = Principal (P) + Interest (I)
Principal
Interest
FV
= P 10,000
= P 1,000 (10,000 x 10%)
= P 11,000
Simple Interest:
B. ABC Corporation deposits P 10,000 in a bank at 10%
interest in a year. How much is the future value of the
principal after 6 months?
Interest (I) = Principal (P) x Rate (R) x Time (T)
Future Value (FV) = Principal (P) + Interest (I)
Principal
Interest
FV
= P 10,000
=P
500 (10,000 x 10% x 6/12 )
= P 10,500
Simple Interest:
C. ABC Corporation deposits P 10,000 in a bank at 10%
interest for 5 years. How much is the future value of
the money after 5 years?
Interest (I) = Principal (P) x Rate (R) x Time (T)
Future Value (FV) = Principal (P) + Interest (I)
Principal
Interest
FV
= P 10,000
= P 5,000 (10,000 x 10% x 5 )
= P 15,000
Compound Interest:
A. ABC Corporation deposits P 10,000 in a bank
compounded annually at 10% interest. How much is
the future value of the principal at the end of year 2?
Year
Amount
Compound
Interest
Future Value
1
10,000
1,000
11,000
2
11,000
1,100
12,100
Total interest
2,100
Compound Interest:
B. ABC Corporation deposits P 10,000 in a bank
compounded annually at 10% interest. How much is
the future value of the principal after 5 years?
Year
Amount
Compound
Interest
Future Value
1
10,000
1,000
11,000
2
11,000
1,100
12,100
3
12,100
1,210
13,310
4
13,310
1,331
14,641
5
14,641
1,464.10
16,105.10
6,105.10
Compound Interest:
B. ABC Corporation deposits P 10,000 in a bank
compounded annually at 10% interest. How much is
the future value of the principal after 5 years?
Alternative Solution:
FV= PV (1 + i )n
= 10,000 (1 + .10)5
= 10,000 (1.61051*)
= 16,105.10
where: FV
PV
i
n
= future value
= initial principal
= interest rate
= period
* 1.61051 is referred to as FVIF (future value interest
factor)
EXERCISE:
CAE Corporation borrowed P 20,000 in a bank at
12% interest. Using simple interest method,
1. How much is the interest after 3 years?
2. What is the FV of the principal after 8 years?
CAE Corporation borrowed P 20,000 in a bank
compounded annually at 12% interest. Using
compound interest method,
1. How much is the interest after 3 years?
2. What is the FV of the principal after 6 months?
3. What is the FV of the principal after 10 years if it is
compounded quarterly?
Future Value (With Intra-period Compounding)
Intra-period Compounding - compounding that occurs
more than once in a year
Examples:
FV
monthly, quarterly, semi-annually
= PV (1 + i/m)m*n
Where:
m refers to the number of times interest is
compounded in a year
Example:
A. ABC Corporation is contemplating to deposit P 10,000
to BPI that pays 10 percent interest compounded
annually. However, the financial manager of ABC
Corporation decided to deposit the money at BDO that
pays 10 percent interest compounded semi-annually.
Questions:
1. What is the FV of P 10,000 should ABC Corporation
opted to deposit it at BPI after 1 year?
2. What is the FV of P 10,000 at BDO after 1 year ?
3. Was the decision of ABC Corporation to deposit the
money at BDO right? By how much was the difference
in future values?
Solution:
A. FV
= PV (1+ i) 1
= 10,000 ( 1.10)1
= 11,000
B. FV
= PV (1+ i/m) nm
= 10,000 ( 1 + (.10/2)2*1
= 10,000 (1 + .05)2
= 10,000 (1.1025)
= 11,025
C. YES
BPI = 11,000
BDO = 11,025
25 difference
Nominal Rate:
 is also known as stated rate
Effective Rate:
 is also called - APR (annual percentage rate)
 is the true interest rate
 arises because of the frequency of
compounding in a year
Nominal Rate = Effective Rate if compounding of
interest happens once in a year
Example:
ABC deposits P 10,000 at BDO that pays a 10 percent
interest rate compounded semi-annually
APR = (1 +i/m) m - 1
= (1 + .10/2)2 -1
= (1 + .05)2 -1
= 1.1025 – 1
= .1025 or 10.25%
Checking:
Principal
Interest
FV
= 10,000
= 1,025 (10,000 x .1025)
= 11,025
SUMMARY EXERCISES:
1. If you invest P 12,000 today, how much will you have
a. in 6 years at 7 percent (ordinary interest)
b. in 15 years at 12 percent (compounded annually)
c. In 25 years at 10 percent (compounded semi-annually)
2. If a bank pays 12 percent nominal interest rate, what is
the effective interest rate assuming quarterly
compounding ?
Future Value of a Stream of Payments
Stream of Payments - compounding of a series or
stream of payments
A. Stream of Unequal Payments
How: Compute the FV of each payment at a specified
future date and then summing all FVs
FV = ϵ PV (1 +i) n-t
where t refers to the no. of periods in which interest is
earned / accrued
Example:
A. A firm plans to deposit P 2,000 today and P 1,500 on
the second year at BPI. The bank pays 10 percent
interest compounded annually. The FV of the account at
the end of four years is?
FV
= 2,000 (1+.10)4 + 1,500 (1+.10)3
= 2,000 (1.4641) + 1,500 (1.331)
= 2,928.20 + 1,996.50
= 4,924.70
Example:
B. A firm plans to deposit P 10,000 on the first year,
P 8,000 on the second year and P 5,000 on the third
year at BDO. The bank pays 8 percent interest
compounded annually. No future deposits or withdrawals
are made. The FV of the account at the end of 5 years is?
FV = 10,000 (1+.08)5 + 8,000 (1+.08)4 + 5,000 (1+.08)3
= 10,000 (1.4693) + 8,000 (1.3605) + 5,000 (1.2597)
= 14,693 + 10,884 + 6,298.5
= 31,875.50
Future Value of a Stream of Payments
A. Stream of Equal Payments
Annuity (Fixed Annuity) – a stream of equal payments
made at regular time intervals
2 Types:
1. Ordinary Annuity (Deferred Annuity) – one in which
payments or receipts occur at the END OF EACH
PERIOD
2. Annuity Due – one in which payments or receipts occur
at the BEGINNING OF EACH PERIOD
Future Value of a Stream of Payments
1. Ordinary Annuity (Deferred Annuity)
FVOA
FVIFAin
= A (FVIFAin)
= (1+i)n -1
i
where FVOA - means future value of ordinary annuity
A - means the amount of the fixed annuity
payment
FVIFAin - future value interest factor of an
ordinary annuity
Example:
A. ABC Corporation deposits P 10,000 at the end of each
year for the next 3 consecutive years in a bank paying 10
percent interest compounded annually. No future
deposits or withdrawals are made. The FV of the account
at the end of the 3rd year is?
FVOA
= A (FVIFAin)
= 10,000 (3.310)
= 33,100
Future Value of a Stream of Payments
1. Annuity Due
FVAD
= A (FVIFADin)
FVIFADin = (1+i)n -1 x (1 +i)
i
where FVAD - means future value of annuity due
A - means the amount of the fixed annuity
payment
FVIFADin - future value interest factor of an
annuitydue
Example:
B. ABC Corporation deposits P 10,000 at the beginning of
each year for 3 consecutive years with a bank paying 10
percent interest compounded annually. No future
deposits or withdrawals are made. The FV of the account
at the end of the 3rd year is?
FVAD
= A (FVIFADin)
= 10,000 (3.641)
= 36,410
Comparing FV of ordinary annuity and annuity due:
33,100
36,410
Ordinary
Annuity
Annuity
Due
Analysis:
The future value for the annuity due is greater
than the ordinary annuity because each
deposit made one year earlier earns interest
one year longer
Present Value:
 discounts money that will be received in the future back in
time to see what it is worth in the present
 the current value of a future amount of money or series of
payments, evaluated at an appropriate discount rate
Discount Rate:
 sometimes called the required rate of return
 the rate of interest that is used to find present values
Discounting:
 the process of determining the present value of a future amount
Present Value:
PV = FV
(1 + i) n
or
PV = FV (1+i) -n
Example:
XYZ expects to receive P 10,000 one year from now. What is the
present value of this amount if the discount rate is 10 percent?
PV = FV
(1 + i) n
= 10,000
(1+.10)1
= 9,090.91
or
PV = FV (1+i) -n
= 10,000 (1+.10)-1
= 10,000 (.9091**)
= 9,090.91
* PVIF
Present Value of Stream of Payments:
Stream of Unequal Payments
How: Compute the PV of an unequal or mixed, stream of
payments separately and then add altogether
PV = ϵ FV (PVIFin)
PVIFin
=
1
(1+i)n
Example:
XYZ expects to receive payments of P 10,000, 11,500
and P 20,000 at the end of one, two and three years
respectively. The present value of this stream of
payments discounted at 10 percent
PV = ϵ FV (PVIFin)
= 10,000 (.909) + 11,500 (.826) + 20,000 (.751)
= 9,090 + 9,499 + 15,020
= 33,609
Present Value of Stream of Payments:
Stream of Equal Payments
A. Ordinary Annuity
PVOA = A (PVIFAin)
PVIFAin =
1 – _1_
__(1+i)n
i
Example: XYZ Corporation expects to receive P 10,000 at year-end for the
next 3 years. The PV of this annuity discounted at 10 percent is? The
PVIFAin is 2.487.
PVOA = 10,000 (2.487)
= 24,870
Present Value of Stream of Payments:
Stream of Equal Payments
B. Annuity Due
PVAD = A (PVIFADin)
PVIFADin = 1 – _1_ (1+i)
__(1+i)n
i
Example:
XYZ Corporation expects to receive P 10,000 at the beginning of the
year for the next 3 years . The PV of this annuity discounted at 10
percent is? The PVIFADin is 2.7355
PVAD = 10,000 (2.7355)
= 27,355
Present Value of a Perpetuity:
Perpetuity
* is an annuity with an infinite life, that is the payments
continue indefinitely
PV of perpetuity = A nnuity
Discount rate
Example:
JBT Corporation wants to deposit an amount of money that will allow it
to withdraw P 1,500 indefinitely at the end of each year without
reducing the amount of the initial deposit. If the bank guarantees to pay
the firm by 10 percent interest on its deposits , the amount to be
deposited NOW is?
\
PV of perpetuity = 1,500
.10
= 15,000
EXERCISE;
The Billy Playhouse wants P 10,000 at the end of each
year for the next 6 years. How much must be deposited
today at 10% to yield this annuity?
What amount must be deposited now in order to
withdraw P 2,000 at the beginning of each year for 5
years if the interest rate is 12% compounded annually?
ABC Company wants to deposit an amount of money
that will allow it to withdraw P 2,000 indefinitely at the
end of each year without decreasing the amount of the
original deposit. If the bank guarantees to pay the firm 5
percent interest , the amount to be deposited NOW is?
Thank you!