2. THE INTEREST RATE
Which would you prefer – Rs10,000
today or Rs10,000 in 5 years?
Obviously, Rs10,000 today.
You already recognize that there is
TIME VALUE TO MONEY!!
3. WHY TIME?
Why is TIME such an important element in
your decision?
TIME allows you the opportunity to postpone
consumption and earn INTEREST
A rupee today represents a greater real purchasing
power than a rupee a year hence
Receiving a rupee a year hence is uncertain so risk is
involved
4. TIME VALUE ADJUSTMENT
Two most common methods of adjusting
cash flows for time value of money:
Compounding—the process of
calculating future values of cash flows
and
Discounting—the process of
calculating present values of cash
flows.
5. TYPES OF INTEREST
Simple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).
Compound Interest
Interest paid (earned) on any previous
interest earned, as well as on the principal
borrowed (lent).
6. SIMPLE INTEREST FORMULA
Formula SI = P0(i)(n)
SI:Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
7. SIMPLE INTEREST EXAMPLE
Assume that you deposit Rs1,000 in an
account earning 7% simple interest for 2
years. What is the accumulated interest at
the end of the 2nd year?
SI = P0(i)(n)
= Rs1,000(.07)(2)
= Rs140
8. SIMPLE INTEREST (FV)
What is the Future Value (FV) of the
deposit?
FV = P0 + SI
= Rs1,000 + Rs140
= Rs 1,140
Future Value is the value at some future time
of a present amount of money, or a series of
payments, evaluated at a given interest rate.
9. SIMPLE INTEREST (PV)
What is the Present Value (PV) of the
previous problem?
The Present Value is simply the
Rs 1,000 you originally deposited.
That is the value today!
Present Value is the current value of a
future amount of money, or a series of
payments, evaluated at a given interest
rate.
10. FUTURE VALUE
SINGLE DEPOSIT (GRAPHIC)
Assume that you deposit Rs 1,000 at
a compound interest rate of 7% for 2
years.
0 1 2
7%
Rs 1,000
FV2
11. Future Value
Single Deposit (Formula)
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 4.90 in Year 2 with
compound over simple interest.
12. GENERAL FUTURE VALUE
FORMULA
FV1 = P0(1+i)1
FV2 = P0(1+i)2
etc.
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n)
13. PROBLEM
Reena wants to know how large her deposit of Rs
10,000 today will become at a compound annual
interest rate of 10% for 5 years.
0 1 2 3 4 5
10%
Rs10,000
FV5
14. SOLUTION
Calculation based on general formula:
FVn = P0 (1+i)n
FV5 = Rs10,000 (1+ 0.10) 5
= Rs 16,105.10
15. DOUBLE YOUR MONEY!!!
Quick! How long does it take to double Rs
5,000 at a compound rate of 12% per
year (approx.)?
We will use the ―Rule-of-72‖.
16. Doubling Period = 72 / Interest Rate
6 years
For accuracy use the “Rule-of-69”.
Doubling Period
=0.35 +(69 / Interest Rate)
6.1 years
17. PRESENT VALUE
SINGLE DEPOSIT (GRAPHIC)
Assume that you need Rs 1,000 in 2 years.
Let’s examine the process to determine how
much you need to deposit today at a discount
rate of 7% compounded annually.
0 1 2
7%
Rs 1,000
PV0 PV1
19. GENERAL PRESENT VALUE
FORMULA
PV0 = FV1 / (1+i)1
PV0 = FV2 / (1+i)2
etc.
General Present Value Formula:
PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n)
20. PROBLEM
Reena wants to know how large of a deposit
to make so that the money will grow to Rs
10,000 in 5 years at a discount rate of 10%.
0 1 2 3 4 5
10%
Rs 10,000
PV0
21. PROBLEM SOLUTION
Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = Rs 10,000 / (1+ 0.10)5
= Rs 6,209.21
22. TYPES OF ANNUITIES
An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
Ordinary Annuity: Payments or receipts
occur at the end of each period.
Annuity Due: Payments or receipts occur at
the beginning of each period.
23. EXAMPLES OF ANNUITIES
Student Loan Payments
Car Loan Payments
Insurance Premiums
Retirement Savings
24. PARTS OF AN ANNUITY
(Ordinary Annuity)
End of End of End of
Period 1 Period 2 Period 3
0 1 2 3
Rs 100 Rs 100 Rs 100
Today
Equal Cash Flows
Each 1 Period Apart
25. PARTS OF AN ANNUITY
(Annuity Due)
Beginning of Beginning of Beginning of
Period 1 Period 2 Period 3
0 1 2 3
Rs 100 Rs 100 Rs 100
Today Equal Cash Flows
Each 1 Period Apart
26. FUTURE VALUE OF ORDINARY
ANNUITY -- FVA
Cash flows occur at the end of the period
0 1 2 n n+1
i% . . .
A A A
A = Periodic
Cash Flow
FVAn
FVAn = A(1+i)n-1 + A(1+i)n-2 +
... + A(1+i)1 + A(1+i)0
27. EXAMPLE OF AN
ORDINARY ANNUITY -- FVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
Rs1,000 Rs1,000 Rs1,000
28. EXAMPLE OF AN
ORDINARY ANNUITY -- FVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
Rs1,000 Rs1,000 Rs1,000
Rs1,070
Rs1,145
FVA3 = 1,000(1.07)2 +
1,000(1.07)1 + 1,000(1.07)0 Rs3,215 =
FVA3
= 1,145 + 1,070 + 1,000
= Rs 3,215
29. General Formula for Calculating
Future Value of an Ordinary Annuity
n 1 n 2
FVAn A(1 i ) A(1 i ) ... A
n
(1 i) 1
A
i
30. FUTURE VALUE OF ANNUITY DUE -
- FVAD
Cash flows occur at the beginning of the period
0 1 2 3 n-1 n
. . .
i%
R R R R R
FVADn = R(1+i)n + R(1+i)n-1 + FVADn
... + R(1+i)2 + R(1+i)1
= FVAn (1+i)
31. EXAMPLE OF AN
ANNUITY DUE -- FVAD
Cash flows occur at the beginning of the period
0 1 2 3 4
7%
1,000 1,000 1,000 1,070
Rs1,145
Rs1,225
FVAD3 = 1,000(1.07)3 + Rs 3,440 =
1,000(1.07)2 + 1,000(1.07)1
FVAD3
= 1,225 + 1,145 + 1,070
= Rs 3,440
32. PRESENT VALUE OF ORDINARY
ANNUITY -- PVA
Cash flows occur at the end of the period
0 1 2 n n+1
i% . . .
R R R
R = Periodic
Cash Flow
PVAn
PVAn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
33. EXAMPLE OF AN
ORDINARY ANNUITY -- PVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
Rs1,000 Rs1,000 Rs1,000
34. EXAMPLE OF AN
ORDINARY ANNUITY -- PVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
Rs1,000 Rs1,000 Rs1,000
934.58
873.44
816.30
Rs 2,624.32 = PVA3 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
35. General Formula for Calculating
Present Value of an Ordinary
Annuity
A A A
PVAn 2
... n
(1 i ) (1 i ) (1 i )
n
(1 i ) 1
A n
i (1 i )
36. PRESENT VALUE OF ANNUITY
DUE -- PVAD
Cash flows occur at the beginning of the period
0 1 2 n-1 n
i% . . .
R R R R
R: Periodic
PVADn Cash Flow
PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1
= PVAn (1+i)
37. EXAMPLE OF AN
ANNUITY DUE -- PVAD
Cash flows occur at the beginning of the period
0 1 2 3 4
7%
1,000.00 1,000 1,000
934.58
873.44
2,808.02 = PVADn
PVADn = 1,000/(1.07)0 + 1,000/(1.07)1 +
1,000/(1.07)2 = Rs 2,808.02
38. MIXED FLOWS EXAMPLE
Reena will receive the set of cash flows
below. What is the Present Value at a
discount rate of 10%?
0 1 2 3 4 5
10%
600 600 400 400 100
PV0
40. SHORTER COMPOUNDING AND
DISCOUNTING PERIODS
General Formula:
FVn = PV0(1 + [i/m])mn
Or
PVn = FV0 /(1+[i/m] )mn
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
41. EXAMPLE
Reena has Rs1,000 to invest for 1 year at an
annual interest rate of 12%.
Annual FV = 1,000(1+ [.12/1])(1)(1) = 1,120
Semi FV = 1,000(1+ [.12/2])(2)(1) = 1,123.6
42. EFFECTIVE VS. NOMINAL RATE OF
INTEREST
Rs. 1000 Rs.1123.6
So,
Rs. 1000 grows @ 12.36% annually
Effective Rate of Interest
m
r = 1 + i/m -1
12% is the nominal rate but 12.36% is the
effective rate.
43. PROBLEM
Basket Wonders (BW) has a Rs1,000 CD
at the bank. The interest rate is 6%
compounded quarterly for 1 year. What is
the Effective Annual Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4 - 1
= 1.0614 - 1 =
= 0.0614 or 6.14%!
44. PERPETUITY
A perpetuity is an annuity with an infinite
number of cash flows.
The present value of cash flows
occurring in the distant future is very
close to zero.
At 10% interest, the PV of Rs 100
cash flow occurring 50 years from
today is Rs 0.85!
45. PRESENT VALUE OF A PERPETUITY
A A A
PVAn 2
... n
(1 i ) (1 i ) (1 i )
When n=
PVperpetuity = [A/(1+i)]
[1-1/(1+i)]
= A(1/i) = A/i
46. PRESENT VALUE OF A PERPETUITY
What is the present value of a perpetuity of
Rs270 per year if the interest rate is 12%
per year?
PV A Rs270
Rs 2250
perpetuity
i 0.12
47. STEPS TO AMORTIZING A LOAN
1. Calculate the payment per period.
2. Determine the interest in Period t.
Loan balance at (t-1) x (i%)
3. Compute principal payment in Period t.
(Payment - interest from Step 2)
4. Determine ending balance in Period t.
(Balance - principal payment from Step 3)
5. Start again at Step 2 and repeat.
48. AMORTIZING A LOAN EXAMPLE
Reena is borrowing Rs10,000 at a compound
annual interest rate of 12%. Amortize the loan if
annual payments are made for 5 years.
Step 1: Payment
PV0 = A(PVIFA i%,n)
Rs10,000 = A(PVIFA 12%,5)
Rs10,000 = A(3.605)
A = Rs10,000 / 3.605 = Rs2,774
49. AMORTIZING A LOAN EXAMPLE
End of Payment Interest Principal Ending
Year Balance
0
1
2
3
4
5
50. AMORTIZING A LOAN EXAMPLE
End of Payment Interest Principal Ending
Year Balance
0 --- --- --- Rs10,000
1 Rs2,774 Rs1,200 Rs1,574 8,426
2
3
4
5
[Last Payment Slightly Higher Due to Rounding]
51. AMORTIZING A LOAN EXAMPLE
End of Payment Interest Principal Ending
Year 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]
52. USEFULNESS OF AMORTIZATION
1. Determine Interest Expense -- Interest expenses
may reduce taxable income of the firm.
2. Calculate Debt Outstanding -- The
quantity of outstanding debt
may be used in financing the day-to -
day activities of the firm.
53. EXERCISE
Ashish recently obtained a Rs.50,000
loan. The loan carries an 8% annual interest.
Amortize the loan if annual payments are
made for 5 years.
55. EXERCISE
Compute the present value of the
following future cash inflows, assuming
a required rate of 10%: Rs. 100 a year
for years 1 through 3, and Rs. 200 a
year from years 6 through 15.
ANS: 1011.75
56. SOLUTION
Cash flows occur at the end of the period
0 1 2 3 6 7 15
. . . . . .
i%
100 100 100 200 200 200
th
Till 5 year
248.70 1228.9
763.05
1011.75
