Time Value of money

1. Time Value of Money

2. TIME allows you the opportunity to postpone
consumption and earn INTEREST.
Why is TIME such an important element in
your decision?

3. You already recognized
that there is
TIME VALUE Of MONEY!!
Which would you prefer -- Rs.10,000 today or Rs.10,000 after 5
years?
Time Value of Money

4. Simple Interest and Compounding Interest
Simple Interest – the interest that is calculated only on the original amount
Compounding Interest – the interest that is received on the original amount as well as on any interest earned
but not withdrawn during earlier periods

5. Compound Interest
Interest paid (earned) on any previous
interest earned, as well as on the principal
borrowed (lent).
Simple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).
Types of interest

6. Present value and Future Value
Present value (PV) is the current value of a future sum of money or stream of cash flows discounted at a
specified rate of return. In the example 100 is the present value of 110.
Future value (FV) is the value of a current sum of money at a specified date in the future based on specified
rate of return. In the example 110 is the future value of 100.

7. Compounding process (Finding out
Future value)
The process when the amount earned on the initial
deposit becomes part of principal amount at the end
of the first compounding period (Eg. 1 year) can be
termed as compounding process.

8. Numerical - Example:
After being placed, let us assume that from your salary you
invested Rs. 1,000 in savings account in a bank at 5% interest
rate compounded annually for 3 years. What will be total
amount appearing in your account at the end of 3rd year?
Assuming you are not withdrawing any money during these
3 years.

9. Numerical - Example:

10. Formula for compounding
Future value of single amount (annual compounding)
FV = 𝑷𝑽 𝟏 + 𝒊 𝒏
OR A = P (1 + i)n
FV = Future value
PV = present value
i= interest rate in decimal OR i/100 .
◦ In excel sheet once can use below formula;
In formula bar : FV(Rate, Nper, Pmt, Pv, Type)
fv is the future value
Rate is the interest rate per period in decimal
Nper is the total number of periods
Pmt can be left blank as no annuity
pv is the present value (Amount has to be in Negative as it is initial cash outflow)
Type is 0 if cash flows occur at the end of the period OR leave blank

11. Example
The fixed deposit scheme of Union Bank of India offers the following interest rates.
Period of Deposit Rate per Annum
135 days to 235 days 6.0%
236 days to < 1 year 6.5%
1 year and above 7.5%
An If you are investing amount of Rs. 10,000 for next 3 years then what will final amount that
you should receive?

12. Example
FV = 𝑷𝑽 𝟏 + 𝒊 𝒏
= 10,000 (1 + 0.075) 3
= 10,000 (1.075) 3
= 10,000 (1.2422)
FV = Rs. 12422

13. Semi-annual and Other Compounding
Periods
Semi-annual Compounding means that there are two compounding periods within the year.
Interest is actually paid after every six months at a rate of one-half of the annual (stated) rate of
interest.
Quarterly Compounding means that there are four compounding periods within the year.
Interest is actually paid after every 3 months (quarter).

14. Numerical - Example:
After being placed, let us assume that from your salary you
invested Rs. 1,000 in savings account in a bank at 6% annual
interest rate for 2 years. What will be total amount
appearing in your account at the end of 2nd year if interest is
paid every six month for this deposit.

15. Numerical - Example:
So, if interest is paid 6 monthly then, 3% interest
rate will be considered for every 6 month
interest calculation.

16. Formula for Semi-annual and Other
Compounding Periods
A = P { 1 + i/m} m*n
m = no. of times per year compounding
i= annual interest rate in decimal
So, for semi-annual m = 2,
for quarterly m = 4,
monthly m = 12,
weekly m = 52
daily m = 365
A = 1000 { 1 + (0.06/2) } 2 * 2
= 1000 { 1.03 } 4
= 1000 { 1.1255}
A = 1125. 50

17. Numerical - Example:
After being placed lets assume that from your salary you
invested Rs. 1,000 in savings account in a bank at 6% annual
interest rate for 2 years. What will be total amount
appearing in your account at the end of 2nd year if interest is
paid every quarter for this deposit.

18. Numerical - Example:
As, interest is paid quarterly, m = 4.
P = 1000, n = 2, I = 0.06
A = P { 1 + i/m} m*n
= 1000 { 1+ (0.06/4)} 4*2
A = 1000 (1.1264)
A = 1126.4

19. Discounting process (Finding out Present
value)
Formula:
P = Present value Principal amount
A = Future value
i = interest rate in decimal
n= no. of years

20. Numerical - Example
You are planning to have Rs. 2,000 at the end of 5 years
from now. You are going to receive interest rate of 10%
annually. What amount you should invest today so that you
can have Rs. 2,000 after 5 years ?

21. Numerical - Example
So,
P = 2000 / (1+0.1)5
P = 2000 / 1.6105
So, currently you need to invest Rs. 1241.85 so that you can receive Rs. 2,000 at the end of 5 years.
OR
You can look at the PVIF table to figure out the value for 10% rate & n = 5 years, you see the value of
0.62092. Now you can simply do A * PVIF value.
So, 2000 * 0.62092
P = 1241.84

22. Present value table

23. Future Value table

24. Present value of series of cash flows
If firm or individual is going to have series of cash flows over a some years, then present value of
all cash flows need to calculated and then totalled it.
Formula:

25. Numerical - Example
The company is going to receive below set of cash flows over 5 years from the project. The
discount rate (interest rate) is assumed to be 10%. Find out the total present value of all the cash
flows.

26. Numerical - Example
The above cash flows are also called discounted cash flows.

27. Annuity
Annuity : It is fixed amount of money paid or received on regular time period.
Annuity : it is recurring cash inflow or cash outflow.
When annuity is at the end of the year it is called annuity in arrears OR ordinary annuity.
Eg.: Every year You receive Rs,10,000 as interest on deposit in bank for 5 years.
Eg.: Every Month You pay installment to Bank for the home loan or car loan. It is called
EMI.(equated monthly installment)

28. Present Value annuity
PVA: It is finding out today’s value for the money paid or money received for regular period of time.
Formula:
C = cash received or paid at regular time period
r = Interest rate
n = No. of years
PV = Present value

29. Numerical - Example
Mr Sharma wishes to determine the present value of the
annuity consisting of cash inflows of Rs. 1,000 every year for
5 years. The rate of interest that is applicable for his
investment is 10 per cent.

30. Numerical - Example
Typical way is as below;

31. Numerical - Example
Alternative way;
PV = 1000 [ 1 – (1/ (1+0.1)5 ) / 0.1 ]
PV = 1000 [ 0.3791 / 0.1 ] So, PV = 1000 (3.791)
PV = 3,791. Now look at the PVIFA (Present value interest factor annuity table) for n = 5 years & r
= 10%...the value is 3.79079.

32. Present Value annuity table

33. Future Value Annuity
FVA: It is finding out future value for the money paid or money received for regular period of
time.
Formula:
C = cash received or paid at regular time period
i = Interest rate (r is also used for interest rate)
n = No. of years

34. Numerical - Example
Mr. Sahu deposits Rs 2,000 at the end of every year
for 5 years in his saving account having 5 per cent
interest compounded annually. He wants to
determine how much sum of money he will have at
the end of the 5th year.

35. Numerical - Example
Alternative way;
FV = 2000 * [ (1 + 0.05)5 - 1 / 0.05 ]
= 2000 * [ 1.2761 -1 / 0.05 ]
= 2000 * 5.525
FV = 11050
Now look at the FVIFA (Future value interest factor annuity table) for n = 5 years & i = 5%...the
value is 5.525.

36. Future value annuity table

37. Present Value of Perpetuity
An annuity that goes on for ever is called a perpetuity.
The present value of a perpetuity of Rs. “C” amount is given by the formula:
Present value of perpetuity = C/I
C = amount
I = Interest rate in decimal

38. Numerical - Example
Mr. Verma wishes to find out the present value
of investments which yield Rs. 500 every year in
perpetuity, discounted at 5 per cent.

39. Numerical - Example
Present value of perpetuity = C/I
= 500/0.05
Present value of perpetuity = 10,000

40. Nominal Rates of Interest and Effective
rate
The nominal rate of interest differs from the effective rate of interest due to the frequency of
compounding (e.g. annual, half-yearly, quarterly, monthly) with the nominal rate.
The coupon rate of interest/ normal rate of interest is called the nominal rate of interest.
The effective rate of interest is higher than nominal rate if it is compounded half-yearly,
quarterly, monthly etc. and increases with an increase in the frequency (half-yearly, quarterly,
monthly )of compounding.
When there is annual compounding nominal rate & effective rate is same.
Formula;
 r = Effective interest rate
 i = Nominal interest rate
 n = No. of compounding periods

41. Nominal Rates of Interest and Effective rate
Mrs. Jain has been provided with nominal interest
rate of 10.25 % on the deposit which is going to have
half-yearly interest payment. What will be effective
interest rate for her?

42. Nominal Rates of Interest and Effective
rate
So,
r = (1 + 0.1025/2) 2 - 1
r = ( 1.0512) 2 - 1
r = 1.1050 – 1
r = 0.1050 So, r = 10.50%

43. Numerical

44. Numerical

45. Numerical

46. Numerical
We have to apply present value annuity;
= 2,50,000 [ 1 – 1/ (1+ 0.1) 10 / 0.1 ]
= 2,50,000 [6.144]
= 15,36,000
So, employee should select option-1 to receive 20,00,000 lump sum right on now as it is higher
than option -2 that is 15,36,000.

47. Numerical

48. Numerical
So, we need to find future value annuity so that firm can invest that amount every year to have
Rs. 10 Cr. at the end of 15 years.
10,00,00,000 = C * [ ( 1+ 0.1) 15 - 1 / 0.1 ]
10,00,00,000 = C * [ 31.7724]
C = 10,00,00,000 / 31.7724
C = 31,47,385
So, firm has to invest Rs. 31,47,385 every year so that at the end of 15 years total Rs. 10 Cr. is
available to pay for bonds.

49. Numerical

50. Numerical
So, firm wants to find out the annuity it should pay every year of which the present value will be
equal to 30,00,000.
30,00,000 = C * [ 1- 1/ (1+0.24) 15 / 0.24 ]
30,00,000 = C * [ 4.0012]
30,00,000 / 4.0012 = C
C = 7,49,775
So, the annual payment the firm should do is Rs. 7,49,775.

51. Numerical

52. Numerical
So, firm wants to find out the annuity it should pay every year of which the present value will be
equal to 5,00,000.
5,00,000 = C * [ 1- 1/ (1+0.16) 5 / 0.16 ]
5,00,000 = C * [ 3.2737]
5,00,000 / 3.2737 = C
C = 1,52,732
So, the annual payment the firm should do is Rs. 1,52,732.

53. Numerical

54. Numerical
5,00,000 = C * [ ( 1+ 0.06) 10 - 1 / 0.06 ]
5,00,000 = C * [1.7908 -1 / 0.06 ]
5,00,000 = C * [ 13.18]
C = 5,00,000 / 13.18
C = 37,936
So, firm has to invest Rs. 37,936 every year so that at the end of 10 years total Rs. 5,00,000 is
available to pay for debentures.