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Time Value of
Money
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Obviously, Rs10,000 today.
You already recognize that there isTIME VALUE TO MONEY!!
The In terest Rate
Which would you preferRs10,000
today or Rs10,000 in 5 years?
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TIMEallows you the oppor tun i tytopostpone consumption and earn
INTEREST
Arupee todayrepresents a greater realpurchas ing powerthan a rupee a year
hence
Why TIME?
Why is TIMEsuch an important
element in your decision?
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Time Value Adjustment
Two most common methods ofadjusting cash flows for time value ofmoney:
Compoundingthe process ofcalculating future valuesof cashflows and
Discountingthe process ofcalculating present valuesof cashflows.
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Types o f In teres t
Compound Interest
Interest paid (earned) on any previousinterest earned, as well as on the
principal borrowed (lent).
Simple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).
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Simp le In terest Formu la
Formula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
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SI = P0
(i)(n)
= Rs1,000(.07)(2)
= Rs140
Simp le In terest Example
Assume that you deposit Rs1,000in an
account earning 7%simple interest for
2years. What is the accumulatedinterestat the end of the 2nd year?
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FV = P0+ SI
= Rs1,000+ Rs140
=Rs 1,140
Future Value is the value at some futuretime of a present amount of money, or a
series of payments, evaluated at a given
interest rate.
Simp le In terest (FV)
What is the Future Value (FV) of the
deposit?
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The Present Value is simply the
Rs 1,000you o r ig inal ly deposi ted.
That is the value today!
Present Value is the current value of afuture amount of money, or a series of
payments, evaluated at a given interest
rate.
Simp le In terest (PV)
What is the Present Value (PV) of the
previous problem?
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Assume that you deposit Rs 1,000at a compound interest rate of 7%
for 2 years.
Futu re Value
Sing le Depos it (Graph ic)
0 1 2
Rs 1,000
FV2
7%
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FV1 = P0(1+i)1 = Rs 1,000(1.07)
= Rs 1,070
FV2 = FV1(1+i)1
= P0 (1+i)(1+i) = Rs1,000(1.07)(1.07)
= P0(1+i)2
= Rs1,000(1.07)2
= Rs1,144.90
You earned an EXTRA Rs 4.90in Year 2 with
compound over simple interest.
Future Value
Sing le Depos it (Formu la)
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FV1= P0(1+i)1
FV2= P0(1+i)
2
General Future Value Formula:
FVn= P0(1+i)n
or FVn= P0(FVIFi,n)
General Fu tu re
Value Formu la
etc.
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Reena wants to know how large her deposit ofRs 10,000today will become at a compound
annual interest rate of 10%for 5 years.
Problem
0 1 2 3 4 5
Rs10,000
FV5
10%
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Solut ion
Calculation based on general formula:
FVn = P0(1+i)n
FV5 = Rs10,000(1+0.10)5
= Rs 16,105.10
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We will use the Rule-of-72.
Doub le Your Money!!!
Quick! How long does it take to double
Rs 5,000 at a compound rate of 12%
per year (approx.)?
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Doubling Period = 72 / Interest Rate
6 years
For accuracy use theRule-of-69
.
Doubling Period
=0.35 +(69 / Interest Rate)
6.1 years
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Assume that you need Rs 1,000in 2 years.Lets examine the process to determinehow much you need to deposit today at a
discount rate of 7% compounded annually.
0 1 2
Rs 1,000
7%
PV1PV0
Present Value
Sing le Depos it (Graph ic)
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PV0= FV2/ (1+i)2 = Rs 1,000/ (1.07)2
= FV2
/ (1+i)2 = Rs 873.44
Present Value
Sing le Depos it (Formu la)
0 1 2
Rs 1,000
7%
PV0
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PV0= FV1/ (1+i)1
PV0= FV2/ (1+i)2
General Present Value Formula:
PV0= FVn/ (1+i)n
or PV0= FVn(PVIFi,n)
General Presen t
Value Formu la
etc.
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Reena wants to know how large of adeposit to make so that the money willgrow to Rs 10,000 in 5 yearsat a discount
rate of 10%.
Problem
0 1 2 3 4 5
Rs 10,000
PV0
10%
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Calculation based on general formula: PV0 = FVn/ (1+i)
n
PV0 = Rs 10,000/ (1+0.10)5= Rs 6,209.21
Prob lem Solu t ion
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Types o f Annui t ies
Ordinary Annuity: Payments or receipts
occur at the endof each period.
Annuity Due: Payments or receipts
occur at the beginningof each period.
An Annu i tyrepresents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
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Parts of an Annui ty
0 1 2 3
Rs 100 Rs 100 Rs 100
(Annuity Due)
Beginning of
Period 1
Beginning of
Period 2
Today Equal Cash Flows
Each 1 Period Apart
Beginning of
Period 3
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FVAn= R(1+i)n-1 + R(1+i)n-2 +
... + R(1+i)1+ R(1+i)0
Ordinary Annu i ty -- FVA
R R R
0 1 2 n n+1
FVAn
R= Periodic
Cash Flow
Cash flows occur at the end of the period
i% . . .
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Example o f an
Ordinary Annu i ty -- FVA
Rs1,000 Rs1,000 Rs1,000
0 1 2 3 4
7%
Cash flows occur at the end of the period
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FVA3=1,000(1.07)2 +1,000(1.07)1 + 1,000(1.07)0
= 1,145+1,070+1,000=Rs 3,215
Example o f an
Ordinary Annu i ty -- FVA
Rs1,000 Rs1,000 Rs1,000
0 1 2 3 4
Rs3,215 =
FVA3
7%
Rs1,070
Rs1,145
Cash flows occur at the end of the period
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General Formu la fo r Calcu lat ing
Fu tu re Value of an Ord inary
Annu i ty
AiAiAFVAn nn ...)1()1( 21
i
i
A
n 1)1(
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FVADn= R(1+i)n + R(1+i)n-1 +... + R(1+i)2+ R(1+i)1
= FVAn (1+i)
Annu ity Due -- FVAD
R R R R R
0 1 2 3 n-1 n
FVADn
i% . . .
Cash flows occur at the beginning of the period
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FVAD3= 1,000(1.07)3 +1,000(1.07)2 + 1,000(1.07)1
= 1,225+1,145+1,070
=Rs 3,440
Example o f an
Annu ity Due -- FVAD
1,000 1,000 1,000 1,070
0 1 2 3 4
Rs 3,440 =FVAD3
7%
Rs1,225
Rs1,145
Cash flows occur at the beginning of the period
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PVAn= R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
Ordinary Annui ty -- PVA
R R R
0 1 2 n n+1
PVAn
R= Periodic
Cash Flow
i% . . .
Cash flows occur at the end of the period
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Example o f an
Ordinary Annu i ty -- PVA
Rs1,000 Rs1,000 Rs1,000
0 1 2 3 4
7%
Cash flows occur at the end of the period
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PVA3= 1,000/(1.07)1
+1,000/(1.07)2 +1,000/(1.07)3
=934.58 + 873.44 + 816.30=2,624.32
Example o f an
Ordinary Annu i ty -- PVA
Rs1,000 Rs1,000 Rs1,000
0 1 2 3 4
Rs 2,624.32 = PVA3
7%
934.58
873.44
816.30
Cash flows occur at the end of the period
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nn
i
A
i
A
i
A
PVA)1(
...)1()1( 2
n
n
ii
iA)1(
1)1(
General Formu la fo r Calcu lat ing
Present Value o f an Ord inaryAnnu i ty
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PVADn= R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1
= PVAn (1+i)
Annu ity Due -- PVAD
R R R R
0 1 2 n-1 n
PVADn
R: Periodic
Cash Flow
i% . . .
Cash flows occur at the beginning of the period
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PVADn= 1,000/(1.07)0 + 1,000/(1.07)1 +
1,000/(1.07)2 = Rs 2,808.02
Example o f an
Annu ity Due -- PVAD
1,000.00 1,000 1,000
0 1 2 3 4
2,808.02 = PVADn
7%
934.58
873.44
Cash flows occur at the beginning of the period
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Reena will receive the set of cashflows below. What is the Present
Value at a discount rate of 10%?
Mixed Flows Examp le
0 1 2 3 4 5
600 600 400 400 100
PV0
10%
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Solut ion
0 1 2 3 4 5
600 600 400 400 100
10%
545.45
495.87
300.53273.21
62.09
Rs 1677.15 = PV0of the Mixed Flow
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General Formula:
FVn= PV0(1 + [i/m])mn
Or=PV0* PVIFi/m,m*n
n: Number of Years
m: Compounding Periods per Year i: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today
Sho rter Discoun t ing Per iods
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Reena has Rs1,000to invest for 1 yearat an annual interest rate of 12%.
Example
Annual FV = 1,000(1+ [.12/1])(1)(1)
= 1,120
Semi FV = 1,000(1+ [.12/2])(2)(1)
= 1,123.6
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Effective vs. Nominal Rate of
InterestRs. 1000 Rs.1123.6
So,Rs. 1000 grows @ 12.36% annually
Effective Rate of Interest
r = 1 + i/mm
- 1
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Basket Wonders (BW) has a Rs1,000 CDat the bank. The interest rate is 6%compounded quarterly for 1 year.
What is the Effective Annual InterestRate (EAR)?
Problem
EAR = ( 1 +6%/ 4)4- 1= 1.0614 - 1 = .0614 or
6.14%!
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Perpetuity
A perpetuity is an annuity with aninfinite number of cash flows.
The present value of cash flows
occurring in the distant future is veryclose to zero.
At 10% interest, the PV of Rs 100
cash flow occurring 50 years fromtoday is Rs 0.85!
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Present Value of a
Perpetuity
nn
i
A
i
A
i
APVA
)1(...
)1()1( 2
When n=
PVperpetuity= [A/(1+i)][1-1/(1+i)]
= A(1/i) = A/i
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Present Value of a
PerpetuityWhat is the present value of a perpetuityof Rs270 per year if the interest rate is
12% per year?
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Present Value of a
Perpetuity
PV Aiperpetuity
Rs2700.12
Rs 2250
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1. Calculate the payment per period.
2. Determine the interestin Period t.
(Loan balance at t-1) x (i% / m )3. Computeprincipal payment in Period t.
(Payment- interestfrom Step 2)
4. Determine ending balance in Period t.(Balance- pr inc ipa l payment f rom Step 3)
5. Start again at Step 2 and repeat.
Steps to Amort izing a Loan
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Reena is borrowing Rs10,000 at a compoundannual interest rate of 12%. Amortize the loan
if annual payments are made for 5 years.
Amort izing a Loan Example
Step 1: Payment
PV0 = R(PVIFA i%,n)
Rs10,000 = R(PVIFA 12%,5)
Rs10,000 = R(3.605)
R= Rs10,000/ 3.605 = Rs2,774
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Amort izing a Loan Example
End ofYear
Payment Interest Principal EndingBalance
0
12
3
4
5
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Amort izing a Loan Example
End of
Year
Payment Interest Principal Ending
Balance
0 --- --- --- Rs10,000
1 Rs2,774 Rs1,200 Rs1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
Rs13,871 Rs3,871 Rs10,000
[Last Payment Slightly Higher Due to Rounding]
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Usefu lness of Amort izat ion
2. Calculate Debt Outstanding -- The
quantity of outstanding debt
may be used in financing theday-to-day activities of the firm.
1. Determine Interest Expense --
Interest expenses may reduce
taxable income of the firm.
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EXERCISE
Ashish recently obtained a
Rs.50,000 loan. The loan carries
an 8% annual interest. Amortizethe loan if annual payments are
made for 5 years.
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SOLUTION50000 5 0.08
12523
TIME PAYMENT INTERESTPRINCIPAL AMOUNTOUTSTANDING
0 50000
1 12523 4000 8523 41477
2 12523 3318 9205 32272
3 12523 2582 9941 22331
4 12523 1786 10737 11594
5 12522 928 11594 0
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EXERCISE
Compute the present value of thefollowing future cash inflows,
assuming a required rate of 10%:Rs. 100 a year for years 1through 3, and Rs. 200 a year
from years 6 through 15.
ANS: 1011.75
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Solution
100 100 100 200 200 200
0 1 2 3 6 7 15
248.70
i% . . .
Cash flows occur at the end of the period
. . .
1228.9
763.05
1011.75
Till 5th
year
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