# Lesson 4: Interest Rates and Rates of Return

Upon completion of this lesson, you should be able to do the following:

• Explain how to use the present value formula in calculating various types of loans.
• Describe the relationship among the market price of securities, market interest rates, and term to maturity.
• Explain the difference between a market interest rate and rate of return.
• Describe the Fisher Equation.

## Introduction

This lesson introduces the interest rates and rates of return. Interest rates are the most closely watched variables in the economy. Interest rates determine whether to consume or save, buy a house, or purchase bonds. Interest rates also affect business decisions to invest in new equipment or invest their money into financial securities. From Lesson 3, you learned the major financial instruments. All these instruments are credit market instruments. All these instruments are loans, where one party lends funds to another party. The only exception is corporate stock. Stock conveys ownership in a corporation and is not a loan. Each credit instrument has different maturities and issued by different companies and governments. Therefore, each credit instrument has an interest rate associated with it. There are hundreds of financial instruments, so there are hundreds of interest rates. The good news is all interest rates tend to move together. If one interest rate increases, the other interest rates increase too. Credit market instruments can be classified into 4 categories.

## Loans

• Simple loan - is commonly used, when banks grant commercial loans to businesses. The loan provides a borrower an amount of funds, which must be paid back at the maturity date plus interest. For example, a bank loans you \$100. (This is called the principle). The maturity is one year. A year later, you pay the bank back the \$100, plus \$10 for interest. The interest rate is 10% and is calculated as follows.

 The Simple Loan What if you paid back the loan for \$100 in 3 years at 10% interest rate? The calculation is below: Year 1: \$100 + \$100 * 0.10 = \$110.00 Year 2: \$110 + \$110 * 0.10 = \$121.00 Year 3: \$121 + \$121 * 0.10 = \$133.10 At the end of the 3rd year you pay the bank \$133.10. One hundred dollars for the principle and \$33.10 for the interest.

i = Interest / Principle = \$10 / \$100 = 0.10

• Discount bond -looking at the example to the right, the financial instrument is a T-bill for \$10,000. There is no interest rate listed on this instrument. Instead, the T-bill is sold at a discount. For example, the federal government could sell this T-bill to you for \$9,500. On August 10, 2002, the Federal government will give you \$10,000 for this instrument. The \$500 difference is the interest on this loan. The most common type of discount bonds are U.S. savings bonds and T-bills. The interest rate is 5.3% if it is a one year loan. The calculation is below:

 Treasury Bill U.S. Government \$10,000 August 10, 2002

i = Interest / Principle = \$500 / \$10,000 = 0.050

• Coupon bond - looking at the example to the right, the financial instrument is a T-note for \$20,000. The \$20,000 is called the face value. The T-note pays 10% interest. At the bottom of the T-note, there are coupons. Every six months, the person who has this instrument will clip off one coupon and send it to the U.S. federal government. The government will then mail a check for the interest payment. When the T-note matures on August 10, 2010, the owner gets \$20,000 and the last interest payment. These loans are very common for corporations and the U.S. federal government. Corporate bonds, T-notes and T-bonds fall into this category. The interest paid every six months is \$1,000. The calculation is below:

Treasury Note
U.S. Government

\$20,000

10%

August 10, 2010

interest = (interest rate per year / number of payments per year) * Principal

interest = ( 0.1 / 2) * \$20,000 = \$1,000

• Fixed-payment loan the borrowers make regular payments, usually every month. Student loans, home mortgages, and car loans fall into this category. Each month, the amount of the payment is the same and the payment covers both the principle and interest.

## Present Value Formula

If you are interested in buying these financial instruments, which one would you buy? There is a formula called the present value that allows the easy comparison of returns for all four loan types. For example, you will put \$1,000 into a savings account at a bank, earning 5% interest each year. The i is the annual interest rate and the formula is:

Present Value (PV) + i * PV = Future Value (FV)

The present value is the \$1,000 and after one year, it earns \$50 ( = \$1,000 * 0.05). You will have \$1,050 in your account.. That is the future value, FV1, after one year. If you do not touch this money, then after year two, you will earn \$52.50 in interest and the future value of your account will be \$1,202.50. This will be FV2. What if this money sat in an account for 20 years? The amount of interest earned is listed below:

Year 1: FV1 = PV ( 1 + i )
Year 2: FV2 = FV1 ( 1 + i )

Year 20: FV20 = FV19 ( 1 + i )

Using the equation for Year 20, substitute FV19 into it.

FV20 = FV18 ( 1 + i )( 1 + i) = FV18 ( 1 + i )2

Now substitute FV18 into this equation. You keep doing this process until you derive the following equation below.

FV20 = PV ( 1 + i )20 = \$1,000 ( 1 + 0.05)20 = \$2,653.30

At the end of the 20th year, the amount will you have in your savings account is \$2,653.30. You have calculated the future value of your savings account based on a fixed interest rate. The present value formula is the exact opposite. If 20 years from now, you will receive \$2,653.30. How much is this money worth to you today? The answer is \$1,000, because you could put this money into a savings account for 20 years at 5% interest and the money will grow into \$2,653.30. You are using the same formula, but now you are solving for the present value, PV.

PV = FV20 / ( 1 + i )20 = \$2,653.30 / ( 1 + 0.05 )20 = \$1,000

The conclusion is a dollar in the future is not as valuable to you as a dollar today, because you can earn interest on this dollar. Let's apply the present value formula to the different loan types. The first loan is a simple loan. The bank lends you \$1,000 at 5% for 20 years. At the end of the 20th year, you will pay the bank \$2,653.30. The principal is \$1,000 and the interest is \$1,653.30. You are using the same formula.

FV20 = PV ( 1 + i )20 = \$1,000 ( 1 + 0.05)20 = \$2,653.30

The discount loan is very similar to a simple loan. The only difference is you are solving for the interest rate. For example, you bought a \$20,000 T-bill for \$15,000 and the maturity is three years from now. You need to calculate the interest rate.

 FV3 = PV ( 1 + i )3 \$20,000 = \$15,000 (1 + i)3 \$20,000 / \$15,000 = (1 + i)3 i = ( \$20,000 / \$15,000 )1/3 -1 = 0.1006

However, this interest rate is for the whole three-year period. The interest rate is expressed per year, so you would have to divide the interest rate by 3. The annual interest rate 3.35%.

The coupon bond is a little different than the discount loan, because the bond pays many payments. For example, you bought a \$1,000 bond with a 10% coupon interest rate. The bond matures in 5 years. You will receive one interest payment each year for a total of 5 payments. How much is this bond worth to you today?

First, you need to calculate the yearly interest payment. Take the face value of the bond and multiply it with the interest rate. Each year, you will receive \$100 for 5 years (\$1,000 * 0.10). At the end of 5 years, you will receive the face value of the bond, the \$1,000. You place all this information into the present value formula. The present value of this bond is worth \$1,000 to you today.

Interest PV = \$100 / 1.10 + \$100 / 1.102 + \$100 / 1.103 + \$100 / 1.104 + \$100 / 1.105 = \$379.08

Face value PV =\$1,000 / 1.105 = \$620.92

Pbond = Face value + interest = \$397.08 + 620.92 = \$1,000

The last loan is a fixed payment loan and it is very similar to the coupon bond. For example, you received a \$50,000 mortgage from the bank and the mortgage is 20 years. The bank is charging you a 12% interest rate for this mortgage and you make one payment each year. The present value (PV) is \$50,000, because you received that today to buy the home. The future value (FV) is your yearly payments. All your payments will be equal. Substitute all this information into the present value formula.

\$50,000 = FV1 / (1.12) + FV2 / (1.12)2 + FV3 / (1.12)3 + . . . . + FV20 / (1.12)20

FV1 = FV2 = FV3 = . . . . + FVn , because all payments are the same.

\$50,000 = FV [ 1 / (1.12) + 1 / (1.12)2 + 1 / (1.12)3 + . . . . + 1 / (1.12)20]

FV = \$50,000 � [ 1 / (1.12) + 1 / (1.12)2 + 1 / (1.12)3 + . . . . + 1 / (1.12)20] = \$6,693.98

As you can see with the mortgage loan, the present value formula can become long. If you paid your mortgage each month, you would have to adjust this equation. You would have to divide the interest rate by 12 (the number of payments each year). Then you would make a total of 420 payments (12 payments per year * 20 years). This equation would end up with 420 terms.  That is why financial institutions use computers to calculate the loan payments. These formulas can become quite long.

## Market Interest Rates and Rate of Return

If you wanted to invest in the financial market, what financial securities would you buy? Economists devised the yield to maturity concept and consider this the most accurate measure of the interest rates. The yield to maturity allows the easy comparison of the four loan types. When paying a financial security, you know the principal, the maturity date, number of payments per year, and the amount of payments. You substitute all this information into the present value formula and solve for the interest rate, which is the yield to maturity. Then you can compare the yield to maturity to other loans and select the security that gives the high yield.

For example, a bank offers to sell you today a commercial loan for \$1,300. The business makes 1 payment per year in the amount of \$500. The loan matures in three years. What is the yield to maturity? First, substitute all this information into the present value formula. Second, solve for i, which is the yield to maturity.

PV = FV1 / (1 + i) + FV2 / (1 + i)2 + FV3 / (1 + i)3

PV : Present value, which is \$1,300.

FV : Future value, the yearly payments of \$500.
i : Yield to maturity, i.e. the market interest rate.

\$1,300 = \$500 / (1 + i) + \$500 / (1 + i)2 + \$500/ (1 + i)3

As you can see, this calculation is very complicated. The yield to maturity is 7.5%. You can also refer to this as the market interest rate of this security. If someone else offered you a security that offered a yield to maturity of 10% and the security has similar risk and liquidity, you would be better off buying that other security.

From the yield to maturity, two rules can be developed.

1. Market interest rate (yield to maturity) and market price of the securities are inversely related.

• For example, you have an offer to buy a T-bond that has a stated interest rate of 10% and a face value of \$10,000. If the market interest rate is 10%, then the market price of the T-bond will be \$10,000. If the market interest rate increased to 12%, then you would not buy the T-bond for \$10,000, because the bond is only earning 10%. The seller will lower the market price of the T-bond, compensating you for the higher interest rate. If the market interest rate decreased to 8%, then nobody would sell you a T-bond for \$10,000, if it earns 10%. The seller will increase the market price of the bond, compensating the seller for the lower market interest rate. The present value formula is used to adjust the market price of the bond to changes in the market interest rate.

2. The shorter the term to maturity of a bond, the less its price will fluctuate for a change in the market interest rate.

• For example, you are comparing two bonds. Each bond has a face value of \$5,000, coupon interest rate of 10%, and interest is paid annually. The only difference is the first bond matures within one year and the other bond matures in 10 years. What happens to the market price of these bonds, if the market interest rate changes to 16%? The bond that matures within a year will have a market price of \$4,741, while the other bond will have a market value of \$3,550. The change in the interest rate had a much bigger impact on the security with the 10 year maturity period. That is why money market securities are so popular; they mature within a year. The calculations are listed below:

Matures in 1 year: P = 0.10 * \$5,000 / 1.16 + \$5,000 / 1.16 = \$4,741.38

Matures in 10 years: P = \$500 / 1.16 + \$500 / 1.162 +. . . . + \$5,500 / 1.1610 = \$3,550.03

The interest rate listed on a security does not tell you how good the investment is. You need to calculate the total rate of return. Total rate of return tells you how good your investment is over the holding period of the security. The return is not an interest rate. It is an interest rate plus the percent change in the security's price when you resell the security. The rate of return will equal the interest rate, when you hold the security until it matures. For example, a bond has a face value of \$2,000, coupon interest rate of 5%, and a 10 year maturity. If you bought this bond for \$2,000 and resold it a year later for \$2,400, then you collected 1 year of interest and a capital gain of \$400 (= \$2,400 - \$2,000). The interest you earned for 1 year is 5%. You would have received \$100 of interest (= 0.05 * \$2,000). However, you sold the security for a higher price, so you have a capital gain. The capital gain is calculated below:

Capital gain: g = ( \$2,400 - \$2,000 ) / \$2,000 = 0.20

The rate of return equals the capital gain plus the interest. So in this case, you earned a return of 25% (20% + 5%). Let's redo the same example. You bought this security for \$2,000 and held it for one year. You earned one year of interest, which is 5%. What happens if the interest rate increased, causing the price of your security to fall to \$1,000? Now you suffer a capital loss.

Capital loss: g = ( \$1,000 - \$2,000 ) / \$2,000 = -0.5

When you add your capital loss to the interest you have earned, your total rate of return is -45% (5% - 50%). This transaction has made you poorer. The capital gains and losses increase when the maturity of securities become longer. For example, a 30-year bond will greatly fluctuate in price, when the interest rate changes. The large swings of the bond's price cause large capital gains and losses, making long-term securities a risky investment.

## Fisher Equation

The interest rates discussed so far has been nominal interest rates. Nominal interest rates have not been adjusted for inflation. Inflation can have a significant influence on the financial markets. Investors and savers are concerned about the real interest rate. The real interest rate is the true cost of borrowing. Using the Fisher equation, you calculate the real interest from the formula below:

 r = i - pe i : nominal interest rate. r : real interest rate. pe : expected inflation rate.

For example, you expect the inflation rate to be zero ( pe = 0 ). If you make a loan for 5% for one year. At the end of Year 1, you have 5% more money in real terms. You can purchase 5% more in goods and services.

r = 5% - 0% = 5%

What if you believe inflation will increase to 5% ( pe = 5% ). If you make a loan for one year for 5%, at the end of year 1, you will have 5% more money, but prices became 5% higher too, so in real terms, your purchasing power does not change. Therefore, the real interest rates reflects the true cost of borrowing and it is a better indicator of incentives to lend and borrow.

r = 5% - 5% = 0%