1 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Fina2802: Investments and Portfolio Analysis Spring, 2008 Dragon Tang Lecture 11 Bond Prices/Yields and Yield Curve February 25/26, 2008 Readings: Chapter 14 Practice Problem Sets: 4,5,7,8,9,10,13,35,36

2 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Prices and Yields Objectives: 1.Analyze the relationship between bond prices and and bond yields. 2.Calculate how bond prices will change over time for a given interest-rate projection. 3.Identify the determinants of bond safety and rating. 4.Analyze how callable, convertible, and sinking fund provisions will affect a bond's equilibrium yield to maturity. 5.Define the yield curve and study its properties

3 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Characteristics Face or par value Coupon rate –Zero coupon bond Compounding and payments –Accrued Interest Indenture

4 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Provisions of Bonds Secured or unsecured Call provision Convertible provision Put provision (putable bonds) Floating rate bonds

5 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Ratings Moody’sS&P Aaa(1,2,3)AAA AaAAA BaaBBB BaBBB CaaCCC CaCCCD Investment grade Speculative grade (junk bonds)

6 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Present Value Value today of an amount to be received in the future FV n = payment received at time n n = number of periods r = interest rate per period PV n = Present value

7 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Present Value. Example Example: What is the present value of $1 to be received (or paid) in 10 years if the annual interest rate is 5%?

8 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing Bond value =Present value of coupons + present value of par value

9 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing Example: Find the value (price) of a 30-year semiannual $1,000 par bond with coupon rate of 8% when similar bonds are also paying 8%. Using formula or PRICE function in Excel.

10 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing Example: Find the value (price) of a 30-year semiannual $1,000 par bond with coupon rate of 8% when similar bonds are also paying 10% (I.e., 5% semi-annually).

11 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing Market interest rates rise  bond prices fall

12 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing Example: Find the value (price) of a 30-year semiannual $1,000 par bond with coupon rate of 8% when similar bonds are also paying 6% (I.e., 3% semi-annually).

13 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing Market interest rates fall  bond prices rise

14 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing and Market Interest Rates Market Maturity CouponInterest Bond Value Rate Rate Price $ 1,000 8% 8%$1, $ 1,000 8% 10%$ $ 1,000 8% 6%$1,276.75

15 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing: Convexity Convexity - An increase in the interest rate results in a price decline that is smaller than the price gain that results from a decrease of equal magnitude in the interest rate. Flatter Bond Price Market Interest Rate

16 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Pricing: Maturity and Coupon The longer the time to maturity/lower coupon rate of a bond, the greater the sensitivity of price fluctuations to the interest rate. Price of a 8% Coupon Bond when Mkt Interest Rate is Maturity 4% 6% 8% 10% 12% 1yr $1,038 $1,019 $1,000 $981 $963 10yrs 1,327 1,148 1, yrs 1,547 1,231 1, yrs 1,695 1,276 1,

17 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Yield to Maturity Yield to maturity is a measure of the average rate of return that will be earned if the bond is held to maturity. YTM assumes reinvestment of the coupons at the yield to maturity. YTM is the solution (IRR) of :

18 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Yield to Maturity Example: Find the YTM of a semiannual bond with a price = $1,276.76, 30 years left to maturity, and a coupon rate of 8%. (Use trial and error method. In Excel: use Solver or YIELD function).

19 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Corporate Bonds Promised versus Expected Yield The promised or stated yield is the maximum possible return if held to maturity. The expected yield weights the promised yield by the probability of future default. The default premium is the differential in promised yield between a corporate bond and a government bond.

20 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Holding Period Return Where B t = Bond Price at time t B t+1 = Bond Price at time t+1

21 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Bond Prices over Time Coupon bonds Premium bonds move towards par as they approach maturity Discount bonds move towards par as they approach maturity

22 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Spot Rates and the Term Structure Spot rate: rate used to discount a cash flow at a point in time – YTM on a zero-coupon bond maturing at that time Term structure of interest rates, represents the variation of spot rates over terms to maturity Yield curve often refers to the variation of YTM over terms to maturity

23 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Treasury Yield Curves

24 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Theories of Term Structure of Interest Rates If future interest rate is uncertain, we need a theory of the term structure of interest rates to explain changes in bond prices and in turn in interest rates Three major theories of yield curve –Expectation Hypothesis –Liquidity Preference Hypothesis –Segmented Market Hypothesis –Combination (Synthesis)

25 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve The Expectation Hypothesis Example: Suppose that 2-year bonds offer yields to maturity of 6%, and 3-year bonds have yields of 7%. What is the expected one-period rate for the third year?

26 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Forward Rates From the current term structure of interest rates, it is possible to glean the market expectation of future interest rates. These expected future spot rates are called forward rates. With forward rates, we can estimate bond prices at a future date. Forward rates are used most often in pricing fixed income derivatives, and in particular, swap contracts.

27 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Figure 15.2 Two 2-Year Investment Programs

28 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Calculating Forward Rates from the Term Structure Consider two alternative investments: –Buy a 3-year zero-coupon bond –Buy a 2-year zero-coupon bond and then after 2 years buy another 1-year zero-coupon bond Investors will be indifferent among these two alternatives if they have the same expected return; i.e., if we end up with $100 either way, the current initial investment should be the same.

29 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Calculating Forward Rates from Term Structure (cont.) Alternative one: –The cost is Alternative two –The cost is Costs in two alternatives should equal, this implies

30 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve An Example 2-year spot rate: 8.995% 3-year spot rate: 9.660% 1-year forward rate 2 years from now is

31 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Figure 15.3 Short Rates versus Spot Rates

32 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve

33 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve

34 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Forward rate Frequently, you will see forward rate expressed as f(t,  ). This indicates a yield on a forward loan agreement struck today. The loan period is from future time t until t+ . Generally,

35 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Future Bond Prices and Forward Rates Recall that when we need to calculate the expected bond price sometime in the future, we need to know the expected spot rates at that time

36 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Expectation Hypothesis Current forward rate equals the market consensus expectation of future short term interest rate. –If the current term structure of interest rates is upward (downward) sloping, then the prediction of future spot rates is that they will be increasing (decreasing). –Example: 6 Mo. spot rate = 6.03, 1 year spot rate = 6.23 then 6-mo. forward rate in 6 mo. is Current long-term rate is a geometric mean of current and future short-term interest rates expected to prevail over the maturity. This implies that an investor is indifferent to holding either short term or long term bonds.

37 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Liquidity Preference Hypothesis Suppose an investor, who prefers liquidity, prefers short term bonds. –There are two bonds available, a one-year zero with yield 4.5%, and a two-year zero 5.0%. As she prefers the one-year bond, after one year, she reinvests in another one-year zero at the then- prevailing yield. The expected return from holding two consecutive one-year bonds is and the expected return from holding the two-year zero is –Then to entice her to hold the two-year bond, it has to be that i.e., there is a positive liquidity premium for the long bond. –This theory assumes that short-horizon investors dominate in the market? (What if long-horizon investors dominate?)

38 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Segmented Market Hypothesis Individual and institutional investors have their preferred maturities as they hedge against their liabilities – Preferred Habitat Theory Long-term bonds and short-term bonds are traded in essentially distinct or segmented markets No longer a popular view, as market participants (and arbitrageurs) invest in all segments to search for more profitable opportunities.

39 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Yield Curve. Combination of Theories Example: Suppose the liquidity preference theory predicted a 4% rate for 1-year securities and a 6% rate for 2-year securities. However, investors expected the interest rate in year 2 fall by 2%. 6% 4% Liquidity Preference Yield Curve Combined Yield Curve Maturity Yield 12

40 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve “Modern” Term Structure Theories Using mathematical models to describe the movement of interest rates directly. Assuming that one or several factors driving the interest rate movement as well as the evolution of the term structure Can be used to price interest rate derivatives, such as options on bonds, swaps, etc.

41 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Summary and Chapter Problems Basic features of bonds Bond yields, in particular, yield-to-maturity Default risk and credit spreads Spot rates and the term structure Forward rates Theories of the term structure Next class: managing bond portfolios

42 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Summary Inverse relationship between bond prices and bond yields Premium and discount bonds Corporate bonds and default risk Term structure of interest rates Expectations theory Liquidity preference theory Next Class: Managing Bond Portfolios