978-1259285295 Chapter 4 Lecture Note

Unlock access to all the studying documents.

View Full Document

Chapter 04 – Future Value, Present Value, and Interest Rates

Chapter 4

Future Value, Present Value, and Interest Rates

Chapter Overview

This chapter continues the exploration of interest rates using the concept of present value and

future value. The concepts are applied to the valuation of bonds, and the relationship between

inflation and interest rates is also considered.

Learning Objectives: Establish and understanding of:

Time and the value of payments

Present value vs. future value

Nominal vs. real interest rates

Important Points of the Chapter

In today’s world, interest rates are of enormous importance to virtually everyone; they link the

present to the future, allowing us to compare payments made on different dates. They also tell us

the future reward for lending today, as well as the cost of borrowing now and repaying in the

future. But to make sound financial decisions we must learn how to calculate and compare

different rates on various financial instruments.

Application of Core Principles

Principle #1: Time. A dollar deposited in an interest-bearing account today will earn a return and

grow into a future value. Future value is the value on some future date of an investment made

today.

Principle #1: Time. The present value is the value today of a payment that is promised to be

made in the future. It is the amount that must be invested today in order to realize a specific

amount on a given future date.

Principle #1: Time. The sooner a payment is to be made the more it is worth, and the rate of

decline in present value is related to the same phenomenon that gives us the rule of 72.

Principle #1: Time. The nominal rate agreed upon by a borrower and lender must be based on

expected inflation over the term of the loan.

4-1

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

Teaching Tips/Student Stumbling Blocks

The biggest challenge in this chapter is the mathematics of calculating future value and

present value. The text does an excellent job in explaining the material, taking students

through the calculations step by step, but you will probably want to reinforce this with a

number of practice exercises.

Emphasize that if someone has funds now those funds can be invested and will earn

interest, growing into future value. Finding present value is, in essence, running that

process in reverse.

Discuss the material in the text with regard to credit card debt (Your Financial World: Pay

Off Your Credit Card Debt as Fast as You Can), and supplement the treatment in the text

with a consideration of the dollar amounts involved. How much does an impulse

purchase really cost someone?

Features in this Chapter

Your Financial World: How Long Does It Take to Double Your Investment?

The Rule of 72 can be used to find out how long it will take your money to double at any rate of

interest. Divide 72 by the rate of interest (as a whole number) and the answer is the number of

years. The Rule of 72 shows the power of compounding, and can be used for anything that is

growing at a constant rate.

Lessons from the Crisis: Risk Taking and the Search for Yield

Investors must understand the risks of what they buy. Many investors underestimate the risk

associated with particular investments. Several things may cause this underestimation. First,

investors may extrapolate from recent patterns that may not accurately reflect the long-term

trends of the risk of the investments. They may also underestimate default risk if using recent

history as their guide. Typically, the risk of default is lower during economic expansions.

Investors may also underestimate risk if their investment managers take risks that are not evident

or are purposely concealed.

Tools of the Trade: Computing Compound Annual Rates

Compounding means earning interest on interest. As a result of compounding we can’t just

multiply a monthly rate by 12 to get an annual rate. Instead we need to compute a 12-month

compounded rate by raising the one-month rate to the twelfth power. Another use for

compounding is the calculation of the average annual rate; if you know how much an investment

has grown over a number of years we can use this calculation to find out the percentage change

per year.

4-2

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

Your Financial World: Should You Buy the New Car Now or Wait?

This feature compares the decision to buy a car now as opposed to waiting a year, and given the

assumptions made in the calculations, shows that by waiting a year the consumer would have

more to spend. Should the consumer buy the car now or wait? The answer depends on how the

person feels about having the extra amount to spend and how much the old car may cost to repair

in the meantime.

Applying the Concept: Early Retirement

Early retirement is very expensive. If someone expects to live to be 85 and has $100,000 a year

in income, in order to retire at 40 he or she would need to accumulate about $2 million in assets.

As a rule of thumb, someone in his or her mid-20s should be putting away 10% of income

toward retirement in order to retire at the same pre-retirement standard of living by age 65. It

takes significant savings to live without a paycheck.

In the News: Pentagon Shows That It Doesn’t Always Pay to Take the Money and Run

People who were downsized from the Defense Department were offered either an annual

payment of $8,000 for 30 years or a lump sun of $50,000 today. Despite being provided with

ample explanatory materials most people chose the lump sum even though the annual payments

were more to their advantage. The reason is that most people put excessive weight on a bird in

the hand. The Treasury saved billions by offering the lump sum option. From this it can be

calculated that the average personal discount rate was about 25 percent. Recognition of this

helps to explain other phenomena, like why people hold high credit card debt.

Lessons of the Article: This article explains a common problem faced by people who

are retiring. Should they take a lump sum payment offered by their employer or

pension fund, or a series of annual payments? Answering this question requires using

present value. The article also describes how most people are extremely impatient,

behaving as if their own personal discount rate is extraordinarily high and how that

explains the willingness to borrow at very high interest rates.

Your Financial World: Pay Off Your Credit Card Debt as Fast as You Can

This feature illustrates the advantage of paying off debts as fast as possible. For example,

someone with $2000 in credit card debt at a rate of 15% paying $50 a month will need 54.3

months to finish paying off the $2000. Increasing the payment to $60 a month will reduce the

time of repayment to 42.5 months, one full year sooner. The lesson is that making large

payments on credit card debt is more important than getting low interest rates. Pay off your

debts as fast as you can; procrastination is expensive.

4-3

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

Applying the Concept: High Interest Rates, Low Interest Rates

Once we realize that nominal interest rates move with expected inflation rates, big swings in

interest rates become less of a mystery. It also makes it easier to understand why interest rates

can be very different in different countries.

Additional Teaching Tools

Kaja Whitehouse, writing for The Wall Street Journal (“Student Loan 101: When Looking to

Consolidate, Discounts are Key,” June 2, 2004), reports that interest rates on certain federal loans

(like the Stafford loans) will drop to historic lows in July, providing an opportunity for people to

consolidate their student loans into one, lower rate loan. The article points out that in addition to

deciding whether or not to consolidate, people must also look for the right consolidator, which

means finding the one that offers the best discounts. Under the Federal Consolidation Loan

Program you will always be quoted the same interest rate and zero fees but the competition is in

the discounts the lenders can offer. Discounts can drop interest rate or lower payments over

time; some lenders even offer cash back to borrowers who make regular payments on time for a

set period of time. A borrower can choose a cash rebate instead of a lower interest rate; as noted

in the article, “This sort of discount may save you less over time, but it could work out better for

people who are unsure of their ability to pay their future bills, or for those who need cash.” The

article also notes that several of the state guaranty agencies provide better-than-average

discounts.

Virtual Tools

Here is an online present value calculator that allows the user to change things like the future

value and the discount rate and see how the present value changes.

http://www.timevalue.com/tcalc.aspx

For more about the state guaranty agencies mentioned in the article above (in “Additional

Teaching Tools,” go to this page on the U.S. Department of Education’s web site.

www.ed.gov

For More Discussion

This chapter points out that paying off credit card debt as soon as possible is worthwhile. Do

people really know the interest rates on their credit cards? Have your students find out; it can be

a real eye-opening exercise!

Chapter Outline

I. Valuing Monetary Payments Now and in the Future

To compare the value of payments made on different dates we need a set of tools called

present value and future value.

4-4

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

A. Future Value and Compound Interest

1. Future value is the value on some future date of an investment made today.

2. To calculate future value we multiply the present value by the interest rate and add

that amount of interest to the present value. For example, $100 earning 5% interest

has a future value of $105 in one year.

a. We can also say that the $100 investment has a yielded a $5 return, which is why

an interest rate can be called a “yield.”

3. The higher the interest rate (or the amount invested) the higher the future value;

remember Core Principle 1: time has value.

4. When we consider investments with interest payments made for more than one year

we need to consider compound interest, or the fact that interest will be paid on

accumulated interest.

5. We can then calculate future value; we multiply the present value by one plus the

interest rate, raised to a power (exponent) that is equal to the number of years.

Remember, using an exponent is really shorthand for performing a series of

multiplications.

6. Fractions of percentage points are called basis points; a basis point is one

one-hundredth of a percentage point.

B. Present Value

How much is a payment promised in the future worth today? We can find out by

calculating present value or present discounted value.

1. The Definition

a. The present value is the value today of a payment that is promised to be made in

the future. It is the amount that must be invested today in order to realize a

specific amount on a given future date.

b. To calculate present value we invert the future value calculation; thus we divide

future value by one plus the interest rate (to find the present value of a payment to

be made one year from now).

c. For payments to be made more than one year from now we divide future value by

one plus the interest rate raised to the nth power where n is the number of years.

d. This tells us that present value is higher the higher the value of the future

payment, the shorter the time until payment, and the lower the interest rate.

2. How Present Value Changes

a. Doubling the future value of the payment without changing the time involved or

the interest rate, results in a doubling of the present value.

b. The sooner a payment is to be made the more it is worth and the rate of decline in

present value is related to the same phenomenon that gives us the rule of 72.

4-5

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

c. Higher interest rates are associated with lower present values no matter what the

size or timing of the payment.

d. At any fixed interest rate, an increase in the time until a payment is made reduces

its present value.

e. Not only does the present value of a future payment fall with the interest rate; the

further in the future the promised payment is to be made, the more the present

value falls.

II. Applying Present Value

To use present value in practice we need to look at a sequence or stream of payments whose

present values must be summed. Present value is additive. To see how this is applied we

will look at internal rate of return and the valuation of bonds.

A. Internal Rate of Return

1. The internal rate of return is the interest rate that equates the present value of an

investment with its cost.

2. It is the interest rate at which the present value of the revenue stream equals the cost

of the investment project. In the calculation we solve for the interest rate.

3. The internal rate of return must be compared to a rate of interest that represents the

cost of funds to make the investment. These funds could be obtained from retained

earnings or borrowing. In either case there is an interest cost.

4. An investment will be profitable if its internal rate of return exceeds the cost of

borrowing.

B. Bonds: The Basics

1. One of the most common uses of the concept of present value is in the valuation of

bonds.

2. A bond is a promise to make a series of payments on specific future dates; it is a legal

contract issued as part of an arrangement to borrow.

3. The most common type is a coupon bond, which makes annual payments called

coupon payments (the percentage rate is called the coupon rate).

4. The bond also specifies a maturity date and has a final payment, which is the

principal, face value, or par value of the bond.

5. The price of a bond is the present value of its payments.

6. To value a bond we need to value the repayment of principal and the payments of

interest.

a. Valuing the Principal Payment: a straightforward application of present value

where n represents the maturity of the bond.

b. Valuing the Coupon Payments: requires calculating the present value of the

payments and then adding them; remember, present value is additive.

c. Valuing the Coupon Payments plus Principal: means combining the above.

4-6

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

7. The value of the coupon bond rises when the yearly coupon payments rise and when

the interest rate falls.

8. Lower interest rates mean higher bond prices and vice versa. The value of a bond

varies inversely with the interest rate used to discount the promised payments.

III. Real and Nominal Interest Rates

1. So far we have been computing the present value using nominal interest rates, or

interest rates expressed in current-dollar terms.

2. But inflation affects the purchasing power of a dollar, so we need to consider the real

interest rate, which is the inflation-adjusted interest rate.

3. The Fisher equation tells us that the nominal interest rate is equal to the real interest

rate plus the expected rate of inflation.

4. The expected rate of inflation can be obtained from forecasts such as those done by the

Federal Reserve Bank of Philadelphia.

Appendix 4A: The Algebra of Present Value Formulas

This appendix shows the derivation of a formula for computing present value, which can be

used for any series of payments.

Using FRED: Codes for Data in This Chapter

Data Series FRED Data Code

One year Treasury bill rate‐TB1YR

Three month Treasury bill rate‐TB3MS

Consumer price index CPIAUCSL

Consumer inflation expectations (one year,

University of Michigan survey)

MICH

Brazil Treasury bill rate INTGSTBRM193N

Brazil consumer price index BRACPIALLMINMEI

China discount rate INTDSRCNM193N

China consumer price index CHNCPIALLMINMEI

10 year Treasury constant maturity rate‐GS10

10 year Treasury inflation indexed security‐ ‐ FII10

5 year Treasury constant maturity rate‐GS5

5 year Treasury inflation indexed security‐ ‐ FII5

Terms Introduced in Chapter 4

basis point

bond

compound interest

coupon bond

coupon payment

4-7

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

coupon rate

face value

fixed-payment loan

future value

internal rate of return

maturity date

nominal interest rate

par value

present value

principal

real interest rate

rule of 72

yield

Lessons of Chapter 4

1. The value of a payment depends on when it is made.

a. Future value is the present value of an initial investment times one plus the interest rate

for each year you hold it. The higher the interest rate, the higher the future value.

b. Present value is equal to the value today of a payment made on a future date.

i. The higher the payment, the higher the present value at a given interest rate.

ii. The higher the interest rate, the lower the present value of a given payment.

iii. The longer the time until the payment is made, the lower the present value of a given

payment at a given interest rate.

iv. For a given increase in the interest rate, the present value of a promised payment falls

more the farther into the future the payment is to be made.

v. In computing present value, the interest rate and the time until the payment is to be

made must be measured in the same time units.

2. Present value can be used to value any stream of future payments.

a. The internal rate of return is the interest rate that equates the present value of the future

payments or profits from an investment with its current cost.

b. A coupon bond is a promise to make periodic interest payments and a final principal

payment on specific future dates.

i. The present value of a bond depends on its coupon rate, date of maturity, and the

current interest rate.

ii. The higher the coupon rate, given the maturity and the interest rate, the higher the

present value of the bond.

iii. The price of a bond is inversely related to the interest rate. The higher the price, the

lower the interest rate that equates the price with the present value of the promised

payments.

3. The real interest rate is the nominal interest rate minus expected inflation. It expresses the

interest rate in terms of purchasing power rather than current dollars.

4-8

Education.

Chapter 04 – Future Value, Present Value, and Interest Rates

4-9

Education.