1. Time Value of Money
P.V. Viswanath

2. Key Concepts
Be able to compute the future value of an investment made today
Be able to compute the present value of cash to be received at some future date
Be able to compute the return on an investment
P.V. Viswanath

3. Chapter Outline Future Value and Compounding
Present Value and Discounting
More on Present and Future Values
P.V. Viswanath

4. Present and Future Value
Present Value – earlier money on a time line
Future Value – later money on a time line
100
100
100
100
100
100 2 3 5 4 6 1
If a project yields $100 a year for 6 years, we may want to know the value of those flows as of year 1; then the year 1 value would be a present value.
If we want to know the value of those flows as of year 6, that year 6 value would be a future value.
If we wanted to know the value of the year 4 payment of $100 as of year 2, then we are thinking of the year 4 money as future value, and the year 2 dollars as present value.
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.
P.V. Viswanath

5. Rates and Prices
A rate is a “price” used to convert earlier money into later money, and vice-versa.
If $1 of today’s money is equal in value to $1.05 of next period’s money, then the conversion rate is 0.05 or 5%.
Equivalently, the price of today’s dollar in terms of next period money is The excess of next period’s monetary value over this period’s value (1.05 – 1.00 or 0.05) is often referred to, as interest.
The price of next period’s money in terms of today’s money would be 1/1.05 or cents.
P.V. Viswanath

6. Rate Terminology
Interest rate – “exchange rate” between earlier money and later money (normally the later money is certain).
Discount Rate – rate used to convert future value to present value.
Compounding rate – rate used to convert present value to future value.
Cost of capital – the rate at which the firm obtains funds for investment.
Opportunity cost of capital – the rate that the firm has to pay investors in order to obtain an additional $ of funds.
Required rate of return – the rate of return that investors demand for providing the firm with funds for investment.
P.V. Viswanath

7. Relation between rates
If capital markets are in equilibrium, the rate that the firm has to pay to obtain additional funds will be equal to the rate that investors will demand for providing those funds. This will be “the” market rate.
Hence this is the rate that should be used to convert future values to present values and vice-versa.
Hence this should be the discount rate used to convert future project (or security) cashflows into present values.
P.V. Viswanath

8. Discount Rates and Risk
In reality there is no single discount rate that can be used to evaluate all future cashflows.
The reason is that future cashflows differ not only in terms of when they occur, but also in terms of riskiness.
Hence, one needs to either convert future risky cashflows into certainty-equivalent cashflows, or, as is more commonly done, add a risk premium to the “certain-future-cashflows” discount rate to get the discount rate appropriate for risky-future-cashflows.
P.V. Viswanath

9. Future Values
Suppose you invest $1000 for one year at 5% per year. What is the future value in one year?
The compounding rate is given as 5%. Hence the value of current dollars in terms of future dollars is 1.05 future dollars per current dollar.
Hence the future value is 1000(1.05) = $1050.
Suppose you leave the money in for another year. How much will you have two years from now?
Now think of money next year as present value and the money in two years as future value. Hence the price of one-year-from-now money in terms of two-years-from-now money is 1.05.
Hence 1050 of one-year-from-now dollars in terms of two years-from-now dollars is 1050(1.05) = 1000 (1.05)(1.05) = 1000(1.05)2 =
Point out that we are just using algebra when deriving the FV formula. We have 1000(1) (.05) = 1000(1+.05)
P.V. Viswanath

10. Future Values: General Formula
FV = PV(1 + r)t
FV = future value
PV = present value
r = period interest rate, expressed as a decimal
T = number of periods
Future value interest factor = (1 + r)t
P.V. Viswanath

11. Effects of Compounding
Simple interest
Compound interest
The notion of compound interest is relevant when money is invested for more than one period.
After one period, the original amount increases by the amount of the interest paid for the use of the money over that period.
After two periods, the borrower has the use of both the original amount invested and the interest accrued for the first period. Hence interest is paid on both quantities.
P.V. Viswanath

12. Figure 4.1
P.V. Viswanath

13. Figure 4.2
P.V. Viswanath

14. Future Values – Example 2
Suppose you invest the $1000 from the previous example for 5 years. How much would you have?
FV = 1000(1.05)5 =
The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.)
It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number.
Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before.
Calculator: N = 5; I/Y = 5; PV = 1000; CPT FV =
P.V. Viswanath

15. Future Values – Example 3
Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?
FV = 10(1.055)200 = 447,189.84
What is the effect of compounding?
Without compounding the future value would have been the original $10 plus the accrued interest of 10(0.055)(200), or = $120.
Compounding caused the future value to be higher by an amount of $447,069.84!
Calculator: N = 200; I/Y = 5.5; PV = 10; CPT FV = -447,198.84
P.V. Viswanath

16. Future Value as a General Growth Formula
Suppose your company expects to increase unit sales of books by 15% per year for the next 5 years. If you currently sell 3 million books in one year, how many books do you expect to sell in 5 years?
FV = 3,000,000(1.15)5 = 6,034,072
Calculator: N = 5; I/Y = 15; PV = 3,000,000 CPT FV = -6,034,072
P.V. Viswanath

17. Present Values
How much do I have to invest today to have some amount in the future?
FV = PV(1 + r)t
Rearrange to solve for PV = FV / (1 + r)t
When we talk about discounting, we mean finding the present value of some future amount.
When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.
Point out that the PV interest factor = 1 / (1 + r)t
P.V. Viswanath

18. PV – One Period Example
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 =
The remaining examples will just use the calculator keys.
P.V. Viswanath

19. Present Values – Example 2
You want to begin saving for your 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
P.V. Viswanath

20. Present Values – Example 3
Your parents set up a trust fund for you 10 years ago that is now worth $19, If the fund earned 7% per year, how much did your parents invest?
PV = 19, / (1.07)10 = 10,000
The actual number computes to – This is a good place to remind the students to pay attention to what the question asked and be reasonable in their answers. A little common sense should tell them that the original amount was 10,000 and that the calculation doesn’t come out exactly because the future value is rounded to cents.
Calculator: N = 10; I/Y = 7; FV = 19,671.51; CPT PV = -10,000
P.V. Viswanath

21. 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 =
P.V. Viswanath

22. 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 =
P.V. Viswanath

23. Quick Quiz
What is the relationship between present value and future value?
Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today?
If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?
Relationship: The mathematical relationship is PV = FV / (1 + r)t. One of the important things for them to take away from this discussion is that the present value is always less than the future value when we have positive rates of interest.
N = 3; I/Y = 6; FV = 15,000; CPT PV = -12,594.29
PV = 15,000 / (1.06)3 = 15,000( ) = 12,594.29
N = 3; I/Y = 8; FV = 15,000; CPT PV = -11, (Difference = )
PV = 15,000 / (1.08)3 = 15,000( ) = 11,907.48
P.V. Viswanath

24. Figure 4.3
P.V. Viswanath

25. The Basic PV Equation - Refresher
PV = FV / (1 + r)t
There are four parts to this equation
PV, FV, r and t
If we know any three, we can solve for the fourth
FV = PV(1+r) t r = (FV/PV)-t – 1 t = ln(FV/PV)  ln(1+r)
P.V. Viswanath

26. Discount Rate – Example 1
You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest?
r = (1200 / 1000)1/5 – 1 = = 3.714%
It is very important at this point to make sure that the students have more than 2 decimal places visible on their calculator.
Efficient key strokes for formula: 1200 / 1000 = yx 5 1/x = - 1 =
If they receive an error when they try to use the financial keys, they probably forgot to enter one of the numbers as a negative.
P.V. Viswanath

27. Discount Rate – Example 2
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%
P.V. Viswanath

28. Discount Rate – Example 3
Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5000 to invest. What interest rate must you earn to have the $75,000 when you need it?
r = (75,000 / 5,000)1/17 – 1 = = 17.27%
Calculator: N = 17; FV = 75,000; PV = 5,000; CPT I/Y = 17.27%
This is a great problem to illustrate how TVM can help you set realistic financial goals and maybe adjust your expectations based on what you can currently afford to save.
P.V. Viswanath

29. Quick Quiz: Part 3
What are some situations where you might want to compute the implied interest rate?
Suppose you are offered the following investment choices:
You can invest $500 today and receive $600 in 5 years. The investment is considered low risk.
You can invest the $500 in a bank account paying 4%.
What is the implied interest rate for the first choice and which investment should you choose?
Implied rate: N = 5; PV = -500; FV = 600; CPT I/Y = 3.714%
r = (600 / 500)1/5 – 1 = 3.714%
Choose the bank account because it pays a higher rate of interest.
How would the decision be different if you were looking at borrowing $500 today and either repaying at 4% or repaying $600? In this case, you would choose to repay $600 because you would be paying a lower rate.
P.V. Viswanath

30. 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.
P.V. Viswanath

31. Number of Periods – Example 1
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 money to pay cash 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
P.V. Viswanath

32. Number of Periods – Example 2
Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?
P.V. Viswanath

33. Example 2 Continued How much do you need to have in the future?
Down payment = .1(150,000) = 15,000
Closing costs = .05(150,000 – 15,000) = 6,750
Total needed = 15, ,750 = 21,750
Using the formula
t = ln(21,750/15,000) / ln(1.075) = 5.14 years
Point out that the closing costs are only paid on the loan amount, not on the total amount paid for the house.
P.V. Viswanath

34. Basic Formulas Refresher
P.V. Viswanath

35. Quick Quiz: Part 4
When might you want to compute the number of periods?
Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?
Calculator: PV = -500; FV = 600; I/Y = 6; CPT N = 3.13 years
Formula: t = ln(600/500) / ln(1.06) = 3.13 years
P.V. Viswanath

36. The Frequency of Compounding
Banks frequently refer to interest rates on loans and deposits in the form of an annual percentage rate (APR) with a certain frequency of compounding.
For example, a loan might carry an APR of 18% per annum with monthly compounding.
This actually means that the loan will carry a monthly rate of interest of 18/12 = 1.5% per month.
The APR is also called the stated rate of interest in contrast to the Effective Rate of Interest, which is effectively the additional dollars the borrower will have to pay if the loan is repaid at the end of a year, over and above the initial principal, assuming no interim repayments.
P.V. Viswanath

37. The Frequency of Compounding
The frequency of compounding affects the future and present values of cash flows. The stated interest rate can deviate significantly from the true interest rate –
For instance, a 10% annual interest rate, if there is semiannual compounding, works out to-
Effective Interest Rate = = or 10.25%
The general formula is Effective Annualized Rate = (1+r/m)m – 1 where m is the frequency of compounding (# times per year), and r is the stated interest rate (or annualized percentage rate (APR) per year
The loan sharks of yesteryear were on to something when they charged daily rates… A 1% daily rate translates into an annual rate of 3678%.
P.V. Viswanath

38. The Frequency of Compounding
Rate t
Formula
Effective Annual Rate
Annual
10% 1 r
10.00%
Semi-Annual 2
(1+r/2)2-1
10.25%
Monthly 12
(1+r/12)12-1
10.47%
Daily
365
(1+r/365)365-1
10.52%
Continuous
er-1
P.V. Viswanath

39. Present Value of an Annuity
The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years]
Good place to link up to the financial calculator. The present value button does wonders…
P.V. Viswanath

40. Example: PV of an Annuity
The present value of an annuity of $1,000 at the end of each year for the next five years, assuming a discount rate of 10% is -
May help to separate the second term in the equation and state it as a present value factor.
In the example above, the present value factor is For those using present value tables, this makes the link to the present value equation.
What if the cash flows had been at the beginning of each year?
Present value = $ $ 1000 (PV(A, 10%, 4 years) = $ 3,791 (1.1)
P.V. Viswanath

41. Annuity, given Present Value
The reverse of this problem, is when the present value is known and the annuity is to be estimated - A(PV,r,n).
Reverses the process. Here, we look for an annuity given a present value (Examples: Figuring out the monthly payment on a mortgage)
P.V. Viswanath

42. Computing Monthly Payment on a Mortgage
Suppose you borrow $200,000 to buy a house on a 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%.
The monthly payments on this loan, with the payments occurring at the end of each month, can be calculated using this equation:
Monthly interest rate on loan = APR/12 = 0.08/12 =
Follow-up questions:
If you pay at the beginning of each month, rather than the end, how much could you shorten the mortgage period?
$ (PV(A,.67%,n)) = ; Trial and error yields n=350: Save 10 months of payments
P.V. Viswanath

43. Future Value of an Annuity
The future value of an end-of-the-period annuity can also be calculated as follows-
This is how much the value will be on the last day (when the last annuity comes due)
P.V. Viswanath

44. An Example
Thus, the future value of $1,000 at the end of each year for the next five years, at the end of the fifth year is (assuming a 10% discount rate) -
Again, worth asking how much the future value would be if the cash flows were at the beginning of each year.
P.V. Viswanath

45. Annuity, given Future Value
If you are given the future value and you are looking for an annuity, you can use the following formula:
Reverses the problem
Note, however, that the two formulas, Annuity, given Future Value and Present Value, given annuity can be derived from each other, quite easily. You may want to simply work with a single formula.
P.V. Viswanath

46. Application : Saving for College Tuition
Assume that you want to send your newborn child to a private college (when he gets to be 18 years old). The tuition costs are $16000/year now and that these costs are expected to rise 5% a year for the next 18 years. Assume that you can invest, after taxes, at 8%.
Expected tuition cost/year 18 years from now = 16000*(1.05)18 = $38,506
PV of four years of tuition costs at $38,506/year = $38,506 * PV(A ,8%,4 years) = $127,537
If you need to set aside a lump sum now, the amount you would need to set aside would be -
Amount one needs to set apart now = $127,357/(1.08)18 = $31,916
If set aside as an annuity each year, starting one year from now -
If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405
The assumptions of constant tuition each year for 4 years, and that the payments are made at the end of each year (no university will accept that payment plan) are purely for simplicity.
P.V. Viswanath

47. Valuing a Straight Bond
You are trying to value a straight bond with a fifteen year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%.
PV of cash flows on bond = * PV(A,8.5%,15 years) / = $
If interest rates rise to 10%,
PV of cash flows on bond = * PV(A,10%,15 years)+ 1000/ = $1,057.05
Percentage change in price = %
If interest rate fall to 7%,
PV of cash flows on bond = * PV(A,7%,15 years)+ 1000/ = $1,341.55
Percentage change in price = %
A straight bond is a combination of an annuity and a simple cash flow. The interest rate used should reflect the risk of the bond. If it is a government bond, it will be a default-free interest rate. If it is a corporate bond, it should be a rate that reflects the default risk.
We have used one interest rate to discount all of the cash flows. A more precise valuation may be obtained by discounting each cash flow at a rate that reflects when that cash flow comes in.. A 1-year rate for the 1-year cash flow, a 2-year rate for the 2-year cash flow…
P.V. Viswanath

48. III. Growing Annuity
A growing annuity is a cash flow growing at a constant rate for a specified period of time. If A is the current cash flow, and g is the expected growth rate, the time line for a growing annuity looks as follows –
This is usually a cash flow that cannot be found on a financial calculator but is very common with equity investments…
P.V. Viswanath

49. Present Value of a Growing Annuity
The present value of a growing annuity can be estimated in all cases, but one - where the growth rate is equal to the discount rate, using the following model:
In that specific case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect.
This formula will work for all cases, except when g=r. When g>r, both the numerator and denominator become negative allowing you to still compute the present value. When g=r, the present value= nA (the growth and discount rate effects cancel out)
P.V. Viswanath

50. The Value of a Gold Mine
Consider the example of a gold mine, where you have the rights to the mine for the next 20 years, over which period you plan to extract 5,000 ounces of gold every year. The price per ounce is $300 currently, but it is expected to increase 3% a year. The appropriate discount rate is 10%. The present value of the gold that will be extracted from this mine can be estimated as follows –
You could always compute the present value by taking each cash flow and discounting back to the present. You should get the same answer.
P.V. Viswanath

51. IV. Perpetuity
A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is-
The best way to think about a perpetuity is not as forever but as a very long time.. The present value of an annuity that last 50 or 60 years converges on this value…
P.V. Viswanath

52. Value of Consol Bond = $60 / .09 = $667
Valuing a Consol Bond
A consol bond is a bond that has no maturity and pays a fixed coupon. Assume that you have a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows -
Value of Consol Bond = $60 / .09 = $667
There are British and Canadian console bonds still in existence…
P.V. Viswanath

53. V. Growing Perpetuities
A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is -
where
CF1 is the expected cash flow next year,
g is the constant growth rate and
r is the discount rate.
This is a cash flow growing at a constant rate forever. Here, g<r. Since the growth rate is forever, it is constrained to be less than or equal to the growth rate of the economy. If we allow for that constraint, g will always be less than r.
P.V. Viswanath