Option Concepts
The specifications of an option include, first, a clear description of what can be bought for a call or sold for a put . For options on stock, each option is usually for 100 shares of a specified stock. Thus a call option on IBM is the option to buy 100 shares of IBM. Second, the exercise price, or strike price, must be specified. This is the price at which the asset can be purchased upon exercise of the option. For IBM stock the exercise price might be 70, which means that each share can be...
Summary Asb
If everybody uses the mean-variance approach to investing, and if everybody has the same estimates of the asset's expected returns, variances, and covariances, then everybody must invest in the same fund F of risky assets and in the risk-free asset Because F is the same for everybody, it follows that, in equilibrium, F must correspond to the market portfolio M the portfolio in which each asset is weighted by its proportion of total market capitalization. This observation is the basis for the...
Exercises 1
1. A simple portfolio Someone who believes that the collection of all stocks satisfies a single-factor model with the market portfolio serving as the factor gives you information on three stocks which make up a portfolio See Table 8 6 in addition, you know that the market portfolio has an expected rate of return of 12 and a standard deviation of 18 The risk-free rate is 5 a What is the portfolio's expected rate of return b Assuming the factor model is accurate, what is the standard deviation of...
FisherWeil Duration
The details work out most nicely for the case of continuous compounding, and we shall present that case first. Given a cash flow sequence v, 1,,.vf , vf,____,x,n and the FIGURE 4.3 Shifted spot rate curves. The original spot rate curve is the middle curve. This curve is shifted upward and downward by an amount k to obtain the other curves It is possible to immunize a portfolio against such shifts for small values of X spot rate curve s,, r0 lt lt , the present value is The Fisher-Weil duration...
Immunization 1
The term structure of interest rates leads directly to a new, more robust method for portfolio immunization. This new method does not depend on selecting bonds with a common yield, as in Chapter .3 indeed, yield does not even enter the calculations, The process is best explained through an example. Example 4.8 A million dollar obligation Suppose that we have a 1 million obligation payable at the end of 5 years, and we wish to invest enough money today to meet this future obligation We wish to...
Quality Ratings
Although bonds offer a supposedly fixed-income stream, they are subject to default if the issuer has financial difficulties or falls into bankruptcy. To characterize the nature of this risk, bonds are rated by rating organizations The two primary rating classifications are issued and published by Moody's and Standard amp Poor's. Their classification schemes are shown in Table 3.4 US Treasury securities are not rated, since they are considered to be essentially free of default risk. Ratings...
Info Srq
This problem cannot be reduced to the solution of a set of linear equations It is termed a quadratic program, since the objective is quadratic and the constraints are linear equalities and inequalities. Special computer programs are available for solving such problems, but small to moderate-sized problems of this type can be solved readily with spreadsheet programs. In the financial industry there are a multitude of special-purpose programs designed to solve this problem for hundreds or even...
Type B Arbitrage
Another form of arbitrage can be identified If an investment has nonpositive cost but has a positive probability of yielding a positive payoff and no probability of yielding a negative payoff, that investment is said to be a type B arbitrage. In other words, a type B arbitrage is a situation where an individual pays nothing or a negative amount and has a chance of getting something An example would be a free lottery ticket you pay nothing for the ticket, but have a chance of winning a prize...
Determining the Spot Rate
The obvious way to determine a spot rate curve is to find the prices of a series of zero-coupon bonds with various maturity dates. Unfortunately the set of available zero-coupon bonds is typically rather sparse, and, indeed, until recently there were essentially no zeros available with long maturities. Thus it is not always practical to determine a complete set of spot rates this way However, the existence of zero-coupon bonds is not necessary for the concept of spot rates to be useful, nor are...
Discount Factors
Another important concept is that of a discount factor between two times The discount factors are, of course, fundamental quantities used in present value calculations, it is useful to apply a double indexing system to the discount factors paralleling the system used for forward rates Accordingly, the symbol dj k denotes the discount factor used to discount cash received at time k back to an equivalent amount of cash at time j The normal, time zero, discount factors are d do i, d2 do 2, . ., d...
yflSS
To use the constant-growth dividend model one must estimate the growth rate g and assign an appropriate value to the discount rate Estimation of g can be based on the history of the firm's dividends and on future prospects, Frequently a value is assigned to r that is larger than the actual risk-free interest rate to reflect the idea that uncertain cash flows should be discounted more heavily than certain cash flows In Chapters 15 and 16, we study better ways to account for uncertainty. Example...
Interdependent Projects
Sometimes various projects are interdependent, the feasibility of one being dependent on whether others are undertaken We formulate a problem of this type by assuming that there are several independent goals, but each goal has more than one possible method of implementation It is these implementation alternatives that define the projects. This formulation generalizes the problems studied in Chapter 2, where there was only one goal such as buying a new car but several ways to achieve that goal...
Duration and Sensitivity
Duration is useful because it measures directly the sensitivity of price to changes in yield. This follows from a simple expression for the derivative of the present value expression. In the case where payments are made m times per year and yield is based on those same periods, we have dPV - lt k m dA 1 V 0 1 i A m We now apply this to the expression for price, Here we have used the fact that the price is equal to the total present value at the yield by definition of yield . We find that dPV,...
Exercises
lr Amortization A debt of 25,000 is to be amortized over 7 years at 1 interest What value of monthly payments will achieve this 2. Cycles and annual worth o Given a cash flow stream X a'q, a,, a2, , v , a new stream of infinite length is made by successively repeating the corresponding finite stream The interest rate is Let P and A be the present value and the annual worth, respectively, of stream X Finally, let P be the present value of stream Find A in terms of and conclude that A can be used...
Mean Blur
We now show how this amplification effect makes the estimation of expected or mean rates nearly impossible. Let us select a basic period length p such as p 1 12 for a monthly period . We shall try to estimate the mean rate of return for this period That is, we assume that the statistical properties of the returns in each of the periods are identical, with mean value F and standard deviation a. We also assume that the individual returns are mutually uncorrected We wish to estimate the common...
Summary 1
if observed yield is plotted as a function of time to maturity for a variety of bonds within a fixed risk class, the result is a scatter of points that can be approximated by a curve the yield curve This curve typically rises gradually with increasing maturity, reflecting the fact that long maturity bonds typically offer higher yields than short maturity bonds. The shape of the yield curve varies continually, and occasionally it may take on an inverted shaped, where yields decrease as the time...
Qualitative Properties of Duration
The duration of a coupon-paying bond is always less than its maturity, but often it is surprisingly short An appreciation for the relation between a bond's duration and other parameters ol the bond can be obtained by examination of Table 3 6. In this table the yield is held fixed at 5 , but various maturities and coupon rates are considered. This procedure approximates the situation of looking through a list of available bonds at a time when all yields hover near 5 . Within a given class say,...
Inflation
Inflation is another factor that often causes confusion, arising from the choice between using actual dollar values to describe cash flows and using values expressed in purchasing power, determined by reducing inflated future dollar values back to a nominal level Inflation is characterized by an increase in general prices with time, inflation can be described quantitatively in terms of an inflation rate . Prices 1 year from now will on average be equal to today's prices multiplied by 1...