1. Business Finance
Michael Dimond

2. The "Magic" Machine Consider the following scenario:
Note: This is not a trick question, merely a framework to help you think about the subject.
Consider the following scenario:
You have the opportunity to buy a machine which is guaranteed to produce $100 per month for the next five years. There are no operating costs and the device will vanish at the end of that time.
How much would you pay for this?
What factors influence your decision?

3. The Time Value of Money (TVM)

4. Compounding
Compounding is the growth of value resulting from some sort of return (such as interest payments) being added to the original amount.
If you put $100 in the bank and receive 10% annual interest
After 1 year: $100 x (1+10%) = $110
After 2 years: $110 x (1+10%) = $121
After 3 years: $121 x (1+10%) = $133.10
The three-year compounding could be rewritten like this:
After 3 years: $100 x (1+10%) x (1+10%) x (1+10%) = $133.10 or
$100 x (1+10%)3 = $133.10
The general formula for compounding:
PV x (1+i)n = FV
where PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate

5. Discounting
Discounting is the opposite of compounding. Instead of growing an amount by a specific rate, we are taking that expected growth out of a future total to find what the starting figure would be.
Since compounding multiplies by (1+i)n, discounting will do the opposite: divide by (1+i)n.
If you will need $ at the end of three years, and you can receive 10% annual interest, how much would you need to deposit today?
$ ÷ (1+10%)3 = $100
The general formula for discounting:
FV ÷ (1+i)n = PV
where PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate

6. Moving parts of compounding & discounting
There are four “moving parts” in a compounding or discounting computation:
PV (Present Value)
FV (Future Value)
n (Number of Periods)
i (Rate of Return per Period)
The general formula for compounding:
PV x (1+i)n = FV
The more periods something is compounded, the greater the future value is.
The general formula for discounting:
FV ÷ (1+i)n = PV
The more periods something is discounted, the smaller the present value is.

7. Time vs Return: Basic TVM
A dollar is worth more now than it will be at any time in the future. The concept is called the Time Value of Money (TVM).
What makes money lose value over time?
How long an investment takes to pay out will affect the price you would pay.
If you require a 12% annual return, how much would you pay for $100 to be given to you in…
1 year?
3 years?
10 years?
The further in the future a cash flow is, the less it is worth.

8. What if compounding happens more frequently?
APR means Annual Percentage Rate
For example: 12% APR means 12% interest rate for the year.
If interest compounds more frequently, divide that rate by the periods per year.
12 % APR compounded…
Annually 1 period/yr 12% ÷ = 12.00% interest/period
Quarterly 4 periods/yr 12% ÷ = 3.00% interest/period
Monthly 12 periods/yr 12% ÷ 12 = 1.00% interest/period
Daily 360 periods/yr 12% ÷ 360 = 0.03% interest/period
Why do financiers use 360 days instead of 365?
After 1 year, how much will $100 be at 12% APR…
compounded at the end of the year? $100 x (1.1200)1 = $112.00
compounded at the end of each quarter? $100 x (1.0300)4 = $112.55
compounded at the end of each month ? $100 x (1.0100)12 = $112.68
compounded at the end of each day ? $100 x (1.0003)360 = $112.75
Remember to watch out for rounding errors:
12/360 = …

9. Effective Annual Rate
The Effective Annual Rate (EAR) is the APR adjusted for the value of compounding.
EAR = (1+APR ÷ n)n - 1
12% APR compounded annually = (1.1200) = 12.00% EAR
12% APR compounded quarterly = (1.0300) = 12.55% EAR
12% APR compounded monthly = (1.0100) = 12.68% EAR
12% APR compounded daily = (1.0003) = 12.75% EAR
Sometimes this is called the APY (Annual Percent Yield)

10. Understanding TVM problems
Time Value of Money scenarios are examined with a timeline.
Each tick mark on the timeline represents the end of one period.
The first tick mark on the left is labeled 0 because zero periods have elapsed. It indicates the present, or the planned beginning of a project.
The last tick mark indicates the end of the last period being analyzed.
Payments and compounding happen at the end of each period.
Consider our basic compounding example: 10 1 2 3 4 5 6 7 8 9 1 2 3
-100
133.10
i = 10%
You could use this diagram to analyze the future value or the present value
PV = -100
i = 10%
n = 3
FV =

11. Understanding TVM problems
You could use this diagram to analyze the future value or the present value.
Notice the cash outflow (money you invested) is shown with a minus sign. Financial calculators require this to give you the correct answer. This is called the sign convention.
100 x (1+0.10)3 =
:. FV = 1 2 3
-100 ?
i = 10%
133.10
133.1 ÷ (1+0.10)3 = 100
:. PV = -100

12. Understanding TVM problems
What if compounding happens more frequently?
What is the future value of $200 invested for 3 years at 8% APR compounded quarterly?
Each tick mark on the timeline represents the end of one period. 3 years x 4 quarters per year = 12 quarters total.
Interest per quarter is the APR / # periods per year. 8% ÷ 4 quarters per year = (8/4)% per period [ie, 2% per period] 10 1 2 3 4 5 6 7 8 9 12 11
-200
253.65
PV = -200
i = (8 / 4) quarterly
n = (3 x 4) quarters
FV =

13. Moving parts of TVM
A TVM problem has one more “moving part” than a simple compounding or discounting problem.
PV (Present Value)
FV (Future Value)
n (Number of Periods)
i (Rate of Return per Period)
PMT (Payment)
There may be payments which happen between the beginning and end of the timeline.
Each payment is discounted separately.
The PV of the stream of cash flows is the sum of the individual PVs.

14. Discounting payments
If you require a 12% annual return, what would you pay for…
…$100 to be delivered in 1 year? ($ )
…$100 to be delivered in 2 years? ($ )
…$100 to be delivered in 3 years? ($ )
…all of the above (i.e. $100 to be paid at the end of each of the next three years)?
By adding together the present values, you find the value of all the cash flows in the stream.
i = 12% 1 2 3 ?
100
100
100
100 ÷ (1+0.12)1
100 ÷ (1+0.12)2
100 ÷ (1+0.12)3

15. Discounting a stream of cash flows
Remember the magic machine?
$100 per month for 5 years. What if you require a 12% annual return? 60 1 2 3 4 56 57 58 59 ?
100
i = 1% monthly (12% APR)
Each payment
has its own
present value. Adding up those PVs gives the total value of the stream of cash flows.
100 ÷ (1+0.01)57
100 ÷ (1+0.01)56
100 ÷ (1+0.01)59
100 ÷ (1+0.01)4
100 ÷ (1+0.01)58
100 ÷ (1+0.01)60
100 ÷ (1+0.01)3
100 ÷ (1+0.01)1
100 ÷ (1+0.01)2
99.01
98.03
97.06
96.10 .
57.28
56.71
56.15
55.60
55.04

16. Timelines & PMTs i and n are always in the same increment.
Monthly periods → monthly rate.
Annual periods → annual rate.
What happens to PV as n increases?
As n increases, PV becomes smaller
100 ÷ = ÷ = 55.04
Value = Sum of PVs
So if you demand a 12% rate of return, the value of the machine’s monthly payments is:
There is also an easier way to compute that value…

17. Ordinary Annuity: FV & PV
A stream of cash flows where all payments are equal is called an Annuity.
In an Ordinary Annuity, each payment happens at the end of the period.
Your financial calculator can solve these easily and quickly.
Find PV given n, i, and PMT
Find FV given n, i, and PMT
For the magic machine, the inputs would be:
PV = ? (This is what we’re solving for)
n = 60 (monthly payments)
i = 12/12 (12% ÷ 12 months)
PMT = 100 (per month)
FV = 0 (This has no value once the final payment is delivered)
Notice that these three items must always be in the same timeframe: monthly annually, daily… whatever is in the scenario

18. Comments about Annuity Due
An Annuity Due has payments which happen at the beginning of each period instead of the end.
Typically used in real estate…
Timelines…
Difference in value…
Calculator setting…
Ordinary Annuity
Annuity Due
i = 12%
i = 12% 1 2 3 1 2 3
100
100
100
100
100
100
100 ÷ (1+0.12)1
100 ÷ (1+0.12)1
100 ÷ (1+0.12)2
100 ÷ (1+0.12)2
100 ÷ (1+0.12)3

19. Ordinary Annuity with an additional payout
What happens if there is a stream of payments, and also a lump sum being paid at the end of the timeline?
Timeline…
Find PV given n, i, PMT & FV