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# Fundamentals Of Bond Valuation

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### Fundamentals Of Bond Valuation

1. 1. FUNDAMENTALS OF BOND VALUATION 1
2. 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 2
3. 3. Bond C coupon pays \$50/year; matures in two years 3
4. 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? 4
5. 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 5
6. 6. YIELD TO MATURITY • CALCULATING YTM: – BOND A – Solving for rA (1 + rA) x \$934.58 = \$1000 rA = 7% 6
7. 7. YIELD TO MATURITY • CALCULATING YTM: – BOND B – Solving for rB (1 + rB) x \$857.34 = \$1000 rB = 8% 7
8. 8. YIELD TO MATURITY • CALCULATING YTM: – BOND C – Solving for rC (1 + rC)+{[(1+ rC)x\$946.93]-\$50 = \$1000 rC = 7.975% 8
9. 9. SPOT RATE • DEFINITION: Measured at a given point in time as the YTM on a pure discount security 9
10. 10. SPOT RATE • SPOT RATE EQUATION: Mt Pt 1 st where Pt = the current market price of a pure discount bond maturing in t years; Mt = the maturity value st = the spot rate 10
11. 11. DISCOUNT FACTORS • EQUATION: Let dt = the discount factor 1 dt 1 st 11
12. 12. DISCOUNT FACTORS • EVALUATING A RISK FREE BOND: – EQUATION n PV d t ct t 1 where ct = the promised cash payments n = the number of payments 12
13. 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 13
14. 14. FORWARD RATE • EXAMPLE OF A FORWARD RATE Let us assume that \$1 paid in one year at a spot rate of 7% has 1 PV \$.9346 1.07 14
15. 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 1 (1 f1, 2 ) PV \$.8573 (1 .07) f1, 2 9.01% 15
16. 16. FORWARD RATE f1,2 is the forward rate from year 1 to year 2 16
17. 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 \$1 1 f1, 2 \$1 (1 s1 ) (1 s2 ) 2 17
18. 18. FORWARD RATE such that (1 s1 ) 1 f1, 2 (1 s2 ) or 2 (1 s1 )(1 f1, 2 ) (1 s2 ) 18
19. 19. FORWARD RATE • More generally for the link between years t- 1 and t: t (1 st ) (1 f1, 2 ) t 1 (1 st ,1 ) • or t 1 t (1 st 1 ) (1 ft 1,t ) (1 st ) 19
20. 20. FORWARD RATES AND DISCOUNT FACTORS • ASSUMPTION: – given a set of spot rates, it is possible to determine a market discount function – equation 1 dt (1 st 1 )t 1 (1 ft 1,t ) 20
21. 21. YIELD CURVES • DEFINITION: a graph that shows the YTM for Treasury securities of various terms (maturities) on a particular date 21
22. 22. YIELD CURVES • TREASURY SECURITIES PRICES – priced in accord with the existing set of spot rates and – associated discount factors 22
23. 23. YIELD CURVES • SPOT RATES FOR TREASURIES – One year is less than two year; – Two year is less than three-year, etc. 23
24. 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 24
25. 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 25
26. 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. 26
27. 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 27
28. 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 28
29. 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 29
30. 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 30
31. 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 31
32. 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 32
33. 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 33
34. 34. TERM STRUCTURE THEORY: Liquidity Preference • BASIC NOTION OF THE THEORY – investors primarily interested in purchasing short-term securities to reduce interest rate risk 34
35. 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 35
36. 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 36
37. 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 37
38. 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 ) 38
39. 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 39
40. 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 40
41. 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 41
42. 42. TERM STRUCTURE THEORY: Liquidity Preference • SHAPES OF THE YIELD CURVE: – a flat yield curve means the market expects interest rates to decline 42
43. 43. TERM STRUCTURE THEORY: Liquidity Preference • SHAPES OF THE YIELD CURVE: – an upward-sloping curve means rates are expected to increase 43
44. 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 44
45. 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 45
46. 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 46
47. 47. TERM STRUCTURE THEORY: Preferred Habitat – When significant differences in yields exist between market segments, investors are willing to leave their desired maturity segment 47
48. 48. TERM STRUCTURE THEORY: Preferred Habitat – Yield differences determined by the supply and demand conditions within the segment 48
49. 49. TERM STRUCTURE THEORY: Preferred Habitat – This theory reflects both • expectations of future spot rates • expectations of a liquidity premium 49