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CHAPTER TWENTY

CHAPTER TWENTY. FUNDAMENTALS OF BOND VALUATION. YIELD TO MATURITY. CALCULATING YIELD TO MATURITY EXAMPLE Imagine three risk-free returns based on three Treasury bonds: Bond A,B are pure discount types; mature in one year . Bond C coupon pays $50/year;. matures in two years.

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CHAPTER TWENTY

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  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,B are pure discount types; mature in one year

  3. Bond C coupon pays $50/year; matures in two years

  4. YIELD TO MATURITY Bond Market Prices: Bond A $934.58 Bond B $857.34 Bond C $946.93 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 rA (1 + rA) x $934.58 = $1000 rA = 7%

  7. YIELD TO MATURITY • CALCULATING YTM: • BOND B • Solving for rB (1 + rB) x $857.34 = $1000 rB = 8%

  8. YIELD TO MATURITY • CALCULATING YTM: • BOND C • Solving for rC (1 + rC)+{[(1+ rC)x$946.93]-$50 = $1000 rC = 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 Pt = the current market price of a pure discount bond maturing in t years; Mt = the maturity value st = the spot rate

  11. DISCOUNT FACTORS • EQUATION: Let dt = the discount factor

  12. DISCOUNT FACTORS • EVALUATING A RISK FREE BOND: • EQUATION where ct = 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 f1,2 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 es1,2 = f1,2 where es1,2 = the expected future spot f1,2 = 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 = es1,2 -f1,2 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 + s1 ) (1 + es1,2 )

  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 + s2 )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+s1)(1 +es1,2)<$1(1 + s2)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

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