Econ 423 Michael SalemiWhy Do Interest Rates Move Together?Class Notes1. Discussion of exercise on the Components Model of interest rates.a. What does the components model say?b. What hypotheses did students formulate as part of the exercise?c. Did the data confirm or contradict the hypotheses?d. Overall, does the data confirm or contradict the components model of interestrates?2. The Expectations Hypothesis of the Term Structure provides another reason why interestrates may move together.a. What is the term structure of interest rates?b. The intuition behind the terms structure is the idea that all bond hold strategiesthat move funds between the same two points in time are substitutes. If they arevery close substitutes, then their expected yields should be the same becauseagents will always choose the strategy with the higher expected yield.c. Holding long maturity bonds is riskier than holding short maturity bonds.d. Many agents dislike risk and will therefore prefer less risky strategies for movingfunds from the present to the future. i. The expectations hypothesis may not hold.ii. The liquidity premium hypothesis modifies the expectations hypothesis bytaking risk into account.e. If the expectations hypothesis is approximately true, or if liquidity premia areconstant, then the term structure of interest rates at a point in time can be used toforecast future interest rates.The following figure is a screen shot of the U.S. Treasury web site that provides the DailyTreasury Yield Curve. It url is:(http://www.ustreas.gov/offices/domestic-finance/debt-management/interest-rate/yield.shtml)1. The slope of the yield curve is said to be the difference between a long maturity rate(generally the yield on a 20 year treasury bond) and a short rate (the yield on a one orthree month treasury bill). What happened to the slope of the yield curve during the firstweek in October?2. Aside from a change in the slope, did the yield curve shift during the first week inOctober?Holding Long Maturity Bonds is Riskier than Holding Short Maturity BondsConsider a bond that matures in M years and pays a coupon of constant value C at the end ofeach year. In reality, the typical bond pays coupons semi-annually but our simplification doesnot affect the results we will obtain.If the annual coupon is constant, then there exists a simpler version of the present value formulafor the price of the bond. Again, let the coupon be C dollars per year, let the par value of the bond (theamount paid to the bond holder at maturity) be F dollars and let the maturity date occur M years in thefuture. If R is the constant discount rate, the present value of the bond is:To prove that the second line follows from the first, rewrite the present value as the difference betweenthe present values of two perpetuities–the first starting in the present and the second starting in yeart+M+1. Then use the rules for computing the present value of perpetuities to show that the result is true.Let F = $1000 and C = $100. The following table gives present value for various values of R and M.Present ValueR\M125304.08 $1018.52 $1035.67 $1079.85 $1225.16 $1250.00.10 $1000.00 $1000.00 $1000.00 $1000.00 $1000.00.12 $982.14 $966.20 $927.90 $838.39 $833.33Question: Why do the data in the above table show that long maturity bonds are riskier than shortmaturity bonds?The Expectations and Liquidity-Premium HypothesesDefinitionsi t = Today’s (time t) interest rate on a one period (one year) bond.i n , t= yield to maturity of an n-year bond at time t.i e t + m= yield to maturity expected at t to occur at time t + m.The HypothesesThe Expectations Hypothesis of the term structure of interest rates is:The hypothesis says that the gross yield at time t on an n-year bond equals the geometric averageof the current one year rate and the one year rates expected for each year between time t and timen. There is an arithmetic approximation to the expectations hypothesis given by:The Liquidity Premium hypothesis modifies the expectations hypothesis by including a liquidityor risk premium in its explanation of long-maturity rates. The formula in its arithmetic form is:Holding constant expectations of future interest rates, the liquidity premium hypothesis says thatthe n-year rate is now greater than the average of expected one-period rates because there is apremium ( l n ,t ) added to compensate asset owners for increased risk and loss of liquidity.Predicting Future Interest Rates Using the above hypotheses one can forecast future one-year interest rates. For the EHthe analysis proceeds as follows.i2 , t = [it + iet + 1 ] / 2 iet+1 = 2 i2 , t - i ti3 , t = [2i2 , t + iet + 2] / 3 iet+2 = 3i 3 , t - 2i 2 , ti4 , t = [3i3 , t + iet + 3] / 4 iet + 3 = 4i 4 , t - 3i 3, t and so forth.Exercise:1. Use the current yield curve to forecast one-year interest rates for the next three years.2. Explain how the liquidity premium hypothesis would alter those