1.
MBA – 401:5
Accounting for Decision Makers (MBA 401) Lecture Title No. 5 Time Value of Money
Document map:
No. MBA401/5/PPT
Version: #1
Status: Draft
Owner: Extralearn

2. Understanding Time Value?
Understanding Time Value?
We say that money has a time value because it can be invested with the expectation of earning a positive rate of return
In other words, “a £ received today is worth more than a £ to be received tomorrow”
That is because today’s £ can be invested so that we have more than one £ tomorrow

3. Terminologies of Time Value
Terminologies of Time Value
Present Value - An amount of money today, or the current value of a future amount
Future Value - An amount of money on a future time
Period - A length of time or duration (year, month, week, day, or even hour)
Interest Rate - The compensation paid to a lender (or investor) for the use of funds expressed as a percentage for a period (normally stated as an annual rate)

4. Abbreviations PV: Present value FV: Future value
Abbreviations
PV: Present value
FV: Future value
Pmt: Per period payment amount
N: Either the total number of cash or The number of a specific period
i: The interest rate per period

5.
Timelines
A timeline is a graphical device used to clarify the timing of the cash flows for an investment
Each tick represents one time period PV FV 1 2 3 4 5
Today

6. Calculating the Future Value
Calculating the Future Value
Suppose that you have an extra £100 today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year?
-100 ? 1 2 3 4 5

7. Calculating the Future Value
Calculating the Future Value
Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2?
-110 ? 1 2 3 4 5

8. Generalising Future Value
Generalising Future Value
Recognizing the pattern that is developing, we can generalize the future value calculations as follows:
If you extended the investment for a third year, you would have:

9.
Compound Interest
Note from the example that the future value is increasing at an increasing rate
In other words, the amount of interest earned each year is increasing
Year 1: £ 10
Year 2: £ 11
Year 3: £ 12.10
The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount

10. Compound Interest - Graph
Compound Interest - Graph

11.
Magic Compounding
On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for £24 worth of beads and other trinkets. Was this a good deal for the Indians?
This happened about 371 years ago, so if they could earn 5% per year, then in 1997 they would have:
If they could have earned 10% per year, in 1997, they would have:
That’s about £ 54,563 Trillion

12. Present Value Calculations
Present Value Calculations
So far, we have seen how to calculate the future value of an investment
But we can turn this around to find the amount that needs to be invested to achieve some desired future value:

13. Present Value - Example
Present Value - Example
Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about £100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?

14.
Annuities
An annuity is a series of nominally equal payments equally spaced in time
Annuities are very common:
Rent
Mortgage payments
Car payment
Pension income
The timeline shows an example of a 5-year, £100 annuity
100
100
100
100
100 1 2 3 4 5

15. Value Additivity How do we find the value (PV or FV) of an annuity?
Value Additivity
How do we find the value (PV or FV) of an annuity?
First, you must understand the principle of value additivity:
The value of any stream of cash flows is equal to the sum of the values of the components
In other words, if we can move the cash flows to the same time period we can simply add them all together to get the total value

16.
PV of Annuity
We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components:

17.
PV of Annuity
Using the example, and assuming a discount rate of 10% per year, we find that the present value is: 1 2 3 4 5
100
62.09
68.30
75.13
82.64
90.91
379.08

18.
PV of Annuity
Actually, there is no need to take the present value of each cash flow separately
We can use a closed-form of the PVA equation instead:

19.
PV of Annuity
We can use this equation to find the present value of our example annuity as follows:
This equation works for all regular annuities, regardless of the number of payments

20.
FV of Annuity
We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components:

21.
FV of Annuity
Using the example, and assuming a discount rate of 10% per year, we find that the future value is: }
146.41
133.10
121.00 =
at year 5
110.00
100
100
100
100
100 1 2 3 4 5

22.
FV of Annuity
Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation:
This equation works for all regular annuities, regardless of the number of payments

23.
FV of Annuity
We can use this equation to find the future value of the example annuity:

24.
Annuities Due
Thus far, the annuities that we have looked at begin their payments at the end of period 1; these are referred to as regular annuities
A annuity due is the same as a regular annuity, except that its cash flows occur at the beginning of the period rather than at the end
5-period Annuity Due
100
100
100
100
100
5-period Regular Annuity
100
100
100
100
100 1 2 3 4 5

25.
PV of Annuity Due
We can find the present value of an annuity due in the same way as we did for a regular annuity, with one exception
Note from the timeline that, if we ignore the first cash flow, the annuity due looks just like a four-period regular annuity
Therefore, we can value an annuity due with:

26.
PV of Annuity Due
Therefore, the present value of our example annuity due is:
Note that this is higher than the PV of the, otherwise equivalent, regular annuity

27.
FV of Annuity Due
To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward:
Pmt
Pmt
Pmt
Pmt
Pmt 1 2 3 4 5

28. FV of Annuity Due The future value of our example annuity is:
FV of Annuity Due
The future value of our example annuity is:
Note that this is higher than the future value of the, otherwise equivalent, regular annuity

29.
Deferred Annuities
A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period
The timeline shows a five-period deferred annuity
100
100
100
100
100 1 2 3 4 5 6 7

30.
PV of Deferred Annuity
We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required
Before we can do this however, there is an important rule to understand:
When using the PVA equation, the resulting PV is always one period before the first payment occurs

31.
PV of Deferred Annuity
To find the PV of a deferred annuity, we first find use the PVA equation, and then discount that result back to period 0
Here we are using a 10% discount rate
PV2 =
PV0 =
100
100
100
100
100 1 2 3 4 5 6 7

32.
PV of Deferred Annuity
Step 1:
Step 2:

33.
FV of Deferred Annuity
The future value of a deferred annuity is calculated in exactly the same way as any other annuity
There are no extra steps at all

34.
Un-even Cash flows
Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity
We can find the present or future value of such a stream by using the principle of value additivity

35. An Example Un-even Cash flows
Un-even Cash flows
An Example
Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?
100
200
300 1 2 3 4 5

36. Another Example Un-even Cash flows
Un-even Cash flows
Another Example
Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?
300
500
700 1 2 3 4 5

37. Non-annual Compounding
Non-annual Compounding
So far we have assumed that the time period is equal to a year
However, there is no reason that a time period can’t be any other length of time
We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time
The only change that must be made is to make sure that the rate of interest is adjusted to the period length

38. Non-annual Compounding
Non-annual Compounding
Suppose that you have £1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds?

39. Non-annual Compounding
Non-annual Compounding
To solve this problem, you need to determine which bank will pay you the most interest
In other words, at which bank will you have the highest future value?
To find out, let’s change our basic FV equation slightly:
In this version of the equation ‘m’ is the number of
compounding periods per year

40. Non-annual Compounding
Non-annual Compounding
We can find the FV for each bank as follows:
First National Bank:
Second National Bank:
Third National Bank:
Obviously, one should choose the Third National Bank

41. Continuous Compounding
Continuous Compounding
There is no reason why we need to stop increasing the compounding frequency at daily
We could compound every hour, minute, or second
We can also compound every instant (i.e., continuously):
Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant equal to about 2.718

42. Continuous Compounding
Continuous Compounding
Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. What is the future value of your £1,000 investment?
This is even better than daily compounding
The basic rule of compounding is: The more frequently interest is compounded, the higher the future value

43. Continuous Compounding
Continuous Compounding
Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. If you plan to leave the money in the account for 5 years, what is the future value of your £1,000 investment?