## YflSS

r -g where D0 is the current dividend

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 5.6 (The XX Corporation) The XX Corporation has just paid a dividend of \$1.37M. The company is expected to grow at 10% for the foreseeable future, and hence most analysts project a similar growth in dividends The discount rate used for this type of company is 15%, What is the value of a share of stock in the XX Corporation?

The total value of all shares is given by (5.2) Hence this value is

Assume that there are 1 million shares outstanding Each share is worth \$30.14 according to this analysis.

Free Cash Flow*

A conceptual difficulty with the dividend discount method is that the dividend rate is set by the board of directors of the firm, and this rate may not be representative of the firm's financial status A different perspective to valuation is obtained by imagining that you were the sole owner and could take out cash as it is earned. From this perspective the value of the firm might be the discounted value of the net earnings stream.

The net earnings of a firm is defined by accounting practice, In the simplest case it is just revenue minus cost, and then minus taxes; but things are rarely this simple Account must be made for depreciation of plant and equipment, payment of interest on debt, taxes, and other factors. The final net earnings figure may have little relation to the cash flow that can be extiacted from the firm

Within the limitations of a deterministic approach, the best way to value a firm is to determine the cash flow stream of maximum present value that can be taken out of the company and distributed to the owners. The corresponding cash flow in any year is termed that year's free cash flow (FCF), Roughly, free cash flow is the cash generated through operations minus the investments necessary to sustain those operations and then anticipated growth,

It is difficult to obtain an accurate measure of the free cash flow. First, it is necessary to assess the firm's potential for generating cash under various policies Second, it is necessary to determine the optimal rate of investment—the rate that will generate the cash flow stream of maximum present value. Usually this optimal rate is merely estimated; but since the relation between growth rate and present value is complex, the estimated rate may be far from the true optimum We shall illustrate the ideal process with a highly idealized example.

Suppose that a company has gross earnings of Yn in year n and decides to invest a portion u of this amount each year in order to attain earnings growth. The growth rate is determined by the function &0<)> which is a property of the firm's characteristics. On a (simplified) accounting basis, depreciation is a fraction a of the current capital account (or .10, for example). In this case the capital C„ follows the formula C„+1 = (1 — o;)C„ With these ideas we can set up a general income statement for a firm, as shown in Table 5 .5

Example 5„7 (Optimal growth) We can go further with the foregoing analysis and calculate K„ and C„ in explicit form. Since y„+i = [I -f gOOlV/», it is easy to see that Y„ =[14- gOOF^o Likewise, it can be shown that

If we ignore the two terms having (I — a)" (since they will nearly cancel) we have hYq[1 + g(it)]' g(u) + a

5.6 valuation of a firm* 127

TABLE 5 5 Free Cash Flow

Income statement

Before-tax cash flow from operations V„

Depreciation o-C,,

After-tax cash flow (after-tax income plus depreciation) 66(!K„ — aC„) + aC„

Sustaining investment iiYn

Free cash flow 66(}'„ — aC„) + aC„ — uYlt

Depreciation is assumed to be a times the amount iit the capital account

Putting the expressions for Yn and C„ in the bottom iine of Table 5 5, we find the free cash flow at time n to be fcf =

This is a growing geometric series. We can use the Gordon formula to calculate its present value at interest rate r. This gives

It is not easy to see by inspection what value of it would be best, Let us consider another example

Example 5.8 (XX Corporation) Assume that the XX Corporation has current earnings of Ko — \$10 million, and the initial capital2 is Co ™ \$19.8 million. The interest rate is / = 15%, the depreciation factor is or = .10, and the relation between investment rate and growth rate is g(u) = A2[l — e5ia"u)]. Notice that g(a) — 0, reflecting the fact that an investment rate of a times earnings just keeps up with the depreciation of capital

Using (5,5) we can find the value of the company for various choices of the investment rate it For example, for u = 0, no investment, the company will slowly shrink, and the present value under that policy will be \$29 million. If it = .10, the company will just maintain its current level, and the present value under that plan will be \$39.6 million. Or if it = .5, the present value will be \$52 million.

It is possible to maximize (5 5) (by trial and error or by a simple optimization routine as is available in some spreadsheet packages). The result is it = 37.7% and g(u) = 9 0%. The corresponding present value is \$58.3 million, This is the company vaiue.

2This value of Co will make the terms that were canceled in deriving (5 3) cancel exactly

Here is a question so consider carefully. Suppose that during the first year, the firm operates according to this plan, investing .37.7% of its gross earnings in new capital. Suppose also, for simplicity, that no dividends are paid that year. What will be the value of the company after I year? Recall that during this year, capital and earnings expand by 9%. Would you guess that the company value will increase by 9% as well? Remember the harmony theorem Actually, the value will increase by the rate of interest, which is 15%. Investors must receive this rate, and they do. The reason this may seem strange is that we assumed that no dividends were paid. The free cash flow that was generated, but not taken out of the company, is held for the year (itself earning 15%), and this must be added to the present value calculation of future cash flows If the free cash flow generated in the first year were distributed as dividends, the company value would increase by 9%, but the total return to investors, including the dividend and the value increase, again would be 15%

Although this example is highly idealized, it indicates the character of a full valuation procedure (under an assumption of certainty). The free cash flow stream must be projected, accounting for future opportunities Furthermore, this cash flow stream must be optimized by proper selection of a capital investment policy. Because the impact of current investment on future free cash flow is complex, effective optimization requires the use of formal models and formal optimization techniques

### 5.7 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 (low 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

1 100 200

2 .300 500

3 200 300

5 150 250

2. (The road®) Refer to the 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 + o^f2 + Qfj/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 at 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.

(n) 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 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 costs 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 reser ves (/;) What is the maximum present value of your profits, and what is the corresponding optimal pumping strategy?

10„ (Multiperiod harmony tiieoremo) 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], , x„, 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

1 + s{ (i + vi)2 0 + c 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*() — 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 Call this value T(). Assuming that interest rates follow expectation dynamics and that V; (aq) > 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\

### REFERENCES

Capital budgeting is a classic topic in financial planning Some good texts are [I-4]; good surveys are [51,  Bond portfolio construction is considered in [6-8] and in other references given for Chapters 3 and 4 Dynamic programming was developed by Bellman (see [9, 10]) The classic reference on stock valuation is  See [12-16] for other presentations, A vivid discussion of how improper analysis techniques led to disastrous overvaluation in the 1980s is in [111

1 Dean, J (1951), Capital Budgeting, Columbia University Press, New York

2 Brealey, R, and S Myers (1984), Principles of Corporate Finance, McGraw-Hill, New

York

3 Bierman, H , .Jr., and S Smidt (1984), The Capital Budgeting Decision, 6th ed , Macmillan,

New York

4 Martin, J D,, S. H. Cox, Jr., and R D MacMinn (Í988), The Theory of Finance, Evidence and Applications, Dryden Press, Chicago, IL

5 Schali, L D , G L Sunciem, and W R Geijsbeek (1978), "Survey and Analysis of Capital

Budgeting Methods," Journal of Finance, 33, 281-287.

6 Weingartner, H M (1966), "Capital Budgeting of Interrelated Projects: Survey and Syn thesis," Management Science, 12, 485-516.

7 Bierwag, G O , G G Kaufman, R Schweitzer, and A Toevs (1981), "The Art of Risk

Management in Bond Portfolios," /out nal of Portfolio Management, 7, 27-36

8 Fabozzi, F.J, and T D Fabozzi (1989), Bond Markets, Analysis and Strategies, Prentice

Hall, Englewood Cliffs, N.J.

9 Bellman, R (1957), Dynamic Programming, Princeton University Press, Princeton, NJ

10 Bellman R , and S Dreyfus (1962), Applied Dynamic Programming, Princeton University

Press, Princeton, NJ

11 Graham, B , D L Dodd, and S Cottle (1962), Security Analysis, McGraw-Hill, New York

12 Williams, J. B (1938), The Theory of Investment Value, North-Holland, Amsterdam, The

Netherlands

13 Gordon, M. J (1959), "Dividends, Earnings, and Stock Prices," Review of Economics and

Statistics, 41, 99-195

14 Molodovsky, N ., C. May, and S Chotüner (1965), "Common Stock Valuation: Principles,

Tables and Application," Financial Analysts Journal, 21, 104-123

15 Foster, G (1986), Financial Statement Analysis, Prentice Hall, Englewood Cliffs, NJ

16 Black, F (1980), "The Magic in Earnings: Economic Earnings versus Accounting Earnings,"

Financial Analysts Journal, 36, 19-24.

17 Klarman, S A. (1991), Margin of Safety. Risk-Averse Value Investing Strategies for the

SINGLE-PERIOD RANDOM CASH

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