0. Introduction to Valuation: The Time Value of Money

1. Lecture Outline Basic time value of money (TVM) relationship
Present value (PV), future value (FV)
Future and Present Values of Multiple Cash Flows
Valuing Level Cash Flows: Annuities and Perpetuities
Comparing Rates: The Effect of Compounding Periods
Loan Types and Loan Amortization

2. Time Line r% 0 1 2 3… t |-----|-----|-----|-----| Period
CF0 CF1 … CFt
Period: year, quarter, month, day
Cash Flow: +/-, at the end of each period
r = Interest rate
“exchange rate” between earlier money and later money
Other names: Discount rate, Cost of capital, Opportunity cost of capital, Required return
It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts.

3. Future Values
Suppose you invest $1000 for one year at 5% per year. What is the future value in one year?
Interest = 1000(.05) = 50
Value in one year (FV1) = principal + interest = = 1050
Suppose you leave the money in for another year. How much will you have two years from now?
Interest = 1050(.05) = 52.50
Value in two year (FV2) = principal + interest = =
Point out that we are just using algebra when deriving the FV formula. We have 1000(1) (.05) = 1000(1+.05)

4. Future Value (FV) Future Value FV = PV(1 + r)t
later value of a CF on a time line
Ending value of an investment
Lump sum (single CF) case
FV = PV(1 + r)t
FV = future value
PV = present value
r = period interest rate, expressed as a decimal
t = number of periods
E.g. FV1 = 1000 (1.05)1 = 1050
FV2 = 1000 (1.05)2 =

5. Effects of Compounding
Simple interest
Interest is not reinvested
E.g. FV with simple interest = = 1100
Simple interest = = 100
Compound interest
Interest is reinvested
FV with compound interest =
Total interest = – 1000 =
Compound interest = total interest – simple interest = – 100 = 2.50
It comes from interest earned on the first interest payment = .05(50) = 2.50

6. Relationships between FV, time and interest rate

7. Future Values – Example
In 1626 Peter Minuit bought Manhattan island from the local Indians for a load of cloth, beads, hatchets, and other items with a total worth of 60 Dutch guilders. In some account that translate to $24. Assuming an interest rate of 8%
FV = 24(1.08)379 = $111 trillion
Interest rate makes a big difference:
At 6%, the FV = $93.6 billion
The current exchange rates put the purchase price at $33.
Price in Dutch Guilder 60
Dutch guilder to Euro
Euro to U$
Price in U$ 60x /1/2122= 33.00

8. Present Value (PV) Present Value FVt = PV0*(1 + r)t
Earlier value of a CF on the time line
Beginning value of an investment
How much do I have to invest today to reach a certain amount in the future?
FVt = PV0*(1 + r)t
Rearrange -> PV0 = FVt / (1 + r)t
Discounting -> finding the PV of some future amount.
In finance, the “value” of something usually refers to the present value.

9. PV – One Period Example Check
Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?
PV = 10,000 / (1.07)1 =
Check
In one year, x (1.07)1 = 10000
We will have the $10,000 we need
The remaining examples will just use the calculator keys.

10. Present Values – Example
You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?
PV = 150,000 / (1.08)17 = 40,540.34
Key strokes: 1.08 yx 17 +/- = x 150,000 =
Calculator: N = 17; I/Y = 8; FV = 150,000; CPT PV = -40,540.34

11. PV – Important Relationship I
For a given interest rate – the longer the time period, the lower the present value
What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%
5 years: PV = 500 / (1.1)5 =
10 years: PV = 500 / (1.1)10 =
Calculator: 5 years: N = 5; I/Y = 10; FV = 500; CPT PV =
N = 10; I/Y = 10; FV = 500; CPT PV =

12. PV – Important Relationship II
For a given time period – the higher the interest rate, the smaller the present value
What is the present value of $500 received in 5 years if the interest rate is 10%? 15%?
Rate = 10%: PV = 500 / (1.1)5 =
Rate = 15%; PV = 500 / (1.15)5 =
Calculator: 10%: N = 5; I/Y = 10; FV = 500; CPT PV =
15%: N = 5; I/Y = 15; FV = 500; CPT PV =

13. Relationships between PV, time and interest rate

14. Discount Rate
Often we will want to know what the implied interest rate is in an investment
Rearrange the basic PV equation and solve for r
FV = PV(1 + r)t
r = (FV / PV)1/t – 1

15. Discount Rate – Example
Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?
r = (20,000 / 10,000)1/6 – 1 = = 12.25%
Calculator: N = 6; FV = 20,000; PV = 10,000; CPT I/Y = 12.25%

16. Finding the Number of Periods
Start with basic equation and solve for t (remember your logs)
FV = PV(1 + r)t
t = ln(FV / PV) / ln(1 + r)
Remind the students that ln is the natural logarithm and can be found on the calculator.
The rule of 72 is a quick way to estimate how long it will take to double your money. # years to double = 72 / r where r is a percent.

17. Number of Periods – Example
You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough cash to pay for the car?
t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
Calculator: I/Y = 10; FV = 20,000; PV = 15,000; CPT N = 3.02 years

18. Multiple Cash Flows: FV Example
Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8% in each year?
FV = 100(1.08) (1.08)2 = =
Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV =
Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV =
Total FV = =

19. FV Example Timeline 1 2 3 4 5 8%
100
300
349.92
136.05
485.97
FV = 100(1.08) (1.08)2 = =

20. Multiple Cash Flows – PV Example
You are considering an investment that will pay you $200 in one year, $400 in two years, $600 in three years and $800 in four years. If you want to earn 12% on your money, how much would you be willing to pay for this investment today?
Calculator:
N = 1; I/Y = 10; FV = 1000; CPT PV =
N = 2; I/Y = 10; FV = 2000; CPT PV =
N = 3; I/Y = 10; FV = 3000; CPT PV =

21. PV Example Timeline
12% 1 2 3 4
200
400
600
800
178.57
318.88
427.07
508.41

22. PV Example Calculations
Find the PV of each cash flow and add them
Year 1 CF: 200 / (1.12)1 =
Year 2 CF: 400 / (1.12)2 =
Year 3 CF: 600 / (1.12)3 =
Year 4 CF: 800 / (1.12)4 =
Total PV = =
The students can read the example in the book.
You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?
Point out that the question could also be phrased as “How much is this investment worth?”
Calculator:
Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV =
Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV =
Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV =
Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV =
Total PV = =
Remember the sign convention. The negative numbers imply that we would have to pay today to receive the cash flows in the future.

23. Decisions, Decisions
Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment?
Solution strategy:
Compute total PV of expected future CFs
Total PV = 40/(1.15) + 75/(1.15)2 = 91.49
Compare PV to cost of investment
Decision?
Year 1 2
CF 40 75
Rate 15% 15% PV
Total PV 91.49

24. Annuities, Perpetuities and Growing Perpetuities
Annuity – finite series of equal payments that occur at regular intervals
If the first payment occurs at the end of the period, it is called an ordinary annuity
If the first payment occurs at the beginning of the period, it is called an annuity due
Value (PV or FV) of annuity due = Value (PV or FV) of ordinary annuity x (1 + interest rate)
Perpetuity – infinite series of equal payments
Growing Perpetuity – infinite series of payments which grow at a constant rate

25. Annuities, Perpetuities and Growing Perpetuities – Basic Formulas
Perpetuity: PV0 = C1 / r
Growing Perpetuity: PV0 = C1 / (r – g)
Annuities:
C is cash flow each peiod
C is constant for annuities and Perpetuities
C1 is the cash flow in the first year for a growing perpetuity
r is the discount rate
g is the constant growth rate for a growing perpetuity
T is the term for the annuity. (Note: perpetuities and growing perpetuities are infinite.)

26. Time Line of An Ordinary Annuity1 2 3 4 5 8%
100
100
Point out that the investment horizon is t = 0 through 5.
Hence, PV (from the formula) results in a value for t=0.
FV (from the formula ) results in a value for t=5.
100
100
100

27. Time Line of An Annuity Due1 2 3 4 5 8%
100
100
100
100
100
Point out that the investment horizon is t = 0 through 5.
Since all cash flows occur one period earlier than an ordinary annuity, PV of annuity due = PV of annuity * (1 + r) and FV of annuity due = FV of annuity *(1+r)

28. Time Line of A Perpetuity1 2 3
Etc. 8%
100
100
Point out that the investment horizon is assumed to begin at t = 0.
Hence, PV (from the formula) results in a value for t=0.
In general, PV(t) = CF beginning in t+1 / r.
100
… forever
Cash flows for a perpetuity always begin one period after the PV
Why?

29. Annuity – Sweepstakes Example
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
Calculator:
30 N; 5 I/Y; 333, PMT; CPT PV = 5,124,150.29

30. Buying a House
You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?

31. Buying a House - Continued
Bank loan
Monthly income = 36,000 / 12 = 3,000
Maximum payment = .28(3,000) = 840
PV = 840[1 – 1/ ] / .005 = 140,105
Total Price
Closing costs = .04(140,105) = 5,604
Down payment = 20,000 – 5604 = 14,396
Total Price = 140, ,396 = 154,501
You might point out that you would probably not offer 154,501. The more likely scenario would be 154,500.
Calculator:
30*12 = 360 N
.5 I/Y
840 PMT
CPT PV = 140,105

32. Finding the Payment
Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8%/12 = % per month). If you take out a 4 year loan, what is your monthly payment?
Number of payments = 4 x 12 = 48
20,000 = C[1 – 1 / ] /
C =
Note if you do not round the monthly rate and actually use 8/12, then the payment will be
Calculator:
4(12) = 48 N; 20,000 PV; I/Y; CPT PMT =

33. Annuity Due
You are saving for a new house and you plan to put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?
FV = 10,000[(1.083 – 1) / .08](1.08)
= (1.08) = 35,061.12
Note that the procedure for changing the calculator to an annuity due is similar on other calculators.
Calculator
2nd BGN 2nd Set (you should see BGN in the display)
3 N
-10,000 PMT
8 I/Y
CPT FV = 35,061.12
2nd BGN 2nd Set (be sure to change it back to an ordinary annuity)
What if it were an ordinary annuity? FV = 32,464 (so receive an additional by starting to save today.)

34. Annuity Due Timeline
32,464
If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.
35,016.12

35. Effective Annual Rate (EAR) versus Annual Percentage Rate (APR)
EAR and APR differs when interest is computed more than once per year
EAR is the actual rate paid (or received) after accounting for compounding that occurs during the year
If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.
APR is the annual rate that is usually quoted
By definition APR = period rate times the number of periods per year
More often, we have the APR but need the periodic rate for calculation:
Period rate = APR / number of periods per year
Where m is the number of compounding periods per year
Using the calculator:
The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.
2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.

36. Things to Remember
You ALWAYS need to make sure that the interest rate and the time period match.
If you are looking at annual periods, you need an annual rate.
If you are looking at monthly periods, you need a monthly rate.

37. EAR - Formula APR is the quoted rate
m is the number times interest is computed per year
EAR is the effective annual rate
Continuous compounding (m is infinite)
EAR = eAPR - 1
Note: The notation in Benninga uses n in place of m. The usual convention for compounding frequency in most textbook is m to avoid confusion because most financial calculators, popular with many analysts, uses “N” to denote the length of an investment.

38. Computing EARs - Example
Suppose you can earn 1% per month on $1 invested today.
What is the APR? 1(12) = 12%
How much are you effectively earning?
FV = 1(1.01)12 =
Rate = ( – 1) / 1 = = 12.68%
Suppose if you put it in another account, you earn 3% per quarter.
What is the APR? 3(4) = 12%
FV = 1(1.03)4 =
Rate = ( – 1) / 1 = = 12.55%
Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.

39. Net Present Value (NPV)
NPV = Total PV of future CFs - PV costs
Example:
A project requires a current investment of $100 and yields future expected cash flows of $21, $34, $40, $33, and $17 in periods 1 through 5, respectively. For these expected cash flows, the appropriate discount rate is 8.0%. Should this project be accepted?
Total PV of future CFs = 21/(1.08) + 34/(1.08)2 + 40/(1.08)3 + 33/(1.08)4 + 17/(1.08)5 =
NPV = – 100 = 16.17

40. Internal Rate of Return
IRR: Internal Rate of Return
Defined as the discount rate that sets NPV to 0
NPV = PV of future CFs – PV Costs (CF0)
NPV = -CF0 + CF1/(1+r)1 + CF2/(1+r) CFt/(1+r)t
NPV = 0 = -CF0 + CF1/(1+IRR)1 + CF2/(1+IRR) CFt/(1+IRR)t
Can only be solved using an iterative method
NPV Profile
A graph showing the relationship between NPV and discount rate
Skip IRR in first session. Discuss in session 3 with sensitivity and breakeven analysis.

41. Spreadsheet Strategies
Use the following formulas for TVM calculations
FV(rate,nper,pmt,pv,type)
PV(rate,nper,pmt,fv,type)
RATE(nper,pmt,pv,fv,type)
NPER(rate,pmt,pv,fv,type)
PMT(rate,nper,pv,fv,type)
The last argument, “type”, specified whether it is an ordinary annuity (enter 0 or omit the argument) or an annuity due (enter 1).
These functions require you to explicitly define the sign (inflow +/outflow -) of each cash flow.
Point out the inflow/outflow assumptions.
When using RATE and NPER, at least one of the cash flows much be of opposite sign.
This assumption is similar to financial calculators.
Click on the tabs at the bottom of the worksheet to move between examples.

42. A Note on Discount Rate
The discount rate represents a fair exchange of present cash flows and future cash flows.
Risk and discount rate
This discount rate should reflect the risk of the cash flows. Higher risk cash flows commend a higher discount rate.
Nominal interest rate, real interest rate and inflation
Nominal rate is what is usually published and quoted
Fisher Effect:
(1 + Nominal) = (1 + real) x (1 + inflation rate)
The Real Rate is not usually published
Time constant and time varying interest rates
In many the previous examples, the discount rate remains the same each year.
If the discount rates differ from year to year, the CFs must be treated as multiple cash flows and each CF evaluated individually.
The Excel Functions for PV, FV, and NPV cannot be used when the discount rate is not constant.

43. Summary of Basic TVM Formulas

44. Summary of Formulas for Level Cash Flows