Lecture Notes for Finance 1 (and More).

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yield to maturity term structure of interest rates 16 CHAPTER 3. PAYMENT STREAMS UNDER CERTAINTY The interpretation of the short rate should be straightforward: Buying 1 d1 units of a maturity 1 zero coupon bond costs 1 d1 d1 = 1 at date 0 and gives a payment at date 1 of 1 d1 = 1 + r0. The forward rate tells us the rate at which we may agree at date 0 to borrow (or lend) between dates t and t+1. To see this, consider the following strategy at time 0 : • Sell 1 zero coupon bond with maturity t. • Buy dt dt+1 zero coupon bonds with maturity t + 1. Note that the amount raised by selling precisely matches the amount used for buying and hence the cash flow from this strategy at time 0 is 0. Now consider what happens if the positions are held to the maturity date of the bonds: At date t the cash flow is then −1 and at date t + 1 the cash flow is dt dt+1 = 1 + f(0, t). Definition 8 The yield (or yield to maturity) at time 0 of a zero coupon bond with maturity t is given as Note that � 1 y(0, t) = dt �1 t − 1. dt(1 + y(0, t)) t = 1. and that one may therefore think of the yield as an ’average interest rate’ earned on a zero coupon bond. In fact, the yield is a geometric average of forward rates: 1 + y(0, t) = ((1 + f(0, 0)) · · ·(1 + f(0, t − 1))) 1 t Definition 9 The term structure of interest rates (or the yield curve) at date 0 is given by (y(0, 1), . . ., y(0, T)). Note that if we have any one of the vector of yields, the vector of forward rates and the vector of discount factors, we may determine the other two. Therefore we could equally well define a term structure of forward rates and a term structure of discount factors. In these notes unless otherwise stated, we think of the term structure of interest rates as the yields of zero coupon bonds as a function of time to maturity. It is important to note that the term structure of interest rate depicts yields of zero coupon bonds. We do however also speak of yields on securities with general positive payment steams:
3.2. ZERO COUPON BONDS AND THE TERM STRUCTURE 17 Definition 10 The yield (or yield to maturity) of a security c ⊤ = (c1, . . .,cT) with c > 0 and price π is the unique solution y > −1of the equation π = T� i=1 ci (1 + y) i. Example 3 (Compounding Periods) In most of the analysis in this chapter the time is “stylized”; it is measured in some unit (which we think of and refer to as “years”) and cash-flows occur at dates {0, 1, 2, . . ., T }. But it is often convenient (and not hard) to work with dates that are not integer multiples of the fundamental time-unit. We quote interest rates in units of years−1 (“per year’), but to any interest rate there should be a number, m, associated stating how often the interest is compounded. By this we mean the following: If you invest 1 $ for n years at the m-compounded rate rm you end up with � 1 + rm m � mn . (3.1) The standard example: If you borrow 1$ in the bank, a 12% interest rate means they will add 1% to you debt each month (i.e. m = 12) and you will end up paying back 1.1268 $ after a year, while if you make a deposit, they will add 12% after a year (i.e. m = 1) and you will of course get 1.12$ back after one year. If we keep rm and n fixed in (3.1) (and then drop the m-subscript) and and let m tend to infinity, it is well known that we get: � lim 1 + m→∞ r �mn = e m nr , and in this case we will call r the continuously compounded interest rate. In other words: If you invest 1 $ and the continuously compounded rate rc for a period of length t, you will get back etrc . Note also that a continuously compounded rate rc can be used to find (uniquely for any m) rm such that 1 $ invested at m-compounding corresponds to 1 $ invested at continuous compounding, i.e. � 1 + rm m � m = e rc . This means that in order to avoid confusion – even in discrete models – there is much to be said in favor of quoting interest rates on a continuously compounded basis. But then again, in the highly stylized discrete models it would be pretty artificial, so we will not do it (rather it will always be m = 1). yield to maturity compounding periods continuously compounded interest rate
Page 1: Lecture Notes for Finance 1 (and Mo
Page 4 and 5: 4 CHAPTER 1. PREFACE
Page 6 and 7: 6 CHAPTER 2. INTRODUCTION A key rol
Page 8 and 9: 8 CHAPTER 2. INTRODUCTION will have
Page 10 and 11: 10 CHAPTER 2. INTRODUCTION
Page 12 and 13: financial market security price sys
Page 14 and 15: 14 CHAPTER 3. PAYMENT STREAMS UNDER
Page 18 and 19: annuity serial loan bullet bond ann
Page 20 and 21: ullet bond serial loan 20 CHAPTER 3
Page 26 and 27: NPV criterion 26 CHAPTER 3. PAYMENT
Page 28 and 29: Gordon’s growth formula capital b
Page 32 and 33: duration, Macaulay convexity 32 CHA
Page 40 and 41: inomial model, one-period stock mon
Page 42 and 43: 42 CHAPTER 4. ARBITRAGE PRICING IN
Page 44 and 45: iskfree asset redundant 44 CHAPTER
Page 48 and 49: utility function equilibrium state-
Page 52 and 53: call option, European 52CHAPTER 5.
Page 54 and 55: eplication hedging derivative secur
Page 56 and 57: measurable adapted process dividend
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Page 62 and 63: martingale equivalent probability m
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66CHAPTER 5. ARBITRAGE PRICING IN T
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splitting index i.e. 68CHAPTER 5. A
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70CHAPTER 5. ARBITRAGE PRICING IN T
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put-call parity frictionless market
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74 CHAPTER 6. OPTION PRICING moneyn
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put-call parity forward price forwa
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78 CHAPTER 6. OPTION PRICING Typica
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80 CHAPTER 6. OPTION PRICING p uS0
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hedging call option, European 82 CH
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84 CHAPTER 6. OPTION PRICING delta
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86 CHAPTER 6. OPTION PRICING S0�
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put option, American 88 CHAPTER 6.
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90 CHAPTER 6. OPTION PRICING In oth
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volatility smile firm debt equity 9
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94 CHAPTER 6. OPTION PRICING clear
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96 CHAPTER 7. THE BLACK-SCHOLES FOR
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Central Limit Theorem Black-Scholes
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100 CHAPTER 7. THE BLACK-SCHOLES FO
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zero coupon bond, ZCB short rate pr
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110 CHAPTER 8. STOCHASTIC INTEREST
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term structure of interest rates, p
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forward contract futures contract 1
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ate of return relative portfolio we
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mean/variance analysis Markowitz an
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minimum variance portfolio minimum
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136 CHAPTER 9. PORTFOLIO THEORY Exp
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138 CHAPTER 9. PORTFOLIO THEORY var
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capital market line, CML tangent po
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142 CHAPTER 9. PORTFOLIO THEORY in
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144 CHAPTER 9. PORTFOLIO THEORY wil
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146 CHAPTER 9. PORTFOLIO THEORY Rol
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148 CHAPTER 9. PORTFOLIO THEORY tha
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150 CHAPTER 9. PORTFOLIO THEORY in
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152 CHAPTER 9. PORTFOLIO THEORY
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portfolio arbitrage opportunity APT
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156CHAPTER 10. FACTOR MODELS OF RET
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Arrow-Debreu prices debt equity bon
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166CHAPTER 11. CORPORATE FINANCE: F
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ankruptcy costs NPV criterion 168CH
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174 CHAPTER 12. EFFICIENT CAPITAL M
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Index (·) + -notation, 41 geometri
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188 INDEX stock holders, 164 swap c
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