**Chapter 3, "Interest Rates and Rates of Return." Or, the Study of Bonds.**

What are Bonds? The Different Types of Bonds.

Future Values of Simple Loans

Who Sells Bonds?

Bonds are financial assets that represent a promise to repay a fixed amount of money.

They are I.O.U.s, or "debt instruments"

For every bond there is a borrower and lender.

For every bond there is a seller and buyer.

There are 4 broad categories of bonds (debt instruments):

Simple loan; the borrower receives from the lender an amount called the principal and agrees to repay the lender the principal plus interest on a specific date (maturity)

Discount bond is where the borrower repays the amount of the loan in a single payment at maturity but receives less than the face value of the bond initially.

Coupon bond requires multiple payments of interest on a regular basis, such as semiannually or annually, and a payment of the face value at maturity.

Fixed payment loans requires the borrower to make regular periodic payments of principal and interest to the lender.

10,000

11,000

borrower

lender

9,091

10,000

borrower

lender

998

1,010

borrower

lender

10

10

10

900

100

borrower

lender

100

100

100

Note: Simple loans = discount loans

Coupon bonds and fixed payment loans are a sum of a series of simple loans (discount bonds)

Bonds

Simple

Discount

Coupon

Fixed Payment

Q: Show that a 30 year fixed payment loan of $100 can be drawn as the sum of 30 simple loans.

Time (months, years, etc)

Q: Who is the seller of the bond, the borrower or lender?

Governments

Firms

Households

State and Local

Municipal Bonds

Federal

Treasury bills: discount bonds with maturities of 28 days (or 4 weeks, about a month), 91 days (or 13 weeks, about 3 months), 182 days (or 26 weeks, about 6 months), and 364 days (or 52 weeks, about 1 year).

Treasury Notes: coupon bonds that mature in one to ten years. They have a coupon payment every six months

Treasury Bonds: coupon bonds mature in 20 to 30 years. They have a coupon payment every six months

Corporate bonds: Coupon bonds

Car

Mortgage

Student

Fixed Rate

Q: A bank wants to sell (borrow) you a $1,000 CD. It promises to pay you back in 1 year plus 6% interest. How much will you recieve in the future from this simple loan?

A: 1000*(1+.06)=1060

Q: A banks wants to sell you $1,000 CD. It promises to pay you back in 2 years plus 6% interest per year compounded for each year. How much will you recieve in the future from this simple loan?

A: 1000*(1+.06)^2=1123.60

Both answers are the future values (FV) of 1000 in 1 and 2 years.

Q: What is the FV of $1,000 in 30 years compounded at 6%?

Banks: CDs

Valuing Simple Loans (Discount Bonds)

What is the Relevant Interest Rate?

Valuing All Bonds

Rule: the FV of $1 in n years is: FV = (1+i)^n, where i is called the interest rate.

Rule: the FV of $X in n years is: FV = (1+i)^n * X

(1+i)^n is also equal to the gross return of $1. By gross return, we mean how much $1 yields in the investment.

Ex: the gross return on a simple loan componded twice at 6% is

(1+.06)^2=1.1236

$1 today will yeild $1.1236 in two years

Q: Your friend wants to borrow money today. He promises to pay back $1,000 next year. How much would you give him today, more or less than 1000?

A: You wouldn't give him more than 1000 as it implies a negative gross return.

Q: Suppose you gave your friend $950 today and received $1,000 next month. What is the gross return on each $?

A: 1000/950=1.05263, we say $1 yeilds 1.05263 future dollars

Rule: The gross return on all bonds is = (what you get)/(what you payed)

The Idea of Valuation

Purchasing bonds (ie saving) has an opportunity cost. The opportunity cost is the foregone gross return on the next best asset. Valuation (pricing) is done by comparision to the opportunity cost

**A: 1000/X=1.06**

**1000/1.06=X**

**943.396=X**

Bond terminology:

Face value, or par value, is the amount to be repaid by the bond issuer (the borrower) at maturity.

The coupon is the annual fixed dollar amount of interest paid by the issuer of the bond to the buyer.

The coupon rate is the value of the coupon expressed as a percentage of the par value of the bond.

The current yield is the value of the coupon expressed as a percentage of the current price.

Maturity is the length of time before the bond expires and the issuer makes the face value payment to the buyer.

Q: What is the par value of the example US T-bill?

Q: What is the maturity of the US T-bill?

Q: What is the par of the British T-bill?

Q: What is the par value of the T-bond?

Q: What is the coupon rate of the corporate bond?

Q: suppose the bank is offering 6% on a CD. If you wanted the loan to your friend to at least match the gross return on the CD at the bank, how much would you give him today?

**Reasoning: In bond terminology, what you pay is called the bond price. Therefore, the gross return to a discount bond is=Par/Price. This should be equal to the gross return of the next best alternative (otherwise nobody would purchase it) that has a gross return of (1+i)^2. Thus, Par/Price=(1+i)^n**

Or, Par/(1+i)^n = Price

Or, Par/(1+i)^n = Price

Q: Your friend wants a 2 year loan to pay you back $1,000. If you wanted the loan to your friend to at least match the gross return on the CD at the bank, how much would you give him today given the gross return at the bank is: (1+.06)^2?

**A: 1000/X=1.06^2**

**1000/1.06^2=X**

**889.996=X**

Time (months, years, etc)

Time (months, years, etc)

Time (months, years, etc)

**Price also called present value (PV):**

Par/(1+i)^n = PV

Par/(1+i)^n = PV

**The present value formula depends on the gross return of the next best alternative. That is, it depends on i. What determines the choice of next best alternative and thus choice of i?**

PV in Pratice; The Relationship Between i and Bond Prices

Economists have found that in the market for bonds i depends on 3 things:

Compensation for inflation: if prices rise, the payments received will buy fewer goods and services.

Compensation for default risk: the borrower might default on the loan

Compensation for the opportunity cost of waiting for the money to be paid back.

Rate of Return versus Yield to Maturity

Real and Nominal Interest Rates

The higher the expectation of inflation/default/waiting, the higher the relevant interest rate i.

In general, inflation/default/waiting place a premium on i.

Q: Which is the Relevant interest rate for a 1 yr loan to your roommate; a 1yr T-bill rate or a 1yr rate on a bond from a 3rd world country?

Q: Which is the Relevant interest rate for a 30 yr Apple Computer corporate bond; a 1yr T-bill rate, 30yr T-bond rate, or a 30yr rate on a Microsoft corporate bond?

10000/(1+.06)=?

10,000

borrower

lender

Discount

1yr

Q: Find the value (PV, Price) of the following 1yr cash flow given the relevant i is 6%:

Q: Find the PV (value, Price) of the following 1yr cash flow given the relevant i is 2%:

10000/(1+.02)=?

10,000

borrower

lender

Discount

1yr

Q: What is the relationship between PV and i?

Q: Find the PV (value, Price) of the following cash flow to be received in yr2 given the relevant i is 6%yr:

10000/(1+.06)^2=?

10,000

borrower

lender

Discount

2yr

Q: What is the relationship between PV and maturiy?

1,010

borrower

lender

10

10

10

Coupon

We want to find the present value of a bond that pays multiple periods, like a coupon bond. To find PV, we break up each cash flow to evaluate. Then, add them to find the total PV.

EX: a 5yr T-note that pays an annual 1% coupon with par value of $1,000. What is it's PV given i=5%?

10

=

borrower

lender

10

Coupon

PV=?

borrower

lender

10

Coupon

borrower

lender

10

Coupon

borrower

lender

Coupon

10

1,010

borrower

lender

Coupon

**PV=10/(1+.05)**

**PV=10/(1+.05)^2**

**PV=10/(1+.05)^3**

**PV=10/(1+.05)^4**

**PV=1010/(1+.05)^5**

**Q: Draw the cash flows of a 8yr T-note that pays an annual 1% coupon with par value of $1,000 given i=5%?**

**Q: What is relationship between the price of coupon bonds and maturity?**

5

1

2

3

4

5

**The financial press quotes either bond prices, yields to maturity, or both**

Yield to maturity is the interest rate that makes PV equal to its market price. That is, yield to maturity is the interest rate that the market is using to evaluate the bond.

**Def: Yield to maturity is the interest rate that makes the present value of the payments from an asset equal to the asset’s price today.**

Note: Whenever participants in financial markets refer to the interest rate on a financial asset, the interest rate is the yield to maturity.

Note: Whenever participants in financial markets refer to the interest rate on a financial asset, the interest rate is the yield to maturity.

**Rule: Yield to maturity is the rate of return you will earn on a bond if you hold it to maturity.**

**Rule: If the bond is sold before maturity, the rate of return the ivestor gets is not necessarily equal to the yield to maturity when you bought it. Why?**

**Q: Suppose you purchase a 2yr 5% coupon bond with a par of $1,000. Its price was 946.5021. Show that its yield is 8%.**

Q: Suppose that you hold it to maturity. What is its annual rate of return?

Q: Instead, you sell the bond after receiving the first coupon. The Price you get is 972.2223. Find the rate of return of the investment.

Q: Instead, the Price you get is 950. Find the rate of return of the investment.

Q: Suppose that you hold it to maturity. What is its annual rate of return?

Q: Instead, you sell the bond after receiving the first coupon. The Price you get is 972.2223. Find the rate of return of the investment.

Q: Instead, the Price you get is 950. Find the rate of return of the investment.

If you never sell your bonds, then why should you care about price fluctuations? If the price of your bond rises, you own a more valuable asset, so your wealth rises. This is called a capital gain. If the price falls, your wealth falls causing a capital loss. Wealth effects are important!

**Def:Nominal interest rate (i) is the rate offered by a bank account or bond**

Def: The ex-ante real interest rate (r) is the nominal minus the expected inflation rate (pi^ex): r=i-pi^ex

Def: The ex-ante real interest rate (r) is the nominal minus the expected inflation rate (pi^ex): r=i-pi^ex

**Q: If your bank pays 5% nominal interest rate and the expected inflation rate is 3%, what is the real ex-ante rate?**

**Def: The ex-post real interest rate r^post is the nominal interest minus the actual inflation (pi) over the loan period:**

r^post=i-pi

r^post=i-pi

**Q: Due to unexpected events, the actual inflation rate was 4%. what was the real ex-post interest rate?**