Time Value of Money
Objectives
• What do we mean by Time value of money
• Present Value, Discounted Value, Annuity
Time Value of Money
• What is Time Value of Money?
– Future Value
– Present Value
• Future Value: Compounding:
Assuming Compounding Done Annually
Principal P 20,000 20,000 20,000
Interest Rate i 10% 10% 10%
No. of Years n 1 2 3 How would you
Future Value FV 22,000 24,200 26,620 do
Interest Amount 2,000 2,200 2,420 Compounding?
Compounding
• Compounding Formula
FVn P * (1 i ) n
• What if compounding is done on monthly basis?
n*t
i
FVn P * 1
t Microsoft Office
Excel Worksheet
Assuming Compounding Done Monthly
Principal P 20,000 20,000 20,000
Interest Rate i 10% 10% 10%
No. of Years n 1 2 3
Times Compounding in a Year t 12 12 12
Maturity Value FV 22,094 24,408 26,964
Interest Amount 2,094 4,408 6,964
Compounding Exercise
• Exercise:
– Prepare a table showing compounding as per
following conditions:
– Rate of Interest - 5%, 12% and 15%
– Compounding 2 & 4 times in a year
– Principal Rs.100,000/-
Discounting
• Present Value
– You have an option to receive Rs. 1,000/- either today or after
one year. Which option you will select? Why?
– Decision will depend upon the present value of money; which
can be calculated by a process called Discounting (opposite of
Compounding)
– Interest Rate and Time of Receipt of money decide Present
Value
– What is the present value of Rs. 1,000/- today and a year later?
To compute Present Value?
Discounting contd…
• Formula to find Present Value of Future Cash Receipt
P
PVn
1 i n
– Where PV = Present Value, P = Principal, i = Rate of Interest, n = Number
of Years after which money is received
• Assuming Rate of Interest is 10%, value of Rs. 1,000/- to be received
after 1 year will be,
1000
909.09
1 10% 1
• Whereas the value of money to be received today will be Rs. 1,000/-
What if you were to choose between:
a. Receive Rs. 1,000/- every year for 3 years, OR
b. Receive Rs. 2,500/- today? (assume 10% annual interest rate)
Discounting of a Series contd…
• How discounting is done for a series of cashflow? e.g.
– Receive Rs. 1,000/- at the end of every year for 3 years OR
– Receive Rs. 2,500/- today Discounting
Annually
– Assume Rate of Interest @10%
Assuming Discounting Done Annually
Principal P 20,000 20,000 20,000
Interest Rate i 10% 10% 10%
Year n 1 2 3
Present Value PV 18,181.82 16,528.93 15,026.30
If cashflow was to occur every 6 months instead of 1 year, what
impact it will have on Present Value?
Periodic Discounting
• What if the receipts are over six months’
interval ? Find Present Value of the money
receipts
– Receive Rs. 1,000/- at the end of every 6 months for 1-1/2 years OR
– Receive Rs. 2,600/- today
– Assume Rate of interest @10%
• Periodic
P
Discounting Formula
PV
i
n Where, P = Principal, i = Rate of
1 Interest,
t
t = Times Payments made in a Year,
n = nth Period (in this case it is half
year)
Periodic Discounting Formula
Expressed mathematically, the equation will look like:
1000 1000 1000 Genericallyexpressed,
Generically expressed,
2723.25 theformula
formulais:
is:
10%
1
10%
2
10%
3 the
1 1 1
2 2 2 SUMofPV
N
xn
Principal
Assuming Discounting Done Semi-Annually
P 1,000 1,000 1,000
n 1 i
n
Interest Rate i 10% 10% 10% 1
t
HY n 1 2 3
Times Discounting in a Year t 2 2
Here, N = 3
2 Here, N = 3
Discount Factor DF 0.9524 0.9070 0.8638
Present Value PV=P*DF 952.38 907.03 863.84
Sum of Present Value 2,723.25
Charting of Cashflow
• For any financial proposition prepare a chart of cashflow: e.g.
Invested in 10% Bonds 01-Jan-04 (1,000) Outflow
Interest received 30-Jun-04 50 Inflow
Interest received 31-Dec-04 50 Inflow
New Bond Purchased from
31-Dec-04 (1,020) Outflow
Open Market
Interest received 30-Jun-05 100 Inflow
Sold Bond in Open Market 30-Jun-05 2,050 Inflow Interest Received + 100
Sold Bond +2,050
Interest Received +50 Total +2,150
01.01.0 31.12.04
4 Timeline
30.06.04 30.06.05
Invested in Bonds Interest Received + 50
(1,000) New Bond Purchased (1,020)
Net ( 970)
Net Present Value
• Net Present Value means the difference between the PV of Cash Inflows &
Cash Outflows
• How do you compute NPV?
– Prepare Cashflow Chart
– Net off Inflow & Outflow for each period separately
• If Inflow > Outflow, positive cash
• If Inflow < Outflow, negative cash
• Find present values of Inflows & Outflows by applying Discount Factor (or
Present Value Factor)
• NPV = (PV of Inflows) LESS (PV of Outflows); Result can be +ve OR -ve
• Continuing with our example of Bond Investment:
Inflow Interest Received
Sold Bond
+ 100
+2,050
Interest Received +50 Total
+2,150
01.01.0 31.12.04
4 Timeline
30.06.04 30.06.05
Invested in Bonds Interest Received + 50
(1,000) New Bond Purchased (1,020)
Outflow Net ( 970)
NPV contd…
• If Cashflows are discounted at say 10%, the sum of PV is 25.05, a positive
number & therefore the IRR has be higher than 10% to make Net Present
Value to zero
Description Date Amount In / Out PV Outflow PV Inflow
Invested in 10% Bonds 01-Jan-04 (1,000) Outflow (1,000.00)
Interest received 30-Jun-04 50 Inflow 47.62
Interest received 31-Dec-04 50 Inflow 45.35
New Bond Purchased from
31-Dec-04 (1,020) Outflow (925.17)
Open Market
Interest received 30-Jun-05 100 Inflow 86.38
Sold Bond in Open Market 30-Jun-05 2,050 Inflow 1,770.87
Sum (1,925.17) 1,950.22
How these values are arrived at?
Net Present Value 25.05
Microsoft Excel
Worksheet
What is IRR?
Internal Rate of Return (IRR)
• Definition: The Rate at which the NPV is Zero. It can also be termed
as “Effective Rate”
• If we want to find out IRR of the bond investment cashflow:
Composit
Description Date
Flow
Invested in Bonds 01-Jan-04 (1,000)
Interest received 30-Jun-04 50
Interest received + New Bond
31-Dec-04 (970)
Purchased
Interest received + Sold Bond 30-Jun-05 2,150
IRR of entire cashflow 11.38%
Internal Rate of Return (IRR)
IRR: the discount rate that results in a zero
NPV for a project.
CF1 CF2 CF3 CFN
NPV 0 CF0 2
3
....
(1 r ) (1 r ) (1 r ) (1 r ) N
The IRR decision rule for an investing project is:
• If IRR is greater than the cost of capital, accept the
project.
• If IRR is less than the cost of capital, reject the
project.
IRR Contd…
• To prove that at IRR of 11.38% the NPV of Investment Cashflow
is zero, see the formula & table:
1000 50 970 2150
0 0
1
2
3
11 .38% 11.38% 11.38% 11 .38%
1 1 1 1
2 2 2 2
Composit NPV at
Description Date PV Factor
Flow IRR
Invested in Bonds 01-Jan-04 (1,000) 1.00000 (1,000.00)
Interest received 30-Jun-04 50 0.94615 47.31
Interest received +
31-Dec-04 (970) (868.34)
New Bond Purchased 0.89520
Interest received +
30-Jun-05 2,150 1,821.04
Sold Bond 0.84699
IRR of entire cashflow 11.38% Sum of PVs 0.00
IRR - Additional Example
• You buy a car costing Rs. 600,000/-
• Banker is willing to finance upto Rs. 500,000/-
• The loan is repayable over 3 years, in Equated
Monthly Installments (EMI) of Rs. 15,000/-
• Installments are payable In Arrears
• What is the IRR?
• How do you express this mathematically? What are
the values of each component in the formula?
• What will be the impact on IRR if the EMIs are
payable In Advance?
• Can we use IRR for computing Interest & Principal
break-up?
IRR - Additional Example contd…
• Plot the cashflow:
– EMI in Arrears
Begin 1 2 3 35 36
+500,000 01.02.200 01.03.200 01.04.200 01.11.200 01.12.200
6 6 6 ……… 8 8
01.01.200 -15,000 -15,000 -15,000 ……… -15,000 -15,000
6
End
Formula Values in Expression
Expression
N
xn 36 15,000n
P n
500,000 n Value of ‘i’
n 1
n 1 i i to be
1 1
t 12 determined
IRR - Additional Example contd…
• Plot the cashflow:
– EMI in Advance
Begin 1 2 3 35 36
+500,000 01.02.200 01.03.200 01.04.200 01.12.200 01.01.200
6 6 6 ……… 8 9
-15,000 -15,000 -15,000 -15,000 ……… -15,000 -15,000
01.01.200 End
6
Formula Values in Expression
Expression
N xn 36 15,000n
P X1 n
500,000 - 15,000 n Value of ‘i’
1 n 2
n 2 i i to be
1 1
t 12 determined
