# 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 static measure of value; it guides us to find the decision or structure with the highest value

One class of problems that can be approached with this combination is capital budgeting problems. In the classic problem of this class, a fixed budget is to be allocated among a set of independent projects in order to maximize net present value. This problem can be solved approximately by selecting projects with the highest benefit-cost ratio The problem can be solved exactly by formulating it as a zero-one optimization problem and using an integer programming package More complex capital budgeting problems having dependencies among projects can be also be solved by the zero-one programming method.

The selection of a bond portfolio to meet certain requirements can be conveniently formulated as an optimization problem—but there are several possible formulations. A particularly simple problem within this class is the cash-matching problem, where a portfolio is constructed to generate a required cash flow in each period. This formulation has the weakness that in some periods extra cash may be generated, beyond that required, and this extra cash is essentially wasted More complex formulations do not have this weakness

To produce excellent results, many investments require deliberate ongoing management The relation between a series of management decisions and the resulting cash flow stream frequently can be modeled as a graph.. (Especially useful types of graphs are trees and lattices ) In such a graph the nodes correspond to states of the process, and a branch leading from a node corresponds to a particular choice made from that node. Associated with each branch is a cash flow value

Optimal dynamic management consists of following the special path of arcs through the graph that produces the greatest present value. This optimal path can be found efficiently by the method of dynamic programming A particularly useful version of dynamic programming for investment problems uses the running method for evaluation of present value

Dynamic programming works backward in time For a problem with n time periods, the running version of the procedure starts by finding the best decision at each of the nodes i at time n - 1 and assigns a V-value, denoted by V„_¡ /, to each such node This K-value is the optimal present value that could be obtained if the investment process were initiated at that node To find that value, each possible arc emanating from node / is examined, The sum of the cash How of the arc and the one-period discounted K-value at the node reached by the arc is evaluated, The V-value of the originating node / is the maximum of those sums After completing this procedure for all the nodes at n — 1, the procedure then steps back to the nodes at time n — 2 Optimal V-values are found for each of those nodes by a procedure that exactly parallels that for the nodes at n ~~ 1 The procedure continues by working backward through all time periods, and it ends when an optimal V-value is assigned to the initial node at time zero.

When operating a venture it is appropriate to maximize the present value. On the other hand, investors may be most interested in the rate of return These criteria might seem to be in conflict, but the harmony theorem states that the criteria are equivalent under the assumption that investors pay the full value for their ownership of the venture.

Present value analysis is commonly used to estimate the value of a firm. One such procedure is the dividend discount method, where the value to a stockholder is assumed to be equal to the present value of the stream of future dividend payments. If dividends are assumed to grow at a rate g per year, a simple formula gives the present value of the resulting stream.

The better method of firm evaluation bases the evaluation on free cash flow, which is the amount of cash that can be taken out of the firm while maintaining optimal operations and investment strategies In idealized form, this method requires that the present value of free cash flow be maximized with respect to all possible management decisions, especially those related to investment that produces earnings growth.

Valuation methods based on present value suffer the defect that future cash flows are treated as if they were known with certainty, when in fact they are usually uncertain The deterministic theory is therefore not adequate. This defect is widely recognized; and to compensate for it, it is common practice to discount predicted, but uncertain, cash flows at higher interest rates than the risk-free rate. There is some theoretical justification for this, but a completely consistent approach to uncertainty is more subtle The exciting story of uncertainty in investment begins with the next chapter and continues throughout the remainder of the text

### EXERCISES

(Capital budgeting) A firm is considering funding several proposed projects that have the financial properties shown in Table 5 6 The available budget is \$600,000. What set of projects would be recommended by the approximate method based on benefit-cost ratios? What is the optimal set of projects?

TABLE 5,6

Financial Properties of Proposed Projects

Outlay Present worth

t too 200

2 .300 500

3 200 300

5 150 250

2. (The road®) Refer to tire transportation alternatives problem of Example 5 2. The bridge at Cay Road is actually part of the road between Augen and Burger Therefore it is not reasonable for the bridge to have fewer lanes than the road itself This means that if projects 2 or 4 are carried out, either projects 6 or 7 must also be carried out Formulate a zero-one programming problem that includes this additional requirement Solve the problem

3. (Two-period budget ©) A company has identified a number of promising projects, as indicated in Table 5 7 The cash flows for the first 2 years are shown (they are all negative)

TABLE 5,7 A List of Projects

Cash flow

TABLE 5,7 A List of Projects

Cash flow

 Project 1 2 NPV 1 -90 -58 150 2 -80 -80 200 3 -50 -100 100 4 -20 -64 100 5 -40 -50 120 6 -80 -20 150 7 -80 -100 240

The cash flows in later years are positive, and the net present value of each project is shown. The company managers have decided that they can allocate up to \$250,000 in each of the first 2 years to fund these projects If less than \$250,000 is used the first year, the balance can be invested at 10% and used to augment the next year's budget Which projects should be funded?

4, (Bond matrix o) The cash matching and other problems can be conveniently represented in matrix form. Suppose there are tn bonds We define for each bond j its associated yearly cash flow stream (column) vector Cj, which is n-dimensional The yearly obligations are likewise represented by the n-dimensional vector y We can stack the C; vectors side by side to form the columns of a bond matrix C Finally we let p and x be /»-dimensional column vectors The cash matching problem can be expressed as minimize p7 x subject to Cx > y x > 0

(b) Show that if all bonds are priced according to a common term structure of interest rates, there is a vector v satisfying

What are the components of v?

(c) Suppose b is a vector whose components represent obligations in each period Show that a portfolio x meeting these obligations exactly satisfies

(<•/) With x and v defined as before, show that the price of the portfolio x is v7b interpret this result

5. (Trinomial lattice) A trinomial lattice is a special case of a trinomial tree From each node three moves are possible: up, middle, and down The special feature of the lattice is that certain pairs of moves lead to identical nodes two periods in the future We can express these equivalences as up-down = down-up = middle-middle middle-down = down-middle middle-up = up-middle

Draw a trinomial lattice spanning three periods How many nodes does it contain? How many nodes are contained in a full trinomial tree of the same number of periods?

6» (A bond project®) You are the manager of XYZ Pension Fund On November 5, 201 I, XYZ must purchase a portfolio of U S Treasury bonds to meet the fund's projected liabilities in the future The bonds available at that time are those of Exercise 4 in Chapter 4. Short selling is not allowed Following the procedure of the earlier exercise, a 4th-order polynomial estimate of the term structure is constructed as t (/) = ao -f a[t + a2t2 + o^/3 4- a.\tA The liabilities of XYZ are as listed in Table 5 8

TABLE 5,0

Liabilities of XYC Pension Fund

TABLE 5,0

Liabilities of XYC Pension Fund

 Liabilities Occur on 15th Feb 2012 \$2,000 Aug 2012 \$20,000 Feb 2013 \$0 Aug 2013 \$25,000 Feb 2014 \$1,000 Aug 2014 \$0 Feb 2015 \$20,000 Aug 2015 \$1,000 Feb 2016 \$15,000

{a) (Simple cash matching) Construct a minimum-cost liability-matching portfolio by buying Treasury bonds assuming that excess periodic cash flows may be held only at zero interest to meet future liabilities

(b) (Complex cash matching) Construct a minimum-cost liability-matching portfolio by buying Treasury bonds assuming that all excess periodic cash flows may be reinvested al the expected interest rates (implied by the current term structure) to meet future liabilities No borrowing is allowed

(c) (Duration matching) Construct a minimum-cost portfolio with present value equal to that of the liability stream Immunize against a change in the term structure parameters Do this for five cases Case 1 is to guard against a change in c*i, case 2 to guard against changes in c^ and a2, and so on

7, (The fishing problem) Find the solution to the fishing problem of Example 5 4 when the interest rate is 33%. Are the decisions different than when the interest rate is 25%? At what critical value of the discount factor does the solution change?

8. (Complexico mine®) Consider the Complexico mine and assume a 10% constant interest rate; also assume the price of gold is constant at \$400/oz.

(fj) Find the value of the mine (not a 10-year lease) if the current deposit is .v0 In particular, how much is the mine worth initially when .vq = 50,000 ounces? [Hint Consider the recursive equation for k\ as k oo ]

(b) For the 10-year lease considered in the text, how much gold remains in the mine at the end of the lease; and how much is the mine worth at that time?

(c) If the mine were not leased, but instead operated optimally by an owner, what would the mine be worth after 10 years?

9. (Little Bear Oil) You have purchased a lease for the Little Bear Oil well This well has initial reserves of 100 thousand barrels of oil. In any year you have three choices of how to operate the well: (a) you can not pump, in which case there is no operating cost and no change in oil reserves; (b) you can pump normally, in which case the operating cost is \$50 thousand and you will pump out 20% of what the reserves were at the beginning of the year; or (c) you can use enhanced pumping using water pressure, in which case the operating cost is \$120 thousand and you will pump out 36% of what the reserves were at the beginning of the year The price of oil is \$10 per barrel and the interest rate is 10% Assume that both your operating cosis and the oil revenues come at the beginning of the year (through advance sales) Your lease is for a period of 3 years

(a) Show how to set up a trinomial lattice to represent the possible states of the oil reserves (/;) What is the maximum present value of your profits, and what is the corresponding optimal pumping strategy?

10„ (Multiperiod harmony theoremo) The value of a firm is the maximum present value of its possible cash flow streams This can be expressed as

where the maximization is with respect to all possible streams a«, v], , a„, and the .v,\s are the spot rates Let a^ be the first cash flow in the optimal plan If the firm chooses an arbitrary plan that results in an initial cash flow of .v() (distributed to the owners), the value of the firm after 1 year is

where now that maximum is with respect to all feasible cash flows that start with a() and the .v(''s are the spot rates after 1 year An investor purchasing the firm at its full fair price has initial cash flow a*q — V() and achieves a value of Vj(a'o) after 1 year Hence the 1-year total return to the investor is

The investor would urge that a{) be chosen to maximize R Caii this value a(). Assuming that interest rates follow expectation dynamics and that V; (Tq) > 0, show that the maximum R is 1 + .?[ and that this return is achieved by the same a-(* that determines Vo

[Hint. Let S be the value of the sum Note that S — 1/(1 4- r) + 5(1 + g}/{ 1 +r) ]

12. (Two-stage growth) It is common practice in security analysis to modify the basic dividend growth model by allowing more than one stage of growth, with the growth factors being different in the different stages As an example consider company Z, which currently distributes dividends of \$I0M annually The dividends are expected to grow at the rate of ¡0% for the next 5 years and at a rate of 5% thereafter.

(a) Using a dividend discount approach with an interest rate of 15%, what is the value of the company?

(b) Find a general formula for the value of a company satisfying a two-stage growth model Assume a growth rate of G for k years, followed by a growth rate of g thereafter, and an initial dividend of D\ 