## Binomial and Trinomial Lattices

A popular method for finding the value of a derivative security is the binomial lattice method of Section 12.6 The method is straightfor ward and leads to reasonably accurate results, even if the time divisions are crude (say, 10 or so time periods over the remaining time interval) However, it is also possible to use other tree and lattice structures. For example, a good choice is to use a trinomial lattice, as shown in Figure 1.3 5. For a given number of time periods, the trinomial lattice has...

## Quadratic Utility

The quadratic utility function can be defined as U(x) ax bx2, where a > 0 and b > 0. This function is shown in Figure 9.6. This utility function is really meaningful only in the range ,v < a b, for it is in this range that the function is increasing. Note also that for b > 0 the function is strictly concave everywhere and thus exhibits risk aversion We assume that all random variables of interest lie in the feasible range v < a b that is, within the meaningful range of the quadratic...

## Spot Rate Forecasts

The basis of this method is to assume that the expectations implied by the current spot rate curve will actually be fulfilled, Under this assumption we can then predict next year's spot rate curve from the current one This new curve implies yet another set of expectations for the following year If we assume that these, too, are fulfilled, we can predict ahead once again Going forward in this way, an entire future of spot rate curves can be predicted. Of course, it is understood that these...

## Discount Factors and Present Value

Once the spot rates have been deter mined, it is natural to define the corresponding discount factors d, for each time point. These are the factors by which future cash flows FIGURE 4.2 Spot rate curve, The yearly rate of interest depends on the length of time funds are heicl FIGURE 4.2 Spot rate curve, The yearly rate of interest depends on the length of time funds are heicl must be multiplied to obtain an equivalent present value. For the various compounding conventions, they are defined as...

## Pricing In A Double Lattice

The double lattice construction does provide a valid representation of the two assets, but there is a problem When a risk-free asset is adjoined, we have four nodes, but only three assets the two risky assets and the risk-free asset There is an extra degree of freedom Therefore the risk-neutral probabilities are not completely specified as they are in the two original small lattices. We must find a way to pin down that extra degree of freedom in the definition of the risk-neutral probabilities,...

## Random Variables

Frequently the amount of money to be obtained when selling an asset is uncertain at the time of'purchase. In that case the return is random and can be described in probabilistic terms In preparation for the study of random returns, we briefly introduce some concepts of probability. (For more detail on basic probability theory, see Appendix A.) Suppose x is a random quantity that can take on any one of a finite number of specific values, say, -vj, xo, , -vm. Assume further that associated with...

## Fisher Weil 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 < < , the present value is The Fisher-Weil...

## Finite Life Streams

Of more practical importance is the case where the payment stream has a finite lifetime Suppose that the stream consists of n periodic payments of amount A, starting at the end of the current period and ending at period n. The pattern of periodic cash flows together with the time indexing system is shown in Figure 3 1 The present value of the finite stream relative to the interest rate r per period is FIGURE 3.1 Time indexing., Time is indexed from 0 lo n. A period is a span between time...

## Costs of Carry

The preceding analysis assumed that there are no storage costs associated with holding the underlying asset This is not always the case Holding a physical asset such as gold entails storage costs, such as vault rental and insurance fees Holding a security may, alternatively, entail negative costs, representing dividend or coupon payments These costs (or incomes) affect the theoretical forward price. We shall use a discrete-time (multiperiod) model to describe this situation The delivery date T...

## 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...

## Replication Synthetic Options And Portfolio Insurance

The derivation of the Black-Scholes equation shows that a derivative security can be duplicated by constructing a portfolio consisting of an appropriate combination of the underlying security and the risk-free asset We say that this portfolio replicates the derivative security, The proportions of stock and the risk-free asset in the portfolio must be adjusted continuously with time, but no additional money need be added or taken away the portfolio is self-financing. This replication can be...

## Utility Functions And Incremental Investment Is Independent Of Initial Wealth

This chapter is devoted to general theory, and hence it is somewhat more abstract than other chapters, but the tools presented are quite powerful The chapter should be reviewed after reading Part 3 and again after reading Part 4 The first part of the chapter presents the basics of expected utility theory Utility functions account for risk aversion in financial decision making, and provide a more general and more useful approach than does the mean-variance framework. In this new approach, an...

## Market Equilibrium

Suppose that everyone is a mean-variance optimizer as described in the previous chapter. Suppose further that everyone agrees on the probabilistic structure of assets that is, everyone assigns to the returns of assets the same mean values, the same variances, and the same covariances Furthermore, assume that there is a unique risk-free rate of borrowing and lending that is available to all, and that there are no transactions costs. With these assumptions what will happen From the one-fund...

## The Characteristic Line

Let us hypothesize a single-factor model for stock returns, with the factor being the market rate of return a . For convenience we can subtract the constant j from this factor and also from the rate of return thereby expressing the model in terms of the excess returns tf and i m rj. The factor model then becomes n - if a, -f - rf eh 8 5 It is conventional to use the notation a, and for the coefficients of this special model, rather than the 's and bf s that are being used more generally. Again...

## Qualitative Nature of Price Yield Curves

Would not buy a bond with a yield of 6 when bank CDs are offering 10 , The general interest rate environment exerts a force on every bond, urging its yield to conform to that of other bonds. However, the only way that the yield of a bond can change is for the bond's price to change, So as yields move, prices move correspondingly But the price change required to match a yield change varies with the structure of the bond its coupon rate and its maturity . So as the yields of various bonds move...

## The Pricing Model

The capital market line relates the expected rate of return of an efficient portfolio to its standard deviation, but it does not show how the expected rate of return of an individual asset relates to its individual risk This relation is expressed by the capital asset pricing model. We state this major result as a theorem The reader may wish merely to glance over the proof at first reading since it is a bit involved. We shall discuss the implications of the result following the proof. The...

## Evaluating Real Investment Opportunities

Options theory can be used to evaluate investment opportunities that are not pure financial instruments. We shall illustrate this by again considering our gold mine lease problems Now, however, the pr ice of gold is assumed to fluctuate randomly, and this fluctuation must be accounted for in our evaluation of the lease prospect. Example 12.7 Simplico gold mine Recall the Simplico gold mine from Chapter 2 Gold can be extracted from this mine at a rate of up to 10,000 ounces per year at a cost of...

## Running Present Value

The present value of a cash flow stream is easily calculated in the term structure framework. One simply multiplies each cash flow by the discount factor associated with the period of the flow and then sums these discounted values that is, present value is obtained by appropriately discounting all future cash flows. There is a special, alternative way to arrange the calculations of present value, which is sometimes quite convenient and which has a useful interpretation This different way is...

## Exercises

I Bull spread An investor who is bullish about a stock believing that it will rise may wish to construct a bull spread for that stock One way to construct such a spread is to buy a call with strike price K and sell a call with the same expiration date but with a strike price of AS gt K Draw the payoff curve for such a spread Is the initial cost of the spread positive or negative 2. Put-call parity Suppose over the period fO, 7j a certain stock pays a dividend whose present value at interest...

## The Cash Matching Problem

A simple optimal portfolio problem is the cash matching problem. To describe this problem, suppose that we face a known sequence of future monetary obligations If we manage a pension fund, these obligations might represent required annuity payments We wish to invest now so that these obligations can be met as they occur and accordingly, we plan to purchase bonds of various maturities and use the coupon payments and redemption values to meet the obligations The simplest approach is to design a...

## The Twofund Theorem

The minimum-variance set has an important property that greatly simplifies its computation. Recall that points in this set satisfy the system of n 2 linear equations Eqs 6.5fi c , which is repeated here CFijWj X7j i 0 for i 1,2, ,, 6 8a Suppose that there are two known solutions, wl ji J, . , w , A.1, x and w2 117, w 2, . , uj , A.2, x2, with expected rates of return 7 and F2, respectively, Let us form a combination by multiplying the first by a and the second by 1 a . By direct substitution,...

## The Harmony Theorem

We know that there is a difference between the present value criterion for selecting investment opportunities and the internal rate of return criterion, and that it is strongly believed by theorists that the present value criterion is the better of the two, provided that account is made for the entire cash flow stream of the investment over all its periods But if you are asked to consider an investment of a fixed amount of dollars say, in your friend's new venture , you probably would not...

## The Security Market Line

The CAPM formula can be expressed in graphical form by regarding the formula as a linear relationship. This relationship is termed the security market line. Two versions are shown in Figure 7 3 Both graphs show the linear variation of r The first expresses it in covariance form, with cov , gt being the horizontal axis. The market portfolio corresponds to the point a on this axis. The second graph shows the relation in beta form, with beta being the horizontal axis In this case the market...

## Real Options

Sometimes options are associated with investment opportunities that are not financial instruments For example, when operating a factory, a manager may have the option of hiring additional employees or buying new equipment. As another example, if one acquires a piece of land, one has the option to drill for oil, and then later the option of extracting oil if oil is found In fact, it is possible to view almost any process that allows control as a process with a series of operational options These...

## R ujji I u22 f wnr

E r ui E n U gt 2E '2 - w E ,, This important result shows how the variance of a portfolio's return can be calculated easily from the covariances of the pairs of asset returns and the asset weights used in the portfolio Recall, an af Example 6-8 Two-asset portfolio Suppose that there are two assets with F 12, 2 J5, lt 7 20, lt r2 .18, and an 01 values typical for two stocks A portfolio is formed with weights tui .25 and ui .75, We can calculate the mean and the variance of the portfolio First...

## Option Combinations And Putcall Parity

It is common to invest in combinations of options in order to implement special hedging or speculative strategies. The payoff curve of such a combination may have any number of connected straight-line segments This overall payoff curve is formed by combining the payoff functions defined by calls, puts, and the underlying stock itself The process is best illustrated by an example and a corresponding graph, Example 12.1 A butterfly spread One of the most interesting combinations of options is the...

## 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

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

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...

## Utility Function Exercises

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...

## 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

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...

## 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...

## Prove To Gavin Jones That The Results He Obtained Were Not Accidents

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,...

## 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...