1. Interest Rates / Bond Prices and Yields
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Interest Rates / Bond Prices and Yields

2. Interest Rates / Bond Valuation
Our goal in these chapters is to discuss the many different interest rates that are commonly reported in the financial press and to understand the basics of bond pricing.

3. U.S. Interest Rate History, 1953-present

4. Money Market Rates, I.
Prime rate - The basic interest rate on short-term loans that the largest commercial banks charge to their most creditworthy corporate customers.
Federal funds rate - Interest rate that banks charge each other for overnight loans of $1 million or more.
Discount rate - The interest rate that the Fed offers to commercial banks for overnight reserve loans.

5. Money Market Rates, II.
Call money rate - The interest rate brokerage firms pay for call money loans from banks. This rate is used as the basis for customer rates on margin loans.
Commercial paper - Short-term, unsecured debt issued by the largest corporations.
Certificate of deposit (CD) - Large-denomination deposits of $100,000 or more at commercial banks for a specified term.
Banker’s acceptance - A postdated check on which a bank has guaranteed payment. Commonly used to finance international trade transactions.

6. Money Market Rates, III.
Eurodollars - U.S. dollar denominated deposits in banks outside the United States.
London Interbank Offered Rate (LIBOR) - Interest rate that international banks charge one another for overnight Eurodollar loans.
Good source for these rates:

7. Money Market Prices and Rates
A Pure Discount Security is an interest-bearing asset:
It makes a single payment of face value at maturity.
It makes no payments before maturity.
There are several different ways market participants quote interest rates.
Banker’s Discount Basis
Bond Equivalent Yields (BEY)
Annual Percentage Rates (APR)
Effective Annual Rates (EAR)
Basis point is 1% of 1%.
If the interest rate increases from 11.25% to 11.27%, this represents a 2 basis point increase. The “11” is referred to as the handle.

8. The Bank Discount Basis
The Bank Discount Basis is a method of quoting interest rates on money market instruments.
It is commonly used for T-bills and banker’s acceptances.
The formula is:
Note that we use 360 days in a year in this (and many other) money market formula.
The term “discount yield” here simply refers to the quoted interest rate.

9. Bond Equivalent Yields
Bond Equivalent Yields (BEY) are another way to quote an interest rate.
You can convert a bank discount yield to a bond equivalent yield using this formula:
Note that this formula is correct only for maturities of six months or less. Moreover, if February 29 occurs within the next 12 months, use 366 days.

10. More Ways to Quote Interest Rates
“Simple” interest basis - Another method to quote interest rates.
Calculated just like annual percentage rates (APRs).
Used for CDs.
The bond equivalent yield on a T-bill with less than six months to maturity is also an APR.
An APR understates the true interest rate, which is usually called the effective annual rate (EAR).

11. Converting APRs to EARs
In general, if we let m be the number of periods in a year, an APR can be converted to an EAR as follows:
EARs are sometimes called effective annual yields, effective yields, or annualized yields.

12. The Treasury Yield Curve
The Treasury yield curve is a plot of Treasury yields against maturities.
It is fundamental to bond market analysis, because it represents the interest rates for default-free lending across the maturity spectrum.

13. The Treasury Yield Curve
Normal – 30 yr. T-bond yields are about 3% higher than 3-month T-bill yields.
Steep – typical at the starting stages of an economic expansion.
Inverted – precedes a recession or significant economic slowdown.
Flat/Humped – rates need to flatten before becoming inverted (if they do).

14. The Term Structure of Interest Rates, I.
The term structure of interest rates is the relationship between time to maturity and the interest rates for default-free, pure discount instruments.
The term structure is sometimes called the “zero-coupon yield curve” to distinguish it from the Treasury yield curve, which is based on coupon bonds.
The term structure can be seen by examining yields on U.S. Treasury STRIPS.

15. U.S. Treasury STRIPS
An asked yield for a U.S. Treasury STRIP is an APR, calculated as two times the true semiannual rate.
Recall:
Therefore, for STRIPS:
M is the number of years to maturity.

16. Bond Basics (Chapter 10)
Two basic yield measures for a bond are its coupon rate and its current yield.

17. Straight Bond Prices and Yield to Maturity
The price of a bond is found by adding together the present value of the bond’s coupon payments and the present value of the bond’s face value.
The Yield to maturity (YTM) of a bond is the discount rate that equates the today’s bond price with the present value of the future cash flows of the bond.

18. The Bond Pricing Formula
The price of a bond is found by adding together the present value of the bond’s coupon payments and the present value of the bond’s face value.
The formula is:
In the formula, C represents the annual coupon payments (in $), FV is the face value of the bond (in $), and M is the maturity of the bond, measured in years.

19. Premium and Discount Bonds, I.
Bonds are given names according to the relationship between the bond’s selling price and its par value.
Premium bonds: price > par value
YTM < coupon rate
Discount bonds: price < par value
YTM > coupon rate
Par bonds: price = par value
YTM = coupon rate

20. Premium and Discount Bonds, II.

21. Premium and Discount Bonds, III.
In general, when the coupon rate and YTM are held constant:
for premium bonds: the longer the term to maturity, the greater the premium over par value.
for discount bonds: the longer the term to maturity, the greater the discount from par value.

22. Relationships among Yield Measures
for premium bonds:
coupon rate > current yield > YTM
for discount bonds:
coupon rate < current yield < YTM
for par value bonds:
coupon rate = current yield = YTM

23. Calculating Yield to Maturity, I.
Suppose we know the current price of a bond, its coupon rate, and its time to maturity. How do we calculate the YTM?
We can use the straight bond formula, trying different yields until we come across the one that produces the current price of the bond.
This is tedious. So, to speed up the calculation, financial calculators and spreadsheets are often used.
We can approximate the YTM using the following equation:

24. A Quick Note on Bond Quotations, I.
We have seen how bond prices are quoted in the financial press, and how to calculate bond prices.
Note: If you buy a bond between coupon dates, you will receive the next coupon payment (and might have to pay taxes on it).
However, when you buy the bond between coupon payments, you must compensate the seller for any accrued interest.

25. A Quick Note on Bond Quotations, II.
The convention in bond price quotes is to ignore accrued interest.
This results in what is commonly called a clean price (i.e., a quoted price net of accrued interest).
Sometimes, this price is also known as a flat price.
The price the buyer actually pays is called the dirty price
This is because accrued interest is added to the clean price.
Note: The price the buyer actually pays is sometimes known as the full price, or invoice price.

26. Callable Bonds
Thus far, we have calculated bond prices assuming that the actual bond maturity is the original stated maturity.
However, most bonds are callable bonds.
A callable bond gives the issuer the option to buy back the bond at a specified call price anytime after an initial call protection period.
Therefore, for callable bonds, YTM may not be useful.

27. Yield to Call
Yield to call (YTC) is a yield measure that assumes a bond will be called at its earliest possible call date.
The formula to price a callable bond is:
In the formula, C is the annual coupon (in $), CP is the call price of the bond, T is the time (in years) to the earliest possible call date, and YTC is the yield to call, with semi-annual coupons.
As with straight bonds, we can solve for the YTC, if we know the price of a callable bond.

28. Interest Rate Risk Holders of bonds face Interest Rate Risk.
Interest Rate Risk is the possibility that changes in interest rates will result in losses in the bond’s value.
The yield actually earned or “realized” on a bond is called the realized yield.
Realized yield is almost never exactly equal to the yield to maturity, or promised yield.

29. Interest Rate Risk and Maturity

30. Malkiel’s Theorems, I.
Bond prices and bond yields move in opposite directions.
As a bond’s yield increases, its price decreases.
Conversely, as a bond’s yield decreases, its price increases.
For a given change in a bond’s YTM, the longer the term to maturity of the bond, the greater the magnitude of the change in the bond’s price.

31. Malkiel’s Theorems, II.
For a given change in a bond’s YTM, the size of the change in the bond’s price increases at a diminishing rate as the bond’s term to maturity lengthens.
For a given change in a bond’s YTM, the absolute magnitude of the resulting change in the bond’s price is inversely related to the bond’s coupon rate.

32. Readings
Chapter 9 up to page 301.
Chapter 10 up to page 334.