Linearity of Pricing and the Certainty Equivalent Form
We now discuss a very important property of the pricing formula namely, that it is linear. This means that the price of the sum of two assets is the sum of their prices, and the price of a multiple of an asset is the same multiple of the price. This is really quite startling because the formula does not look linear at all at least for sums . For example, if 1 rf Pi Fm - rf ' 1 ij P20'm if it does not seem obvious that where f l 2 is the beta of a new asset, which is the sum of assets 1 and 2,...
Capm As A Pricing Formula
The CAPM is a pricing model. However, the standard CAPM formula does not contain prices explicitly only expected rates of return To see why the CAPM is called a pricing model we must go back to the definition of return Suppose that an asset is purchased at price P and later sold at price Q The rate of return is then r Q P P. Flere P is known and Q is random. Putting this in the CAPM formula, we have This gives the price of the asset according to the CAPM We highlight this important result...
Savings Deposits
Probably the most familiar fixed-income instrument is an interest-bearing bank deposit. These are offered by commercial banks, savings and loan institutions, and credit unions In the United States most such deposits are guaranteed by agencies of the federal government, The simplest demand deposit pays a rate of interest that varies with market conditions. Over an extended period of time, such a deposit is not strictly of a fixed-income type nevertheless, we place it in the fixed-income category...
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...
Bond Details
Bonds represent by far the greatest monetary value of fixed-income securities and are, as a class, the most liquid of these securities. We devote special attention to bonds, both because of their practical importance as investment vehicles and because of their theoretical value, which will be exploited heavily in Chapter 4 We describe the general structure and trading mechanics of bonds in this section and then discuss in the following few sections some methods by which bonds are analyzed Our...
Tight Markets
At any one time it is possible to define several different forward contracts on a given commodity, each contract having a different delivery date If the commodity is a physical commodity such as soybean meal, the preceding theory implies that the forward prices of these various contracts will increase smoothly as the delivery date is increased because the value of F in 10,2 increases with M In fact, however, this is frequently not the case. Consider, for example, the prices for soybean...
Explicit Formula
In the case where all coupon payments are identical which is the normal case for bonds there is an explicit formula for the sum of the series that appears in the numerator of the expression for the Macaulay duration We skip the algebra here and just give the result. Macaulay duration formula The Macaulay duration for a bond with a coupon rate c per period, vie Id y pet period, m periods per year, and exactly n periods remaining, is Example 3.7 Duration of a 30-year par bond Consider the 10 ,...
Hedging Nonlinear Risk
In our examples so far the risk being hedged was linear, in the sense that final wealth v was a linear function of an underlying market variable, such as a commodity price, The general theory of hedging does not depend on this assumption, and indeed nonlinear risks frequently occur For example, immunization of a bond portfolio with T-bills see Exercise 15 is a nonlinear hedging problem because the change in the value of a bond portfolio is a nonlinear function of the future T-bill price....
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...
The Capital Market Line
Given the preceding conclusion that the single efficient fund of risky assets is the market portfolio, we can label this fund on the T a diagram with an M for market. The efficient set therefore consists of a single straight line, emanating from the risk-free point and passing through the market portfolio. This line, shown in Figure 7 1, is called the capital market line. This line shows the relation between the expected rate of return and the risk of return as measured by the standard...
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 ....
Random Walks And Wiener Processes
in Section 11,7 we will shorten the period length in a multiplicative model and take the limit as this length goes to zero. This will produce a model in continuous time, In preparation for that step, we introduce special random functions of time, called random walks and Wiener processes Suppose that we have N periods of length At We define the additive process ' - z tk tk V t k tk A for k 0, 1,2,. ,. , N This process is termed a random walk. In these equations tk is a normal random variable...
Exercises Ndk
Gold futures The current price of gold is 412 per ounce The storage cost is 2 per ounce per year, payable quarterly in advance. Assuming a constant interest rate of 9 compounded quarterly, what is the theoretical forward price of gold for delivery in 9 months 2. Proportional carrying charges o Suppose that a forward contract on an asset is written at time zero and there are M periods until delivery Suppose that the carrying charge in period k is qS k , where S k is the spot price of the asset...
Cycle Problems
When using interest rate theory to evaluate ongoing repeatable activities, it is essential that alternatives be compared over the same time horizon, The difficulties that can arise from not doing this are illustrated in the tree cutting example The two alternatives in that example have different cycle lengths, but the nature of the possible repetition of the cycles was not clearly spelled out originally We illustrate here two ways to account properly for different cycle lengths The first is to...
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 - ffl cft 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...
Summary Bgc
Interest rate theory is probably the most widely used financial tool. It is used to determine the value of projects, to allocate money among alternatives, to design complex bond portfolios, to determine how to manage investments effectively, and even to determine the value of a firm Interest rate theory is most powerful when it is combined with general problem-solving methods, particularly methods of optimization. With the aid of such methods, interest rate theory provides more than just a...
Cfi
procedure, either by recording them at the nodes as the V-values are computed, or by working forward, using the known future V-values. The running dynamic programming method can be written very succinctly by a recurrence relation Define cki to be the cash flow generated by moving from node kt 0 to node k 4- 1, a . The recursion procedure is Vki - maximize dkVk 1 An example will make all of this clear Example 5.4 Fishing problem Suppose that you own both a lake and a fishing boat as an...
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...
Summary Kmk
The study of one-period investment situations is based on asset and portfolio returns Both total returns and rates of return are used The return of an asset may be uncertain, in which case it is useful to consider it formally as a random variable. The probabilistic properties of such random returns can be summarized by their expected values, their variances, and their covariances with each other A portfolio is defined by allocating fractions of initial wealth to individual assets, The fractions...
Risk Aversion
Another principle of investment science is risk aversion. Suppose two possible investments have the same cost, and both are expected to return the same amount somewhat greater than the initial cost , where the term expected is defined in a probabilistic sense explained in Chapter 6 However, the return is certain for one of these investments and uncertain for the other. Individuals seeking investment rather than outright speculation will elect the first certain alternative over the second risky...
Project Choice
A firm can use the CAPM as a basis for deciding which projects it should carry out. Suppose, for example, that a potential project requires an initial outlay of P and will generate a net amount Q after I year As usual, P is known and Q is random, with expected value Q It is natural to define the net present value NPV of this project by the formula This formula is based on the certainty equivalent form of the CAPM the first negative term is the initial outlay and the second term is the certainty...
Summary Xut
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...
Systematic Risk
The CAPM implies a special structural property for the return of an asset, and this property provides further insight as to why beta is the most important measure of risk, To develop this result we write the random rate of return of asset as FIGURE 73 Security market line. The expected rate of return increases linearly as the covariance with the market increases or, equivalently, as 3 increases This is just an arbitrary equation at this point, The random variable j is chosen to make it true,...
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'0, .v,, .v2, , .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...
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...
Deterministic Cash Flow Streams
The simplest cash flow streams are those that are deterministic that is, not random, but definite . The first part of the book treats these. Such cash flows can be represented by sequences such as 1, 0, 3 , as discussed earlier. Investments of this type, either with one or with several periods, are analyzed mainly with various concepts of interest rate. Accordingly, interest rate theory is emphasized in this fust part of the book This theory provides a basis for a fairly deep understanding of...