comprehensive explaination of the concept of time value of money

1. TIME VALUE OF MONEY

2. CONCEPT
• Time value of money is the economic principle that implies money
received today has greater value than the same amount of money
received in future.
• Time value of money describes the greater benefit of receiving money
now rather than later.
• time value of money underpins the concept of interest, and can be used
to compare investments, such as loans, bonds, mortgages, leases and
savings.

3. • Preferring to get certain sum of money now, than
receiving the same amount at some point in future.
Also known as
TIME
PREFERENCE OF
MONEY.

4. Not certain of cash receipts.
Prefer present consumption of goods due to :-
• Urgency of goods.
• Risk of not being in a position to consume
these goods in future.(as money is the means)
Increase the total cash inflow in future by making
appropriate investments.
REASONS FOR TIME VALUE OF MONEY
RISK
PREFERENCE FOR
CONSUMPTION
INVESTMENT
OPPORTUNITIES

5. ADJUSTING CASH FLOWS FOR TIME VALUE OF MONEY
DISCOUNTING
Process of calculating
present value of
cashflows.
COMPOUNDING
Process of calculating
future value of
cashflows.

6. COMPOUNDING
 Techniques of determining the future value of an investment made in
present.
 FUTURE VALUE :- The value of money at a future date with a given
interest rate is called future value.
COMPOUNDING TECHNIQUES
FUTURE
VALUE OF
SINGLE CASH
FLOW.
FUTURE
VALUE OF AN
ANNUITY
SINKING
FUND

7. 𝒊 = interest rate per period
𝒏= number of years
F = Future value or compound value
P = Present amount invested
If the present amount “P” is
invested at “i” rate of interest for
“n” years, then the future value
"𝑭 𝒏" (viz., principal plus interest)
at the end of n years will be
𝑭 𝒏 = 𝑷(𝟏 + 𝒊) 𝒏
SITUATION-1:- FUTURE VALUE OF SINGLE CASH FLOW

8. GIVEN :- 𝑷 = 1000; 𝒊 = 5%; 𝒏 = 10 years

9. GIVEN :-
• 𝒑 = 1000
• 𝒊 = 5%
• 𝒏 = 10 years
THEN :-
𝐹𝑛 = 𝑷 × 𝑪𝑽𝑭(𝒏,𝒊)
𝐹𝟏𝟎 = 𝑷 × 𝑪𝑽𝑭(𝟏𝟎,𝟎.𝟎𝟓)
𝐹𝟏𝟎 = 1000 × 1.629
𝐹𝟏𝟎 = 1629
≫ 𝐹𝑛 = 𝑷 × 𝑪𝑽𝑭(𝒏,𝒊)
𝐹𝑛 = 𝑃(𝟏 + 𝒊) 𝒏
(𝟏 + 𝒊) 𝒏 = 𝐂𝐎𝐌𝐏𝐎𝐔𝐍𝐃 𝐕𝐀𝐋𝐔𝐄 𝐅𝐀𝐂𝐓𝐎𝐄 𝐂𝐕𝐅 for a given
interest rate, 𝒊 and the time period, 𝒏.
Compound value factor of a lumpsum of ₹ 1

10. ANNUITY :- A fixed payment/receipt each year for a specified number of
years.
SITUATION-2:- FUTURE VALUE OF AN ANNUITY
Suppose ₹ 1 is deposited in a saving A/c at the end of each year for 4 years at a rate of
interest of 6%. How much would this annuity accumulate at the end of fourth year?
Period 0 1 2 3
Deposit at the end of
year (₹) 1 1 1 1
Yields no interest
Yields interest for 1 year
Yields interest for 2 years
Yields interest for 3 years
Value after 4 years
1.000
1.060
1.124
1.191
4.375FUTURE SUM AFTER 4 YEARS

11. GIVEN:-
𝐴 = 100
𝑖 = 10%
𝑛 = 3 years

12. 𝑭 𝒏 = 𝑨
(𝟏 + 𝒊) 𝒏−𝟏
𝒊
COMPOUND VALUE FACTOR
OF AN ANNUITY (CVFA) for
“n” number of years at “i” rate
of interest.
⟹ 𝑭 𝒏 = 𝑨 × 𝑪𝑽𝑭𝑨 𝒏,𝒊
GIVEN :-
• 𝐴 = 100
• 𝑖 = 10%
• 𝑛 = 3 years
THEN :-
𝐹𝟑 = 𝑨 × 𝑪𝑽𝑭𝑨(𝟑,𝟎.𝟏𝟎)
𝐹𝟑 = 100 × 3.310
𝐹𝟑 = 331.00
Compound value factor of an annuity of a lumpsum of ₹ 1

13. SINKING FUND :- A fund created out of fixed payment each period to
accumulate to a future sum after a specified period.
SITUATION-3 :- SINKING FUND
Suppose MR. X wants to accumulate ₹10,000 at the end of fourth year. How much
amount should MR. X deposit at an interest rate of 6% such that it grows to ₹10,000 at
the end of 4th year.
SINKING FUND REQUIRED DEPOSITS = ₹ 10,000 At the end of FOUR years
Interest rate
of 6%
𝝌 𝝌 𝝌 𝝌
1st year 2nd year 3rd year 4th year

14. Sinking fund (annuity) =
Future value
Compound value factor of an annuity of Re 1
As we know, 𝑭 𝒏= 𝑨 × 𝑪𝑽𝑭𝑨 𝒏,𝒊
⟹ 𝑨 = 𝑭 𝒏 ×
𝟏
𝑪𝑽𝑭𝑨(𝒏,𝒊)
⟹ 𝑨 = 𝑭 𝒏 ×
𝒊
(𝟏+𝒊) 𝒏−𝟏
; since,𝐶𝑉𝐹𝐴(𝑛,𝑖) =
(1+𝑖) 𝑛 −1
𝑖
Also,
𝟏
𝑪𝑽𝑭𝑨 𝒏,𝒊
= SINKING FUND FACTOR (SFF)
∴ 𝑨 = 𝑭 𝒏 × 𝑺𝑭𝑭 𝒏,𝒊

15. GIVEN:-
• 𝐹3= 331
• 𝑖 = 10%
• 𝑛 = 3 years
• 𝐴 = ?
Using, 𝑨 = 𝑭 𝒏 ×
𝟏
𝑪𝑽𝑭𝑨(𝒏,𝒊)
𝐴 = 𝐹3 ×
𝟏
𝑪𝑽𝑭𝑨(𝟑,𝟎.𝟏𝟎)
𝐴 = 331 ×
1
3.310
𝐴 = 100
Compound value factor of an annuity of a lumpsum of ₹ 1

16. DISCOUNTING
 Techniques of determining the present value of a some of money to be received in
future.
 PRESENT VALUE :- The worth of money today, that is receivable or payable in
future is called the present value.
 Present value is determined by applying a discount rate to the given future value.
DISCOUNTING TECHNIQUES
PRESENT
VALUE OF
SINGLE CASH
FLOW.
PRESENT
VALUE OF AN
ANNUITY
PRESENT
VALUE OF
UNEVEN
CASHFLOWS
PRESENT
VALUE OF
PERPETUITY

17. Let “I” represent the interest rate per period, “n” the numbe o periods, “F” the Future
Value of Cashflow and “P” the present value of cash flow .
We know, 𝑭 𝒏 = 𝑷(𝟏 + 𝒊) 𝒏
SITUATION-1:- PRESENT VALUE OF SINGLE CASH FLOW
𝒊= interest rate per period
𝒏= number of years
𝑭= Future value or compound value
𝑷 = present value

18. Suppose:-
• 𝒊= 10%
• 𝒏= 10 years
• 𝑭 𝟏𝟎= ₹
50,000
• 𝑷 = ?
⟹ 𝑃 =
𝐹𝑛
(1 + 𝑖) 𝑛
⟹ 𝑃 =
𝐹20
(1 + 𝑖)10
⟹ 𝑃 =
50,000
(1 + 0.10)10
⟹ 𝑷 =
𝟓𝟎,𝟎𝟎𝟎
𝟏.10 10 = 19300
⟹ 𝑷 = 𝑭 𝒏 ×
𝟏
(𝟏 + 𝒊) 𝒏
𝟏
(𝟏+𝒊) 𝒏 = discount factor or Present
Value Factor

19. ⟹ 𝑷 = 𝑭 𝒏 ×
𝟏
(𝟏 + 𝒊) 𝒏
Suppose:-
• 𝒊= 10%
• 𝒏= 10 years
• 𝑭 𝟏𝟎= ₹ 50000
⟹ 𝑷 = 𝑭 𝒏 × 𝑷𝑽𝑭(𝒏,𝒊)
SOLUTION:-
⟹ 𝑷 = 𝑭 𝒏 × 𝑷𝑽𝑭(𝒏,𝒊)
⟹ 𝑷 = 𝟓𝟎𝟎𝟎𝟎 × 𝟎. 𝟑𝟖𝟔
⟹ 𝑷 = 𝟏𝟗𝟑𝟎𝟎
Present value factor of lumpsum of ₹ 1

20. Mr. X
Has an investment opportunity of RECEIVING an
annuity of ₹ A for n years at an interest rate of 𝒊%.
The present value 𝒑 can be calculate as follows:-
Consider,
𝑷 =
𝑨
(𝟏 + 𝒊)
+
𝑨
(𝟏 + 𝒊) 𝟐
+
𝑨
(𝟏 + 𝒊) 𝟑
+−−−−−− +
𝑨
(𝟏 + 𝒊) 𝒏
𝑷 = 𝑨
𝟏
(𝟏 + 𝒊)
+
𝟏
(𝟏 + 𝒊) 𝟐 +
𝟏
(𝟏 + 𝒊) 𝟑 +−−−−−− +
𝟏
(𝟏 + 𝒊) 𝒏
𝑷 = 𝑨
𝟏−
𝟏
(𝟏+𝒊) 𝒏
𝒊
= 𝑨
(𝟏+𝒊) 𝒏−𝟏
𝒊(𝟏+𝒊) 𝒏
𝑷 = 𝑨
𝟏
𝒊
−
𝟏
𝒊(𝟏 + 𝒊) 𝒏
SITUATION-2 :- PRESENT VALUE OF ANNUITY

21. 𝑷 = 𝑨
𝟏
𝒊
−
𝟏
𝒊(𝟏 + 𝒊) 𝒏
𝟏
𝒊
−
𝟏
𝒊(𝟏+𝒊) 𝒏 is the Present Value Factor of an Annuity (PVFA).
𝑷 = 𝑨 × 𝑷𝑽𝑭𝑨 𝒏,𝒊
Suppose:-
• 𝒊 = 𝟏𝟎%
• 𝒏 = 4 years
• A = ₹ 5000
SOLUTION:-
⟹ 𝐏 = 𝐀 × 𝐏𝐕𝐅𝑨(𝐧,𝐢)
⟹ 𝐏 = 𝟓𝟎𝟎𝟎 × 𝟑. 𝟏𝟕𝟎
⟹ 𝐏 = 𝟏𝟓𝟖𝟓𝟎
Present value factor of an annuity of ₹ 1

22. SITUATION-3 :- PRESENT VALUE OF PERPETUITY
• PERPETUITY :- Perpetuity is an annuity that occurs indefinitely.
For e.g. :- In the case of irredeemable preference shares, the company is
expected to pay preference dividend perpetually.
𝑃 = 𝐴
1
𝑖
−
1
𝑖(1+𝑖) 𝑛 ;
𝑃 = 𝐴
1
𝑖
;
𝑃 =
𝐴
𝑖
{In perpetuity the time period is so large that time period “n”
approaches infinity, ∞ and expression (𝟏 + 𝒊) 𝒏
in the equation tends
to become zero}
Since, (𝟏 + 𝒊) 𝒏
in the equation tends
to become zero.
∴ 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒

23. ∴ 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒
𝑃 =
𝐴
𝑖
EXAMPLE:-
An investor expects a perpetual sum of ₹ 500 annually from his investment. What is
the present value of this perpetuity if the interest rate is 10%
GIVEN: 𝐴 = 500 ; 𝑖 = 10% ;
Using,𝑃 =
𝐴
𝑖
𝑃 =
500
0.10
= ₹ 5000

24. Prepared and presented by:-
RIYA ARORA
ROLL NO. -47