1. Future Value, Present Value, and Interest Rates Chapter 4
Chapter Four
Future Value, Present Value,
and Interest Rates
Chapter 4

2. Learning Objectives Develop an understanding of
Time and the value of payments
Present value versus future value
Nominal versus real interest rates
Interest rates link the present to the future.
Tell the future reward for lending today.
Tell the cost of borrowing now and repaying later.

3. Valuing Monetary Payments Now and in the Future
We need a set of tools:
Future value
Present value

4. Future Value and Compound Interest
Future value: the value on some future date of an investment made today.
$100 invested today at 5% interest gives $105 in a year.
So the future value of $100 today at 5% interest is $105 one year from now.
The $100 investment “yields” $5, which is why interest rates are called yields.
This is an example of a simple loan of $100 for a year at 5% interest.

5. Future Value and Compound Interest
Formula –
If the present value (PV) is $100 and the interest rate (i) is 5%, then the future value (FV) one year from now is:
$105 = [$100 + $100(0.05)] = $100( )= $100(1.05)
The higher the interest rate, the higher the future value.
In general: FV = PV + PV x i = PV(1 + i)

6. Future Value and Compound Interest
Most financial instruments are not this simple.
We must consider compound interest to compute the value repaid more than one year from now.
Compound interest is the interest on the interest.

7. Future Value and Compound Interest
Suppose you deposit $100 (PV) in a bank saving account for two years at 5%(i) yearly interest rate?
The future value is:
FV = $100 + $100(0.05) + $100(0.05) + $5(0.05) = $110.25
FV = $100(1.05)(1.05) = $100(1.05)2
In general: FVn = PV(1 + i)n, where n is time.

8. Future Value and Compound Interest
Computing the future value of $100 at 5% annual interest

9. Future Value and Compound Interest
i and n must be in the same units.
If n is annual, i must be annual
If the annual interest rate is 5%, what is the monthly rate?
Assume im is the one-month interest rate and n is the number of months, then a deposit made for one year will have a future value of
PV(1 + im)12. (NOTE: i and n are monthly)

10. Future Value and Compound Interest
We know that in one year the future value is $100(1.05) so we can solve for im:
(1 + im)12 = (1.05)
(1 + im) = (1.05)1/12 = , which is 0.407%
Fractions of percentage points are called basis points.
A basis point is one one-hundredth of a percentage point
1 percent is 100 basis points
0.407% is 40.7 basis points

11. Present Value
Present value is the value today (in the present) of a payment that is promised to be made in the future.
Or, present value is the amount that must be invested today in order to realize a specific amount on a given future date.

12. Present Value Solve the Future Value Formula for PV:
FV = PV x (1+i), so
This is just the future value calculation inverted.

13. Present Value
We can generalize the process as we did for future value. Present Value of payment received n years in the future:
Present value is higher:
The higher future value of the payment, FVn
The shorter time period until payment, n.
The lower the interest rate, i.
Present value is the single most important relationship in our study of financial instruments.

14. Present Value of $100 Payment
Higher interest rates are associated with lower present values, no matter what the size or timing of the payment.
At any fixed interest rate, an increase in the time reduces its present value.

15. Compound annual rate = 6.17%
Computing Compound Annual Returns
We can turn a monthly growth rate into a compound-annual rate.
Investment grows 0.5% per month
What is the compound annual rate? (1.005)12 =
Compound annual rate = 6.17%
(Note: 6.17% > 12 x 0.05 = 6.0%)
Monthly Rate =
0.005
Com Annual Rate
Simple
PV =
100
Interest 1
100.50
100.5 2
101.00 3
101.51
101.50 4
102.02
102.00 5
102.53
102.50 6
103.04
103.00 7
103.55
103.50 8
104.07
104.00 9
104.59
104.50 10
105.11
105.00 11
105.64
105.50 12
106.17
106.00

16. Computing Compound Annual Rates
We can also use this to compute the percentage change per year when we know how much an investment has grown over a number of years.
Suppose an investment has increased 20 percent over five years: from $100 to $120.
FVn = PV(1 + i)n
120 = 100(1 + i)5
Solve for i
i =

17. Computing Compound Annual Rates
i = => i =3.71%

18. Internal Rate of Return
Imagine that you own a firm and you are considering purchasing a new machine.
Machine costs $1 million and can produce 4000 units of product per year.
You sell the product for $30 per unit, generating $120,000 in added revenue per year.
Keeping this simple, assume the machine is the only input and you have certainty about the revenue (very simple), no maintenance (very, very simple) and a 10 year lifespan.

19. Internal Rate of Return
Question: if you borrow $1 million to buy the machine, is the 10 year revenue stream enough to make the payments?
We need to compare internal rate of return (IRR) to the cost of buying the machine.
IRR is the interest rate that equates the present value of an investment with its cost.

20. Internal Rate of Return
Balance the cost of the machine against the PV of the future stream of revenue.
$1 million today versus $120,000 a year for ten years.
Solve for i - the internal rate of return.

21. Internal Rate of Return: Example
Solving for i, i = or 3.46%
So long as the interest rate at which you borrow money is less than 3.46%, then you should buy the machine
Or, if IRR is greater than opportunity cost, you should buy the machine.

22. Internal Rate of Return: Example
Suppose you are let go from your job and your employer offers you two options:
An annual payment of $8000 per year for 30 years or
A lump sum of $50,000 today.
Which do you take?
In the 1990s, when the Defense Department downsized, they offered many personnel a similar deal.
NOTE – a fixed payment for a fixed number of years is called an annuity

23. Internal Rate of Return: Example
Economic theory suggests that a person must compare their discount rate with the current rate of borrowing or lending.
The military pamphlet gave the present value of the annuity using a 7% discount rate, the interest on money markets at the time. The PV= $99,272.

24. Internal Rate of Return: Example
About 3/4 of military personnel took the lump sum, which was ½ the PV of the annuity.
Studying other separation packages, the study finds that people’s discount rate varies from 17 to 20%.
The discount rate that equates PV of the $8,000 annual annuity to $50,000 is 15.8%.

25. Bond Basics
A bond is a promise to make a series of payments on specific future dates.
Bonds create obligations, and are therefore legal contracts that:
Require the borrower to make payments to the lender, and
Specify what happens if the borrower fails to do so.

26. Bond Basics The most common type of bond is a coupon bond.
Issuer is required to make annual payments, called coupon payments (C).
The stated annual interest the borrower pays is called the coupon rate (ic).
The date on which the payments stop and the loan is repaid (n), is the maturity date or term to maturity.
The final payment is called the principal, face value, or par value of the bond.

27. Coupon Bond: the good-ole days
Called a coupon bond as buyer would receive a certificate with a number of dated coupons attached.
Principal
Coupons

28. Valuing the Principal
Assume a bond has a principal (FV) payment of $1000 and its maturity date is n years in the future.
The present value of the bond principal is:

29. Valuing the Coupon Payments
The present value expression gives the formula for the string of yearly coupon payments made over n years.
The longer the payments go, the higher their total value.
The higher the interest rate, the lower the present value.

30. Valuing the Coupon Payments plus Principal
We combine the previous two equations to get the price of a coupon bond:
The value of the coupon bond, PCB, rises when
The yearly coupon payments, C, rise and
The interest rate, i, falls.

31. Real and Nominal Interest Rates
Nominal Interest Rates (i)
The interest rate expressed in current-dollar terms.
Real Interest Rates (r)
The inflation adjusted interest rate
Borrowers care about the resources required to repay.
Lenders care about the purchasing power of the payments they received.
Neither cares solely about the number of dollars, they care about what the dollars buy.

32. Real and Nominal Interest Rates
The nominal interest rate you agree on (i) must be based on expected inflation (e) over the term of the loan plus the real interest rate you agree on (r).
i = r + e
This is the Fisher Equation.
The higher expected inflation, the higher the nominal interest rate.
This equation is an approximation that works well when expected inflation and the real interest rate are low.
Exact formula: (1 + i) = (1 + r)(1 + πe)
(1 + i) = 1 + r + πe +r πe
Subtract 1 from each side and ignore the cross-term.

33. Inflation and Nominal Interest Rates
Mankiw

34. Inflation and Nominal Interest rates

35. Nominal Interest Rate, Inflation Rate and Real Interest Rate

36. Real and Nominal Interest Rates
Financial markets quote nominal interest rates.
When people use the term interest rate, they are referring to the nominal rate.
The real interest rate is estimated using the Fisher equation:
r = i - e

37. Real and Nominal Interest Rates
Ex ante real interest rate is adjusted for expected changes in the price level (πe)
Ex post real interest rate is adjusted for actual changes in the price level (π)
Fisher Equation:
i = r + πe
From this we get -
rex ante = i - πe
rex post = i - π

38. Real and Nominal Interest Rates
Real Interest Rate - interest rate that is adjusted for expected changes in the price level
r = i –πe
if i = 5% and πe = 3%; r = 5% - 3% = 2%
if i = 8% and πe = 10%; r = 8% - 10% = -2%

39. A measure in inflationary expectations
i = r + πe
πe = i - r

40. Algebra - Annuity
To compute the payment, we will use the present-value formula. If we call the size of the monthly payments C, then we need to solve the following formula: