Presentation on theme: "The Term Structure of Interest Rates"— Presentation transcript:
1 The Term Structure of Interest Rates Chapter 12The Term Structure of Interest Rates
2 Overview of Term Structure The relationship between yield to maturity and maturity.Yield curve - a graph of the yields on bonds relative to the number of years to maturityInformation on expected future short term rates can be implied from yield curve.Three major theories are proposed to explain the observed yield curve.
4 Yield to MaturityMeasure of rate of return that accounts for both current income and the price increase over the lifeYTM is the discount rate that makes the present value of a bond’s payments equal to its priceProxy for average return
7 Bond Pricing Yields on different maturity bonds are not all equal Need to consider each bond cash flow as a stand- alone zero-coupon bond when valuing coupon bondsExample: 1-year maturity T-bond paying semiannual coupons can be split into a 6-month maturity zero and a 12-month zero. If each cash flow can be sold off as a separate security, the value of the whole bond should be the same as the value of its cash flows bought piece by piece.
8 Bond PricingBond stripping and bond reconstitution offer opportunities for arbitrageLaw of one price, identical cash flow bundles must sell for identical pricesValue each stripped cash flowdiscount by using the yield appropriate to its particular maturity
9 Yields and Prices to Maturities on Zero-Coupon Bonds ($1,000 Face Value)
10 Bond PricingTreat each of the bond’s payments as a stand- alone zero-coupon security(bond stripping : portfolio of three zeros)Pure yield curveRelationship between YTM and time to maturity for zero- coupon bondsOn-the-run yield curvePlot of yield as a function of maturity for recently issued coupon bonds selling at or near par value
12 Yield Curve Under Certainty If interest rates are certainConsidering two strategiesBuying the 2-year zero, YTM=6%, hold until maturity. Price=890Invest 890 in a 1-year zero, YTM=5%. Reinvest the proceeds in another 1-year bond
13 Yield Curve Under Certainty The proceeds after 2 years to either strategy must be equal
14 Yield Curve Under Certainty Spot rateThe rate that prevails today for a time period corresponding to the zero’s maturityShort rateFor a given time interval (e.g. 1 year) refers to the interest rate for that interval available at different points in time2-year spot rate is an average (geometric) of today’s short rate and next year’s short rate
15 Yield Curve Under Certainty An upward sloping yield curve is evidence that short-term rates are going to be higher next yearWhen next year’s short rate (r2=7.01%) is greater than this year’s short rate, the average of the two rates is higher than today’s rate
17 Holding period returns When interest rate with certainty, all bonds must offer identical rates of return over any holding periodCalculate HPR for 1-year maturity zero- coupon bond (YTM=5%)The first 1-year HPR for 2-year maturity zero-coupon bond (YTM=6%)
18 Holding period returns For 1-year maturity bondRate of return=( )/952.38=5%For 2-year maturity bondPrice of today=890One year later, when next year’s interest rate=7.01%, sell it for 1000/1.0701=934.49Rate of return=( )/890=5%
19 Forward Rates from Observed Rates No access to short-term interest rate quotations for coming years---infer future short rates from yield curve of zerosTwo alternatives get same final payoff3-year zero2-year zero, reinvest in 1-year bond
20 Forward Rates from Observed Rates fn = one-year forward rate for period nyn = yield for a security with a maturity of n
21 Forward Rates from Observed Rates future short rates are uncertainForward interest ratedefined as the break-even interest rate that equates the return on an n-period zero-coupon bond to that of an (n-1)-period zero-coupon bond rolled over into a 1-year bond in year nCalculated from today’s data, interest rate that actually will prevail in the future need not equal the forward rate.
22 Example of Forward Rates 4 yr = 8.00% 3yr = 7.00% fn = ?(1.08)4 = (1.07)3 (1+fn)(1.3605) / (1.2250) = (1+fn)fn = or 11.06%
23 Interest Rate Uncertainty and 12.3Interest Rate Uncertainty andForward Rates
24 Interest Rate Uncertainty What can we say when future interest rates are not known todaySuppose that today’s rate is 5% and the expected short rate for the following year is E(r2) = 6% then:The rate of return on the 2-year bond is risky for if next year’s interest rate turns out to be above expectations, the price will lower and vice versa
25 Short-Term Investors Short-term-horizon investors If invest only for 1 yearcertain return=( )/952.38=5%If invest for 2-year zero, if expect the 1-year rate be 6% at the end of the first yearthe price will be 1000/1.06=943.4the first-year’s expected rate of return also is 5%=( )/898.47but the 2nd year’s rate is risky, is not certainIf >6%, bond price<943.4If <6%, bond price>943.42-year bond must offer an expected rate of return greater than riskless 5% return , sell at price lower than
26 Short-Term InvestorsIf the investors will hold the bond when it falls toExpected holding period return for the first year=( )/881.83=7%risk premium=7%-5%=2%Forward rate:Liquidity premium compensates short-term investors for the uncertainty about the price at which they will be able to sell their long-term bonds
27 Short-Term InvestorsShort-term Investors require a risk premium to hold a longer-term bondThis liquidity premium compensates short-term investors for the uncertainty about future pricesIf most individuals are short-term investors, bonds must have prices that make f2 greater than E(r2)
28 Long-Term Investor Wish to invest a full 2-year period Purchase 2-year zeros at , guaranteed YTM=8.995%If roll over two 1-year investments, an investment of grow in 2 years to beThe investor will requireOffered as a reward for bearing interest rate risk
29 Long-Term Investor Wish to invest a full 2-year period Purchase 2-year zeros at 890, guaranteed YTM=6%If roll over two 1-year investments, an investment of grow in 2 years to beThe investor will requireOffered as a reward for bearing interest rate risk
30 Theories of the Term Structure 12.4Theories of the Term Structure
31 Theories of Term Structure Expectations theoriesLiquidity Preference theoriesUpward bias over expectationsMarket Segmentation
32 Expectations TheoryObserved long-term rate is a function of today’s short-term rate and expected future short-term rates.Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.An upward-sloping yield curve if investors anticipate increases in interest ratesUpward slope means that the market is expecting higher future short term ratesDownward slope means that the market is expecting lower future short term rates
33 Liquidity Preference Theory Short-term investors dominate the marketForward rates contain a liquidity premium and are not equal to expected future short-term rates.Investors will demand a premium for the risk associated with long-term bonds.The yield curve has an upward bias built into the long-term rates because of the risk premium.
34 Market SegmentationShort- and long-term bonds are traded in distinct markets.Trading in the distinct segments determines the various rates.Observed rates are not directly influenced by expectations.
35 Expected One-Year Rates in Coming Years Year Interest Rate0 (today) 8%1 10%2 11%3 11%8% % % %
36 Pricing of Bonds PVn = Present Value of $1 in n periods r1 = One-year rate for period 1r2 = One-year rate for period 2rn = One-year rate for period n
37 Pricing of Bonds using Expected Rates Price of 1-year maturity bondPrice of 2-year maturity bondPrice of 3-year maturity bond
38 Pricing of Bonds using Expected Rates YTM is average rate that is applied to discount all of the bond’s payments
39 Bond Prices using Expected Rates Time to Maturity Price of Zero* Yield to Maturity1 $ %* $1,000 Par value zero
41 Bond Prices using Expected Rates YTM, average of the interest rates in each period (geometric)
42 Interpreting the Term Structure 12.4Interpreting the Term Structure
43 Direct relationship between YTM and forward rate Under certaintyUncertainThe yield curve is upward sloping at any maturity date, n, for which the forward rate for the coming period is greater than the yield at that maturity
44 If yield curve is rising, must exceed Example YTM on 3-year zero is 9%, YTM on 4-year zeroIf thenIf , then
45 Given an upward-sloping yield curve, What account for the higher forward rate? Expectations of increases in can result in rising yield curve; converse is not true .