MS-E2114 Investment Science Exercise 4/2016, Solutions

Transcription

1 Capital budgeting problems can be solved based on, for example, the benet-cost ratio (that is, present value of benets per present value of the costs) or the net present value (the present value of benets - present value of the costs). According to the benet-cost ratio, a project is benecial if the ratio is greater than one, i.e., the present value of the benets is greater than that of the costs. Similarly, when evaluating projects based on net present value, a project is worth carrying out if NPV>. If there are more projects available than there is capital to fund them, an approximate solution to the capital budgeting problem is obtained by ordering the projects by their benet-cost ratio and selecting them one-by-one until the budget limit is reached. If the benets a j of the projects are independent, the capital budgeting can be written as max m a i x i x i {, 1}, i = 1,..., m x X F, where x = [x 1 x 2... x m ] represents the project portfolio such that x i = 1 if project i will be implemented and x i = if not. X F is the feasible set of the projects, which is dened by, for example, budget limits and project interdependencies. If the project interdependencies aect the benets of the projects, the object function of the optimization problem has to be modied correspondingly. If the only constraint is the budget C and the cost of each project i is c i, the feasible set is dened as X = {x m c ix i C}. If projects i and j are mutually exclusive (that is, only one of them can be implemented), a constraint x i + x j 1 has to be subjected. A rm can be evaluated by, for example, based on the paid dividends. Using a constant-growth dividend model, the value of a rm can be dened as the present value of the dividend stream. Suppose a constant rate g, rst dividend D 1 paid at the end of rst period and a constant interest rate r. The present value of the dividends is V = D r + D 1(1 + g) (1 + r) 2 + D 1(1 + q) 2 (1 + r) 3 + = D 1 where the last equation is the Gordon formula. k=1 (1 + g) k 1 (1 + r) k V = D 1 r g,

2 1. (L5.1) (Capital budgeting) A rm is considering funding several proposed projects that have the nancial properties shown in Table 1. The available budget is 6 e. What set of projects would be recommended by the approximate method based on benet-cost ratios? What is the optimal set of projects (using net present value)? Table 1: Financial properties of the proposed projects. Outlay Present value of benets Project (1 e) (1 e) The projects can be ranked based on their benet-cost ratio φ or their net present value, dened as Present value of benets φ =, NPV=Present value of benets - Investment cost Investment cost The projects ranked by their benet-cost ratios are presented in Table 2 below. Based on the benet-cost Table 2: Projects ranked by the benet-cost ratios φ. Outlay Present value of benets φ Project (1 e) (1 e) ratios, projects 1,2 and 5 are selected, having total investment cost of 55 < 6 e. The total net present value of the projects of this approximate solution is NPV=4 e. Those projects that create the greatest total net present value of the project portfolio comprise the optimal project portfolio. The project selection can be formulated as an optimization problem as follows: max 5 NPV i x i 5 c i x i 6 e, x i {, 1}, i = 1,..., 5, where x = [x 1 x 2... x m ] (x i = 1 if i selected and, otherwise) presents the selections of projects in the project portfolio, NPV i is the net present value and c i the investment cost of project i. This optimization problem can be solved with, for example, Solver of Excel. We nd the solution to this problem to

3 be the projects 1,2 and 5. Thus, the approximate solution using benet-cost ratios provided the optimal solution in this case. 2. (Two-period budget) A company has identied a number of promising projects, as indicated in Table 3. The cash ows for the rst 2 years are shown (they are all negative). Table 3: A list of projects. Cash ow (1 e) Project year 1 year 2 Present value of benets (years 3 ) The cash ows in later years are positive, and the present values of the benets of each project are shown. The company managers have decided that they can allocate up to 25 e in the rst 2 years to fund these projects. If less than 25 e is used the rst year, the balance can be invested at 1% and used to augment the next year's budget. Which projects should be funded? Formulate the problem as an optimization problem. First we calculate the net present values of the projects at 1% interest rate: Table 4: Net present values of the projects at 1% interest rate. Project year 1 year 2 Present value of benets (years 3 ) NPV (1 e) We dene binary variables x i, i = 1,..., 7 so that x i = 1, if project i is selected and x i =, otherwise. The project selection problem can be formulated as and optimization problem. The objective function of the problem is

4 7 f(x) = NPV i x i = 2.25x x x x x x x 7, where NPV i is the net present value of project i. In general, inequalities can be written as equations by introducing slack variables. For example, we can write x C as C x s + =, where s +. Using this method, the budget constraints for the rst two years can be set using slack variables s + i (i = 1, 2), which dene the amount of budget remaining in each years. Hence, we write the budget constraint for the rst year as 25 9x 1 8x 2 5x 3 2x 4 4x 5 8x 6 8x 7 s + 1 =. The remaining balance s + 1 from the rst year can be invested at 1% interest to be used in the budget of the second year. The budget for the second year is then 1.1s + 1. We can then write the budget constraint for the second year as 1.1s x 1 8x 2 1x 3 64x 4 5x 5 2x 6 1x 7 s + 2 =. We formulate the optimization problem of the project selection as follows: max f(x) x 25 9x 1 8x 2 5x 3 2x 4 4x 5 8x 6 8x 7 s + 1 = 1.1s x 1 8x 2 1x 3 64x 4 5x 5 2x 6 1x 7 s + 2 = x i {, 1} i = 1,..., 7 s + 1, s+ 2. s + 1 is the remaining budget from the total budget 25 e after the expenses of the rst year, and which can be invested at 1% interest. s + 2 is the excess capital that remains unused after two years. We solve this optimization problem using Solver of Excel. The solution is x 4 = x 7 = 1, x 1 = x 2 = x 3 = x 5 = x 6 =, s + 1 = 15, and s+ 2 = 1. Hence we select projects 4 and 7.

5 3. Suppose that we face a known sequence of future monetary obligations. In cash ow matching problem, we design a portfolio that will provide the necessary cash as required for the obligations. We formulate this optimization problem in matrix form as follows. Let the number of bonds be m and the time horizon be n. The cash ow stream of bond j can be denoted as a c j n 1 and the yearly obligations as b n 1. We denote the bond matrix that has columns of the cash ows c j as C n m. Furthermore, the prices of the bonds can be denoted as p m 1 and the numbers of the bonds in the portfolio as x m 1. These notations give the cash ow matching problem as min p T x Cx b x. a) The cash ow structure of a cash ow matching problem is presented in Table 3. Dene C, b, p and x. b) Suppose the bonds are priced according to a conventional spot rate curve. The price vector p can be then written as C T v = p, where v n 1 is a vector of the discount rates. Moreover, if the portfolio x matches the obligations exactly, we have Cx = b. Show that the price p T x of the portfolio is v T b and interpret this. c) The optimization problem presented above seeks a solution that matches the obligations each year exactly. If the cash ows cannot be matched exactly, the present value of the portfolio is greater than the present value of the obligations. How does this model dier from immunization of a portfolio? What factor of portfolio immunization is neglected in this approach? Which approach is better? Table 5: Bonds of exercise 3. Bonds Year equired Actual p x Cost

6 a) The bond matrix C and vectors of obligations b, bond prices p and numbers of bonds in the portfolio x can be directly read from Table 5: C = , b = 8 1, p = , x = b) We have C T v = p and Cx = b, where elements of v are [v] k = 1/(1 + s k ) k Because (AB) T = B T A T, we can write C T v = p p T = v T C. Hence, the price of the portfolio can be written as: p T x = v T Cx = v T b. Interpretation: If the cash ows of the portfolio match the obligations exactly, the present value of the project portfolio (which equals the price of the portfolio) matches the present value of the obligations. c) In immunization the present value and rst order derivative (quasi-modied duration) of the portfolio is matched with those of the obligation stream. The cash ows of an immunized portfolio do not necessarily have to match those of the obligations, and instead the assets in the portfolio are sold when needed to pay the obligations. In cash ow matching, the positive cash ows from the bonds always suce to pay the obligations, regardless of the interest rates. However, the present value of the portfolio is not matched to that of the obligations. From part b) of this exercise can be seen that if Cx > b, then p T x = v T Cx > v T b, and hence the present value (price) of the portfolio exceeds that of the obligation stream when the cash ows received from the bonds exceed the obligations. This problem of cash ow matching can be diminished by, for example, introducing articial bonds that are consistent with the forward rates, or by allowing extra cash to be put "under the matress".

7 4. When pricing nancial instruments, the dividend discount model can be extended by taking more growth phases into account. Consider Nokia Corp. that paid 1439M e of dividends in year 23. Suppose that the dividends grow at a constant rate G = 1.3 in the rst ve years (that is, during years 24-28), and the dividends grow at rate g = 1.5 from year 29 onwards. a) Formulate the general formula for two-stage dividend discount model for valuing a publicly traded company. The growth rate is constant G for k years and then g from year k + 1 onwards. The dividend of the rst year D is paid immediately. b) What is the market value of Nokia Corp., if it is valued solely based on the shared dividends? Assume a constant interest rate r =.1 and that rst dividend is paid immediately. a) Growth rate is G for the rst k years, the rst dividends are D and the discount factor is 1/ = 1/(1+r). Hence, the present value of the dividends in the rst k years is P V 1 = D + D G + + D ( G ) k = D k i= ( ) G i (1) After k years, the growth rate changes to g. The present value of the paid dividends from year k+1 onwards is then ( ) G k ( ) g G k ( g ) 2 P V 2 = D + D +. (2) Combining equations (1) and (2) yields the present value of the whole stream as P V = D + D G + + D ( G ) k 1 ( ) G k [ + D 1 + g ( g ) ] 2 r + +. (3) The formula for a sum of rst n terms of a geometric series is a + ar + + ar n 1 = a(1 r n )/(1 r) and the formula for the geometric sum when n goes to innity is a + ar + ar 2 + = a/(1 r). The rst formula applies for r 1 and second for r < 1. Using these formulas (assuming G and g < ), the present value of the dividend stream becomes [ ( 1 G ) k ( ) ] G k 1 P V = D 1 G + 1 g (4) b) We substitute the values G = 1.3, g = 1.5, = 1.1, D = 1439, and k = 5 into (4) and get the market value of Nokia Corp. as Me. The price per share (the present value divided by the number of shares) is then e. For comparison, the average price of Nokia stocks was e in 23.