Notes for Session 6
What are the components of the yield? We will decompose the yield into price appreciation yield and coupon yield to better understand the relationship between bond price and bond yield on a given day. We will do an example of the decomposition of bond yield in our lecture session. We then discuss the inverse relationship between price and the annual yieldto maturity (simply called the “yield”) for a bond. This important insight helps us understand how the Fed, by buying and selling of the government bonds an instrument of monetary policy influences the rates of interest in an economy and with it, helps bring the level of economic activity and the inflation rate within the desirable range (low inflation and full employment). Questions that you need to think about include the following: How do we determine the price of an existing bond on any given day, if we have the information on market interest rate for that day? From the formula in the previous session’s notes, we have: Yield = [(Par Value + Coupon – Price)/Price]*100 We now derive the formula for calculating the price of a bond on a given day. Multiply both sides of the above relationship by Price/100, or P/100: P* Yield/100 = Par Value + Coupon – P Since the par value is $100, we have P* Yield/100 = $100 + $coupon – P Now add P to both sides: P + P* Yield/100 = $100 + $coupon Factor out P from the left hand side:
P[1 + Yield/100] = $100 +$ coupon P = ($100 + $coupon) / (1+Yield/100) Note that the P in the above formula is in fact the “Present Value” (or the present discounted value) calculation for the price of an asset that pays some income in the future (one year from now). Now suppose, you buy a bond on the day that it is issued when the market interest rate (i.e., yield) is 4%. What would be the price? Remember that the fixed coupon is determined as the market interest rate on the day the bond is issued. What is the price of this bond on the day of issue? Now suppose you have purchased the newly issued bond and one day later, you decide to sell it. You call your bond broker who informs you that today interest rates have inched up a bit, they are now 4.5%. Would you be able to sell your bond at $100, more than $100, or less than $100? What is the price of your bond now? Now take a different scenario. Suppose that when you decide to sell your bond, you find out that market interest rate has fallen, to say, 3.5%. What is the price of your bond now? If market interest rates rise or fall, what happens to bond prices? From the exercise in part 1, above, what is the relationship between bond prices and market interest rates? As the market interest rate rises, bond prices fall and vice versa. This is an important relationship that we will use in our analysis of monetary policy. In conducting monetary policy to fix the economy, our Fed sells or purchases bonds from the public, thereby changing bond prices and hence changing interest rates, a key macroeconomic variable whose changes bring about changes in the level of economic activity in the economy. Price risk of a bond: The price of a bond varies in inverse relationship to market interest rates. As market interest rates go up, the price of an existing bond falls (a loss to the holder of the bond). Do you expect the interest rate to be generally higher on a long term or a short term bond? Why?
Note: A short term bond is defined as a bond whose period of maturity is short—up to one year. A long term bond, for our purposes is defined as any bond with a maturity longer than one year. There are bonds with maturity periods of 2, 3, 4….30 years. A long term bond, say a 4 year bond, pays the fixed interest –the coupon rate—at the end of years 1, 2, and 3 and then at the end of 4th year, it pays the full principal and one last coupon payment. The market interest rate for a short term bond/loan is called a short term interest rate and the interest rate for a long term bond/loan is called the long term interest rate. Bonds are “nominally denominated” fixed income assets. That is, they pay fixed dollars of interest (and pay back the principal) in some future date. As such, you would expect the rate of inflation—the rate of increase in the overall price level to influence interest rates and, inversely, the value (price) of bonds. The question is why we should care about inflation and how we should analyze its effects. In chapter 4, we first analyze the concept of the Consumer Price Index (CPI). We define the concept, discuss its uses and of course derive the inflation rate in the CPI as a frequently used measure of inflation in the U.S. We then discuss the economic effects of high inflation in general, and, the effects of inflation on interest rates. This discussion involves important insights about the perils of high inflation in an economy and explains the vigilance of many central banks around the world to try and keep the rate of inflation low. You can find comprehensive information on the “Urban CPI” in the U.S. on the website of the Bureau of Labor Statistics (BLS). The web address is: www.bls.gov/. Questions that you need to think about include the following: 1. How is the CPI index constructed and how do you interpret this measure? The Consumer Price Index measures the cost of consumer goods in the U.S. relative to some base period. The base period is a year (or a period longer than a year) in the past, conventionally chosen. A representative bundle of goods purchased by consumers is chosen and the prices in this bundle are monitored every month by the Bureau of Labor Statistics. The incoming price information on various items in the bundle is then entered in the formula for CPI to yield an updated measure of cost of living for a typical consumer.
Calculating the CPI A simple example: 1992: base year. 2008: current year Item Apples Bus rides Tshirts
Quantity in 1992 10 12 8
Price in 1992 $ 1 per Lb. $ 2 per ride $6 per shirt
Price in 2008 $ 0.75per Lb. $ 3 per ride $8 per shirt
To calculate the CPI: 1. Calculate the P*Q in 1992 = 10*$1 + 12*$2 + 8*$6 = $ 82 2. Calculate the P*Q (quantity of 1992) for 2008 = 10*$.75 + 12*$3 + 8*$8 = $107.5 3. To calculate the CPI for 2008: [(sum of P2008 *Q1992 )/ (sum of P1992 *Q1992)]* 100 = 107.5/82=131.09 What does the number 131.09 mean? What are the units in this measurement (i.e., does 131.09 mean $131.09 or 131.09 tons, or 131.09 feet…)? What is the value of CPI in the base year? What is the inflation rate in the CPI between 1992 and 2008?