1. Time Value of Money by Binam Ghimire

2. Learning Objectives Concept of TVM
Represent the cash flows occurred in different time periods using cash flow time line
Calculate the present value and future value of given streams of cash flows with and without using table
Identify the impact of time period and required rate of return on present value and future value
Prepare amortised schedule for amortised term loan
Compare interest rates quoted over different time intervals
Understand the real and nominal cash flows

3. Source: Total Access http://www.totalaccess.co.uk/Case_Studies/big_ben

4. Warm Up: The School Maths
Adding a percent to an amount
A computer costs £ VAT 15%. What is the cost after VAT is added?
Reverse percentage
The price of a computer is £230 after VAT of 15%. What is the price before VAT is added?

5. Time value of Money: Concept and Significance
Theory behind TVM: “One pound today is worth more than one pound tomorrow”
Deals with values of cash flows occurred at different points in time
Every sum of money received now has a value for reinvestment

6. Time value of Money: Concept and Significance
Why is APR a legal requirement?
36.5% p.a.* buy today pay after 12 months
* Condition apply see page 115 for detail
On page 115: payable 0.1 % interest per day
Image source: google

7. Time value of Money: Concept and Significance
Having Cash NOW is valuable
Inflation
People’s preference to current consumption
TVM Usage: Firms, Households, Banks, Insurance Cos.
TVM in Corporate Finance: Economic Welfare of Shareholders
Investment Decisions
Financing Decisions

8. Cash Flow Time Line
Cash Flow Time Line is an important tool used to understand the timing of cash flows. It is a graphical presentation of cash flows occurring at different points of time, and is helpful for analysing the time value of cash flows

9. Cash Flow Time Line
The corresponding cash flows are placed below the scale as shown in the following time line of cash flows:
The time line of cash flow is also used to denote the interest rate that each cash flow earns

10. Future Value
Future Value (FV) is the sum of investment amount and interest earned on the investment
Interest may be earned simple and compound
Accordingly FV can be calculated using simple and compound interest rate methods

11. Future Value: Simple Interest
Interest earned only on the original investment is Simple Interest. For example, a savings account with a bank in which bank posts the interest every year. The interest for principle £P deposited at r % interest p.a. for t number of years will equal to I (where I = p x n x r ÷ 100). So the FV of an investment will be P + I
We aim to maximise the benefit from any monetary transaction. So in real world, interest is paid (savings) or charged (loan) more than once during the tenure of the savings or loan. As such the Simple Interest does not exist

12. Future Value : Compound Interest
Compound Interest is interest earned on principle plus interest from the previous period
FV of £ 100,000 investment for one year, rate of interest: 12% and when interest is 1) simple interest 2) compounded monthly and daily are shown in figure next

13. Future Value : Compound Interest Effect

14. Future Value : Compound Interest different intervals
Hence, with Compound interest, FV grows when the frequency of interest payment is higher. (i.e. total amount of interest is growing every next period).
Note that when interest is per annum and to be calculated for mth times within a year the formula changes to:
And if it is for more than one year, the formula is
Where, n=no. of years
Effective interest rate formula
£100 at 12% p.a. at the end = 112 simple
£100 at 12% p.a. half yearly =
£100 x 0.12/2 = = 106
£106 x 0.12/2 =
Devise a formula
£100 x (1.06)^2
£112.36
So the effective interest rate is 12.36%
Now let us make a formula for this
100 x (1+ APR) = 100 x (1+0.12/2)^2
APR = (1 + r/m)^m -1

15. Compound Interest different intervals
has offered 5 % interest charged half yearly
has offered 4.5% interest charged every 3 months
The name of the banks are used only to illustrate the APR and the numbers are imaginary. Logo source: website of respective banks.
APR = (1 + r/m)^m -1
You want to borrow money
Which bank will you prefer and why?

16. Future Value : Compound Interest Table
Below FV of £ 1 at the end of various numbers of years have been calculated for different rates of interest:
The table is known as FV Compound Interest table [(1+r)t]

17. Future Value : Graphical View
FV of a sum of money has positive relation with the time period and interest rate

18. Future Value: FV, Time Period and Interest rate
FV increases with rate of interest and time
The higher the interest rate higher will be the FV
More the number of years the more will be the FV
Higher the no. of frequency of payment (within the year) higher will be the FV

19. Future Value : Compound Interest in Excel
In Microsoft Excel, FV function wizard can be used to calculate FV
Check the relationship between FV and r and t by making line chart in Microsoft Excel for r=5, 10 & 15 percent for t= 0,2,4,6,8,10,12 & 14 years

20. Present Value
Present value (PV) is today’s value of a future cash flow
FV is only available in the future
Can we get the FV now? i.e. can we convert the FV into PV?
You may if you sacrifice 1) the interest that you could have earned had you invested the money and 2) adjust for the time period that you don’t want to wait. Hence,
The sacrifice is reflected in the formula above. When t and r get bigger PV gets smaller

21. Present Value: Discount Rate
You bought a television. The payment term is single payment of £ 2,000 to be made after 2 years. If you can earn 8% on your money how much money should you set aside today in order to make final payment when due in 2 years?
Note that to calculate PV, we discounted FV at the interest rate r. The calculation is therefore termed a discounted cash flow. The interest rate is known as Discount rate

22. Present Value: Discount Factor
The PV formula may be converted as follows
The expression 1/(1+r)t basically measures the PV of £1 received in year t. The expression is known as Discount Factor. Using the PV of £ 1, a table can be constructed for FV to be received after t years at different discount rates

23. Present Value : Discount Factor Table
Below PV of £ 1 at the end of various numbers of years have been calculated for different rates of interest:
The table is known as PV Discount Factor table [1/(1+r)t]

24. Present Value: Graphical View
The PV of a sum of money has inverse relation with the time period and interest rate

25. Present Value: PV, Time Period and Interest rate
PV increases when rate of interest falls. The lower the interest rate higher will be the PV
Similarly, PV increases when the time period decreases
In the example of TV purchase above, note that £1, invested for 2 years at 8% will prove just enough to buy your computer

26. Present Value: PV, Time Period and Interest rate
The longer the time period available for payment the less you need to invest today (i.e. lower PV value). For example, if the payment for TV was only required in the third year, the PV would be:
The relationship between PV and Interest Rate is also the same i.e. higher interest rate lower PV

27. Multiple Cash Flow: FV and PV for uneven Cash Flows
Most real word investments will involve many cash flows over time. This is also known as Stream of cash flows. Calculation of FV and PV of a stream of cash flows is more important and common in finance
To calculate the FV of a stream of uneven cash flows, we may calculate what each cash flow is worth at that future date, and then add up the values
To find the PV of a stream of uneven cash flows, we may calculate what each FV is worth today and then add up the values

28. Multiple Cash Flow: Future Value of Uneven Cash Flow
A security provides the following Cash Flows
If the interest rate is 10% the FV will be
The third column above is the factor from FV of £ 1 table

29. Multiple Cash Flow: Present Value of Uneven Cash Flow
The PV for the same Cash Flow above assuming same rate as discount rate will then be
The third column above is the factor from PV of £ 1 table

30. Multiple Cash Flow: FV and PV for even Cash Flows
In finance, it is more common to find a stream of EQUAL cash flows at same time intervals. For example, a home mortgage or a car loan might require to make equal monthly payments for the life of the loan. Any such sequence of equally spaced, level cash flows is called an Annuity

31. Multiple Cash Flow: Annuity and Annuity Due
Annuity may be Ordinary Annuity and Annuity Due
In case of an ordinary annuity, each equal payment is made at the end of each interval of time throughout the period
In case of Annuity Due, equal payments are made at the beginning of each interval throughout the period

32. Multiple Cash Flow: Annuity and Annuity Due
For example, if an individual promises to pay £ 1,000 at the end of each of three years for amortisation of a loan, then it is called an ordinary annuity. If it were the annuity due, each payment would be made at the beginning of each of the three years

33. Multiple Cash Flow: Future Value of an Annuity
Future Value for the Ordinary Annuity (FVA) will be

34. Multiple Cash Flow: Future Value of an Annuity
Formula,
where C is the Cash payment from annuity
If payment value is £ 1, it would be:
The table showing future value of £ 1 for various years at different interest rates is called FV Annuity table

35. Multiple Cash Flow: Future Value of an Annuity Due
Future value for the Annuity Due (FVAD) will be
Hence,

36. Multiple Cash Flow: Perpetuity, Delayed Annuity &Annuity

37. Multiple Cash Flow: Perpetuity
If the payment stream lasts forever, it is Perpetuity
Example: Consol
For a perpetual stream of £C payment every year the PV = C/r
So a perpetual stream of £1 payment every year the PV = 1/r

38. Multiple Cash Flow: Delayed Perpetuity
Delayed perpetuity is one in which the payment (C) only starts after some time t
PV of a delayed perpetuity for £ 1 payment starting after time t will be

39. Multiple Cash Flow: Present Value of an Annuity
Annuity is basically a difference between an Annuity (immediate) and a delayed perpetuity
PV of an Annuity (t year) will therefore be PV of immediate Perpetuity – PV of Delayed Perpetuity starting after t year
This is also called t-year annuity factor. The PV table constructed for various time t and interest rate r for payment £1 each year is called PV Annuity factor table

40. Multiple Cash Flow: PV of an Annuity and Annuity Due
For Payment £ C every year, PV of t year annuity is therefore Payment x annuity factor or
Or:

41. Multiple Cash Flow: PV of an Annuity and Annuity Due
For Payment £ C every year, PV of t year annuity is therefore Payment x annuity factor or
Present value of Annuity Due simply requires the use of the formula:
PV of ordinary annuity x (1+r)

42. Multiple Cash Flow: PV of an Annuity and Annuity Due

43. Multiple Cash Flow: Amortised Loans
Loan that is to be repaid in equal periodic installments including both principal and interest is known as Amortised Loan
Let us suppose a loan of £ 10,000 is to be repaid in four equal installments including principal and 10 percent interest per annum
The lender needs to set the payments so that a present value of £ 10,000 is received
Present Value = Mortgage Payment x t year annuity Factor

44. Multiple Cash Flow: Amortised Loans
Mortgage Payment =
The loan amortisation schedule is shown next

45. Multiple Cash Flow: Amortised Loan Schedule

46. Cash Flow and Inflation
Note that cash flows are affected by the level of inflation in the country
Generally, the rate of interest quoted in the market are nominal interest rate which does not take into consideration the effect of price changes. The real interest rate can be calculated using the formula:

47. Cash Flow and Inflation: Example
What is the real interest rate when you deposit £ 1,000 in a bank at interest rate of 5%. Suppose that the Inflation is 5% as well
What will it be if interest rate is 6% and inflation is only 2 %?
Discounting real payment by real interest rate and Nominal Payment by Nominal interest rate will always give the same answer. We must not mix up real and nominal

48. Thank You