TIME VALUE OF MONEY
Cash Flow Time Lines
► Cash flow time line: An important tool used
in time value of money analysis, it is a
graphical representation used to show the
timing of cash flows.
► Outflow: A payment or disbursement of cash
for expenses, investment and so on.
► Inflow:
A receipt of cash from an investment,
an employer or other sources.
Future Value
► Future Value: The amount to which a
cash flow or series of cash flows will
grow over a given period of time when
compounded at a given interest rate.
► FVn = PV (1+i)n
► Here, PV=$100, i=5%,n=1
► Example: FV1= $100(1+.05)1 = $105
FV2= $100(1+.05)2= $110.25
FV5= $100(1+.05)5 =$127.63
Cont…..
► Compound Interest: Interest earned on
interest.
► Compounding: The process of determining
the value of a cash flow or series of cash
flows some time in the future when
compound interest is applied.
► In fact, at the period interest earned is
translated into principle amount. It starts to
earn interest in future time period.
► If interest is compounded more often, then
money grows faster given all other things
remaining the same.
Cont…..
► In banking system, interest is charged
(given) more than once in a year. Usually
banks pay interest in every six months,
known as semiannual compounding.
► Example: Assume that Tk.100 is placed
into an account at an interest rate of 6
percent and left there for 3 years.
In case of annual compounding,
► FV3= Tk.100 (1+.06)3 = Tk.119.10
In case of semiannual compounding,
► FV6= Tk.100 (1+.03)6 = Tk.119.41
Present Value
► Present Value: The value today of a future cash flow or series
of cash flows.
► Discounting: The process of finding the present value of a cash
flow or a series of cash flows; the reverse of compounding.
► Here, FVn=127.63, i=5%, n=5
FVn
PV = ----------------
(1+i)n
127.63
Example: PV= -----------------
(1+.o5)5
127.63
= --------------- = $100
1.2763
Future Value of an Annuity
► Annuity: A series of payments of an equal amount at fixed
intervals for a specified number of periods.
► Ordinary Annuity: An annuity whose payments occur at the
end of each period.
(1+i)n -1
FVAn= PMT ------------
i
PMT= Annuity Payment
i=Interest Rate per period
N=Number of period
Cont…..
► PV=$400, i=10% & n=10 years
(1+.10)10 -1
FVA10= 400 ---------------------
.10
1.59
= 400 ---------------
.10
= 6374.96
Cont…..
► Annuity Due: An annuity whose
payments occur at the beginning of
each period.
{(1+i)n -1}
FVA(DUE)n=PMT [ --------------- X
(1+i)]
i
Present Value of an Annuity
► PVAn: The present value of an ordinary annuity
with n payments.
1
1 - -------------
(1+i)n
PVAn=PMT ----------------------
i
► Annuity Due:
1
1- ----------
(1+i)n
PVA(DUE)n= PMT {-------------------} X (1+i)
i
Cont…..
► Example: PMT=$100, I=5% & n=3 years
1
1- -------------
(1.05)3
PVA(DUE)n= PMT {------------------------} X (1.05)
0.05
= $100[(2.72325)(1.05)]
= $100(2.85941)
= $285.941
Perpetuities
► Perpetuity: A Stream of equal payments expected
to continue forever.
Payment
Present Value of perpetuity (PVP)= -------------------
Interest Rate
PMT
=-------------
i
► Consol: A perpetual bond issued by the government
to consolidate past debts; in general, any perpetual
bond.
Uneven Cash Flow Streams
► Uneven Cash Flow Stream: A series of cash flows in
which the amount varies from one period to the next.
► Payment (PMT): This term designates constant cash
flows
► Cash Flow (CF): This term designates cash flows in
general, including uneven cash flows.
► Present Value of Uneven Cash Flow Stream
n 1 n
PV= ∑ CFt [-------------] = ∑ CFt (PVIF i, t)
t=1 (1+i)t t=1
Cont…..
► Future Value of Uneven Cash Flow Stream
► Terminal Value: The future value of a cash flow
stream.
n n
FVn= ∑ CFt (1+i)n-1 = ∑ CFt (FVIF i, n-t)
t=1 t=1
Semiannual and Other Compounding
Period
► Annual Compounding: The arithmetic process of
determining the final value of a cash flow or series
of cash flows when interest is added once a year.
► Semiannual Compounding: The arithmetic process
of determining the final value of a cash flow or
series of cash flows when interest is added twice a
year.
► Simple (Quoted) Interest Rate: The constructed or
quoted interest rate that is used to compute the
interest paid per period
Cont…..
► Effective Annual Rate (EAR): The annual Rate of
interest actually being earned as opposed to the
quoted rate, considering the compounding of
interest.
iSIMPLE
EAR= (1+ ----------)m -1.0
m
Here,
iSIMPLE= Simple or Quoted Interest Rate
m= Number of compounding periods per year
Cont…..
0.10
► Effected Annual Rate (EAR)= (1+---------)2 -1.0
2
= (1.05)2-1.0
= 1.1025-1.0
= 0.1025
= 10.25%
Annual Percentage Rate (APR)= The periodic Rate X
The number of
period per year
Amortized Loans
► Amortized Loan: A loan that is repaid in equal
payments over its life.
► Amortization Schedule: A schedule shows precisely
how a loan will be repaid. It gives the required
payment on each payment date and a breakdown
of the payment, showing how much is interest and
how much is repayment of Principal.
Comparison of Different Types of Interest Rate
► Simple or Quoted Rate (iSIMPLE): The rate quoted by
borrowers and lenders that is used to determine the
rate earned per compounding period (periodic rate).
► Periodic Rate: The rate changed by a lender or paid
by a borrower each interest period (e.g., monthly,
quarterly, annually and so on)
iSIMPLE
Periodic Rate= --------------
m
Cont…..
► Annual Percentage Rate (APR): The rate reported to
borrowers- it is the periodic rate times the number of periods
in the year; thus interest compounding is not considered.
iSIMPLE= (Periodic Rate) X (m) = APR
► Effective Annual Rate (EAR): The annual rate earned or paid
considering interest compounding during year (i.e., the annual
rate that equates to a given periodic rate compounded for m
periods during the year)
iSIMPLE
EAR= (1+ --------------)m -1.0
m
=(1+ Periodic rate)m -1
