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The Term Structure of Interest Rates.  The relationship between yield to maturity and maturity.  Yield curve - a graph of the yields on bonds relative.

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Presentation on theme: "The Term Structure of Interest Rates.  The relationship between yield to maturity and maturity.  Yield curve - a graph of the yields on bonds relative."— Presentation transcript:

1 The Term Structure of Interest Rates

2  The relationship between yield to maturity and maturity.  Yield curve - a graph of the yields on bonds relative to the number of years to maturity  Information on expected future short term rates can be implied from yield curve.  Three major theories are proposed to explain the observed yield curve.

3 Treasury Yield Curves

4  Measure of rate of return that accounts for both current income and the price increase over the life  YTM is the discount rate that makes the present value of a bond ’ s payments equal to its price ◦ Proxy for average return

5 Solve the bond formula for r

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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 bonds ◦ Example: 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 Pricing  Bond stripping and bond reconstitution offer opportunities for arbitrage  Law of one price, identical cash flow bundles must sell for identical prices  Value each stripped cash flow ◦ discount by using the yield appropriate to its particular maturity

9 Yields and Prices to Maturities on Zero- Coupon Bonds ($1,000 Face Value)

10  Treat each of the bond ’ s payments as a stand- alone zero-coupon security (bond stripping : portfolio of three zeros)  Pure yield curve ◦ Relationship between YTM and time to maturity for zero- coupon bonds  On-the-run yield curve ◦ Plot of yield as a function of maturity for recently issued coupon bonds selling at or near par value

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12  If interest rates are certain  Considering two strategies ◦ Buying the 2-year zero, YTM=6%, hold until maturity. Price=890 ◦ Invest 890 in a 1-year zero, YTM=5%. Reinvest the proceeds in another 1-year bond

13 ◦ The proceeds after 2 years to either strategy must be equal

14  Spot rate ◦ The rate that prevails today for a time period corresponding to the zero ’ s maturity  Short rate ◦ For a given time interval (e.g. 1 year) refers to the interest rate for that interval available at different points in time  2-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 year  When next year ’ s short rate ( r 2 =7.01%) is greater than this year ’ s short rate, the average of the two rates is higher than today ’ s rate

16 Short Rates versus Spot Rates

17  When interest rate with certainty, all bonds must offer identical rates of return over any holding period  Calculate 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  For 1-year maturity bond ◦ Rate of return=( )/952.38=5%  For 2-year maturity bond ◦ Price of today=890 ◦ One year later, when next year ’ s interest rate=7.01%, sell it for 1000/1.0701= ◦ Rate of return=( )/890=5%

19  No access to short-term interest rate quotations for coming years---infer future short rates from yield curve of zeros  Two alternatives get same final payoff ◦ 3-year zero ◦ 2-year zero, reinvest in 1-year bond

20  f n = one-year forward rate for period n  y n = yield for a security with a maturity of n

21  future short rates are uncertain  Forward interest rate ◦ defined 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 n ◦ Calculated 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+f n ) (1.3605) / (1.2250) = (1+f n ) f n =.1106 or 11.06%

23 12.3 Interest Rate Uncertainty and Forward Rates

24 Interest Rate Uncertainty  What can we say when future interest rates are not known today  Suppose that today ’ s rate is 5% and the expected short rate for the following year is E(r 2 ) = 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-horizon investors ◦ If invest only for 1 year certain return=( )/952.38=5% ◦ If invest for 2-year zero, if expect the 1-year rate be 6% at the end of the first year  the price will be 1000/1.06=943.4  the first-year’s expected rate of return also is 5%=( )/  but the 2nd year’s rate is risky, is not certain  If >6%, bond price<943.4  If  2-year bond must offer an expected rate of return greater than riskless 5% return, sell at price lower than

26  If the investors will hold the bond when it falls to ◦ Expected 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 Investors  Short-term Investors require a risk premium to hold a longer-term bond  This liquidity premium compensates short-term investors for the uncertainty about future prices  If most individuals are short-term investors, bonds must have prices that make f 2 greater than E( r 2 )

28  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 be ◦ The investor will require ◦ Offered as a reward for bearing interest rate risk

29  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 be ◦ The investor will require ◦ Offered as a reward for bearing interest rate risk

30 12.4 Theories of the Term Structure

31  Expectations theories  Liquidity Preference theories ◦ Upward bias over expectations  Market Segmentation

32  Observed 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 rates ◦ Upward slope means that the market is expecting higher future short term rates ◦ Downward slope means that the market is expecting lower future short term rates

33  Short-term investors dominate the market  Forward 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  Short- 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  YearInterest Rate  0 (today) 8%  110%  211%  311%  8% 10% 11% 11%

36  PV n = Present Value of $1 in n periods  r 1 = One-year rate for period 1  r 2 = One-year rate for period 2  r n = One-year rate for period n

37 Price of 1-year maturity bond Price of 2-year maturity bond Price of 3-year maturity bond

38 YTM is average rate that is applied to discount all of the bond’s payments

39 Time to Maturity Price of Zero*Yield to Maturity 1$ % * $1,000 Par value zero

40 8% 10% 11% 11% y1=8% y2=8.995% y3=9.660% y4=9.993% expected Short rate in each year YTM for various maturities (Current spot rate)

41  YTM, average of the interest rates in each period (geometric)

42 12.4 Interpreting the Term Structure

43  Direct relationship between YTM and forward rate  Under certainty  Uncertain  The 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 zero ◦ If then ◦ If, 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.

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