1 CHAPTER TWENTY FUNDAMENTALS OF BOND VALUATION

2 YIELD TO MATURITY CALCULATING YIELD TO MATURITY EXAMPLE –Imagine three risk-free returns based on three Treasury bonds: Bond A,Bare pure discount types; mature in one year

3 Bond Ccoupon pays $50/year; matures in two years

4 YIELD TO MATURITY Bond Market Prices: Bond A$ Bond B$ Bond C$ WHAT IS THE YIELD-TO-MATURITY OF THE THREE BONDS?

5 YIELD TO MATURITY YIELD-TO-MATURITY (YTM) –Definition: the single interest rate* that would enable investor to obtain all payments promised by the security. –very similar to the internal rate of return (IRR) measure * with interest compounded at some specified interval

6 YIELD TO MATURITY CALCULATING YTM: –BOND A –Solving for r A (1 + r A ) x $ = $1000 r A = 7%

7 YIELD TO MATURITY CALCULATING YTM: –BOND B –Solving for r B (1 + r B ) x $ = $1000 r B = 8%

8 YIELD TO MATURITY CALCULATING YTM: –BOND C –Solving for r C (1 + r C )+{[(1+ r C )x$946.93]-$50 = $1000 r C = 7.975%

9 SPOT RATE DEFINITION: Measured at a given point in time as the YTM on a pure discount security

10 SPOT RATE SPOT RATE EQUATION: where P t = the current market price of a pure discount bond maturing in t years; M t = the maturity value s t = the spot rate

11 DISCOUNT FACTORS EQUATION: Let d t = the discount factor

12 DISCOUNT FACTORS EVALUATING A RISK FREE BOND: –EQUATION where c t = the promised cash payments n = the number of payments

13 FORWARD RATE DEFINITION: the interest rate today that will be paid on money to be –borrowed at some specific future date and –to be repaid at a specific more distant future date

14 FORWARD RATE EXAMPLE OF A FORWARD RATE Let us assume that $1 paid in one year at a spot rate of 7% has

15 FORWARD RATE EXAMPLE OF A FORWARD RATE Let us assume that $1 paid in two years at a spot rate of 7% has a

16 FORWARD RATE f 1,2 is the forward rate from year 1 to year 2

17 FORWARD RATE To show the link between the spot rate in year 1 and the spot rate in year 2 and the forward rate from year 1 to year 2

18 FORWARD RATE such that or

19 FORWARD RATE More generally for the link between years t- 1 and t: or

20 FORWARD RATES AND DISCOUNT FACTORS ASSUMPTION: –given a set of spot rates, it is possible to determine a market discount function –equation

21 YIELD CURVES DEFINITION: a graph that shows the YTM for Treasury securities of various terms (maturities) on a particular date

22 YIELD CURVES TREASURY SECURITIES PRICES –priced in accord with the existing set of spot rates and –associated discount factors

23 YIELD CURVES SPOT RATES FOR TREASURIES –One year is less than two year; –Two year is less than three-year, etc.

24 YIELD CURVES YIELD CURVES AND TERM STRUCTURE –yield curve provides an estimate of the current TERM STRUCTURE OF INTEREST RATES yields change daily as YTM changes

25 TERM STRUCTURE THEORIES THE FOUR THEORIES 1.THE UNBIASED EXPECTATION THEORY 2. THE LIQUIDITY PREFERENCE THEORY 3. MARKET SEGMENTATION THEORY 4. PREFERRED HABITAT THEORY

26 TERM STRUCTURE THEORIES THEORY 1: UNBIASED EXPECTATIONS –Basic Theory: the forward rate represents the average opinion of the expected future spot rate for the period in question –in other words, the forward rate is an unbiased estimate of the future spot rate.

27 TERM STRUCTURE THEORY: Unbiased Expectations THEORY 1: UNBIASED EXPECTATIONS –A Set of Rising Spot Rates the market believes spot rates will rise in the future –the expected future spot rate equals the forward rate –in equilibrium es 1,2 = f 1,2 where es 1,2 = the expected future spot f 1,2 = the forward rate

28 TERM STRUCTURE THEORY: Unbiased Expectations THE THEORY STATES: –The longer the term, the higher the spot rate, and –If investors expect higher rates, then the yield curve is upward sloping and vice-versa

29 TERM STRUCTURE THEORY: Unbiased Expectations CHANGING SPOT RATES AND INFLATION –Why do investors expect rates to rise or fall in the future? spot rates = nominal rates –because we know that the nominal rate is the real rate plus the expected rate of inflation

30 TERM STRUCTURE THEORY: Unbiased Expectations CHANGING SPOT RATES AND INFLATION –Why do investors expect rates to rise or fall in the future? if either the spot or the nominal rate is expected to change in the future, the spot rate will change

31 TERM STRUCTURE THEORY: Unbiased Expectations CHANGING SPOT RATES AND INFLATION –Why do investors expect rates to rise or fall in the future? if either the spot or the nominal rate is expected to change in the future, the spot rate will change

32 TERM STRUCTURE THEORY: Unbiased Expectations –Current conditions influence the shape of the yield curve, such that if deflation expected, the term structure and yield curve are downward sloping if inflation expected, the term structure and yield curve are upward sloping

33 TERM STRUCTURE THEORY: Unbiased Expectations PROBLEMS WITH THIS THEORY: –upward-sloping yield curves occur more frequently –the majority of the time, investors expect spot rates to rise –not realistic position

34 TERM STRUCTURE THEORY: Liquidity Preference BASIC NOTION OF THE THEORY –investors primarily interested in purchasing short-term securities to reduce interest rate risk

35 TERM STRUCTURE THEORY: Liquidity Preference BASIC NOTION OF THE THEORY –Price Risk maturity strategy is more risky than a rollover strategy to convince investors to buy longer-term securities, borrowers must pay a risk premium to the investor

36 TERM STRUCTURE THEORY: Liquidity Preference BASIC NOTION OF THE THEORY –Liquidity Premium DEFINITION: the difference between the forward rate and the expected future rate

37 TERM STRUCTURE THEORY: Liquidity Preference BASIC NOTION OF THE THEORY –Liquidity Premium Equation L = es 1,2 - f 1,2 where L is the liquidity premium

38 TERM STRUCTURE THEORY: Liquidity Preference How does this theory explain the shape of the yield curve? –rollover strategy at the end of 2 years $1 has an expected value of $1 x (1 + s 1 ) (1 + es 1,2 )

39 TERM STRUCTURE THEORY: Liquidity Preference How does this theory explain the shape of the yield curve? –whereas a maturity strategy holds that $1 x (1 + s 2 ) 2 –which implies with a maturity strategy, you must have a higher rate of return

40 TERM STRUCTURE THEORY: Liquidity Preference How does this theory explain the shape of the yield curve? –Key Idea to the theory: The Inequality holds $1(1+s 1 )(1 +es 1,2 )<$1(1 + s 2 ) 2

41 TERM STRUCTURE THEORY: Liquidity Preference SHAPES OF THE YIELD CURVE: –a downward-sloping curve means the market believes interest rates are going to decline

42 TERM STRUCTURE THEORY: Liquidity Preference SHAPES OF THE YIELD CURVE: –a flat yield curve means the market expects interest rates to decline

43 TERM STRUCTURE THEORY: Liquidity Preference SHAPES OF THE YIELD CURVE: –an upward-sloping curve means rates are expected to increase

44 TERM STRUCTURE THEORY: Market Segmentation BASIC NOTION OF THE THEORY –various investors and borrowers are restricted by law, preference or custom to certain securities

45 TERM STRUCTURE THEORY: Liquidity Preference WHAT EXPLAINS THE SHAPE OF THE YIELD CURVE? –Upward-sloping curves mean that supply and demand intersect for short-term is at a lower rate than longer-term funds –cause: relatively greater demand for longer- term funds or a relative greater supply of shorter-term funds

46 TERM STRUCTURE THEORY: Preferred Habitat BASIC NOTION OF THE THEORY: –Investors and borrowers have segments of the market in which they prefer to operate

47 TERM STRUCTURE THEORY: Preferred Habitat –When significant differences in yields exist between market segments, investors are willing to leave their desired maturity segment

48 TERM STRUCTURE THEORY: Preferred Habitat –Yield differences determined by the supply and demand conditions within the segment

49 TERM STRUCTURE THEORY: Preferred Habitat –This theory reflects both expectations of future spot rates expectations of a liquidity premium