MS-E2114 Investment Science Lecture 4: Applied interest rate analysis A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 2/43
This lecture Lectures 2 & 3: We have determined interest rates from fixed income securities and computed the present value of cash flows This lecture: How can interest rate analysis be used to make better investment decisions? From descriptive ( what is ) to prescriptive analysis ( what should ) 3/43
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 4/43
Capital budgeting Task: Allocate capital C among m investment projects b i = Benefit of project i c i = Cost of project i Benefits and costs are aggregated and expressed in monetary terms ( e ) Projects are assumed to be Lumpy go / no-go investments E.g. bridges - it does not make sense to build half a bridge This differs from for securities which are sold and bought in small amounts Independent of each other The benefits and costs of a given project do not depend those of other projects (i.e., synergies, cannibalization impacts not considered) Not tradeable in established markets E.g., projects for developing prototypes or new products 5/43
Capital budgeting The budgeted capital C need not be a hard constraint There is a tendency to deplete the budget even if the last projects to be funded offer but a small marginal benefit Conversely, it would make sense to increase C if there are excellent projects that the available budget C does not allow to be funded A In theory, one could raise unlimited funding from the markets and fund all projects with positive NPV B In practice, there are limits Banks limit the amount of credit they provide More credit taken risk of default grows required interest rate rises In large organizations, investment decisions are typically part of divisional budgeting Impacts of changing the budget C should be explored Soft constraints can be introduced by imposing penalties for budget extensions, for instance 6/43
Solving the capital budgeting problem Projects denoted by x {0, 1} m { 1, if project i is funded x i = 0, otherwise Optimization formulation max x {0,1} m subject to m b i x i i=1 m c i x i C i=1 x {0, 1} m This is a binary 0-1 optimization problem Essentially a knapsack (packing) problem 7/43
Solving the capital budgeting problem Determining the exact solution may involve computational challenges There are hard-to-solve instances with m = 50...100 Yet many instances with m = 10 000 can be solved in milliseconds with state-of-the-art optimization methods Small and easy instances can be solved with Excel Solver An approximate solution can be generated by using benefit-to-cost ratios r i = b i /c i Fund projects one by one in decreasing order of ratios r i (i.e., starting from the project whose ratio is highest) until the budget C has been depleted 8/43
Solving the capital budgeting problem Project Cost Benefit r i Benefitto-cost solution Optimal solution 1 100 300 3.00 1 1 2 20 50 2.50 1 0 3 150 350 2.33 1 1 4 50 110 2.20 1 1 5 50 100 2.00 1 1 6 150 250 1.67 0 1 7 150 200 1.33 0 0 Cost 370 500 Benefit 910 1110 NPV 540 610 9/43
Modelling dependencies Project dependencies modelled through linear constraints E.g., project i has n i variants of which but one can be funded: m n i max b ij x ij x subject to i=1 j=i m n i c ij x ij C i=1 n i j=i x ij 1, j=i x ij {0, 1}, i = 1, 2,..., m j = 1, 2,..., n i, i = 1, 2,..., m b ij, c ij = benefit and cost of variant j of project i x ij = decision to fund variant j of project i 10/43
Modelling dependencies Examples of other extensions Enabler: If project j cannot be started unless the enabling project i is implemented, the constraint x j x i must hold E.g. a follow-up feature j that builds on another feature i If at least/at most/exactly k projects from the set i 1, i 2,..., i l must be selected, the respective constraints are k x i1 + x i2 + + x il k = k These extensions can be solved with integer optimization methods Small and easy problems can be solved with Excel Solver 11/43
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 12/43
Portfolio optimization In (financial) portfolio optimization the objective is to build a portfolio consisting of market tradable securities Securities are traded in the markets The price dynamics of securities has to be taken into account Some financial portfolio optimization problems can be solved much in the same way as capital budgeting problems 13/43
Cash matching Fixed liabilities: Obligation to pay y i euro in period i = 1, 2,..., n Task: Determine the bond portfolio whose cash flow meets or exceeds these liabilities for the smallest purchasing price c ij = cash flow of bond j in period i p j = price of bond j min x {0,1} m subject to m p j x j j=1 m c ij x j y i, j=1 x j 0, i = 1, 2,..., n j = 1, 2,..., m 14/43
Cash matching No short selling or issuing new bonds (x j 0) For shorting, the non-negativity constraint can be eliminated No reinvestment of excess cash flows Add artificial bonds with cash flows (0, 0,..., 0, 1, 1 + r i, 0,..., 0) where r i is the short rate for period i The solution of the continuous problem may contain non-integer bond investments Introduce integer constraints x j {0, 1, 2,... } Emphasis on cost minimization may expose to a large interest rate risk Add an immunization constraint (see Lectures 2 & 3) 15/43
Example: Basic cash matching problem Use the following 10 bonds to match the liabilities Input Bond cash flows c ij Liability Portfolio cash flow i \j 1 2 3 4 5 6 7 8 9 10 1 10 7 8 6 7 5 10 8 7 100 100 171.74 2 10 7 8 6 7 5 10 8 107 200 200 3 10 7 8 6 7 5 110 108 800 800 4 10 7 8 6 7 105 100 119.34 5 10 7 8 106 107 800 800 6 110 107 108 1200 1200 p 109 94.8 99.5 93.1 97.2 92.9 110 104 102 95.2 2381.14 Cost x 0 11.2 0 6.81 0 0 0 6.30 0.28 0 Variables Objective 16/43
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 17/43
Dynamic cash flows In dynamic problems, decisions and the information that supports them depend on previous decisions Example: Oil well as an investment Oil well can be drilled only if the site has been acquired Information about profitability can be obtained through testing How much is the test worth? Based on the results of the test, should one buy or not? Dynamic decisions can be structured as decision trees or lattices Lattice = Tree with recombining branches 18/43
Dynamic cash flows Consider a dynamic investment problem such that: Initial cost of investment is 100 e The investment yields cash flows of 300 e or 0 e with equal probability With a cost of 10 e, an expert can be consulted to assess whether the investment is profitable (300 e ) or not (0 e ) The expert knows for sure which one of these is the case The problem can be represented as a decision tree with three kinds of nodes Decision node Chance node Value node 19/43
... Dynamic cash flows Consult expert Yes Expert statement Cost 10 Profitable Not profitable Invest Invest Yes No Yes No 190=300-100-10-10 -110-10 No Invest Yes No 50=-100 +0.5 300 (300-100). 0 20/43
... Example: Dynamic programming Yes Expert statement Cost 10 Profitable Optimal 190 Invest Optimal -10 Yes No Yes 190=300-100-10-10 -110 Consult expert Not profitable Invest No -10 No Invest Yes 50=0.5 (300-100). No 0 21/43
... Example: Dynamic programming Consult expert Yes Profitable Expected value 0.5 190+0.5 (-10)=90 Cost 10 Not profitable Optimal 190 Invest Optimal -10 Invest Yes No Yes No 190=300-100-10-10 -110-10 No Optimal 50 Invest Yes 50=0.5 (300-100). No 0 22/43
... Example: Dynamic programming Yes Expected value 90 Cost 10 Profitable Optimal 190 Invest Optimal -10 Yes No Yes 190=300-100-10-10 -110 Consult expert Not profitable Invest No -10 No Optimal 50 Invest Yes 50=0.5 (300-100). No 0 23/43
Lattices n-periods price goes up by factor u or down by factor d Current price P0 u 2 P 0 up 0 i 1 2 3 4 udp 0 Binary tree P (3-periods) 0 udp 0 dp 0 d 2 P 0 u 3 P 0 u 2 dp 0 u 2 dp 0 ud 2 P 0 u 2 dp 0 ud 2 P 0 ud 2 dp 0 d 3 P 0 Binary lattice (equivalent to the tree) P 0 u 3 P 0 u 2 dp 0 ud 2 P 0 d 3 P 0 24/43
Dynamic programming Helps solve complex decision trees and simple lattices A binomial tree with n periods has 2 n paths Traversing all paths becomes impossible with large n Solution principle: What is optimal decision at node u? Decision leads to node v D(u) Cash flow of c v Optimal cash flow V (u) = max v D(u) {c v + V (v)} Solve recursively from the end to the beginning u c v1 c v2 c v3 v1 v2 v3 D(u) 25/43
Running present value The preceding slide had no discounting Discounting can be included through V (u) = max v D(u) {c v + d tu,t v V (v)} Discount factor dtu,t v = 1/f tu,t v f tu,tv = forward rate t u, t v times of decisions u, v Decisions in every period short rate rk = f k,k+1 u c v1 c v2 c v3 v1 v2 v3 D(u) Time t u t v1 = t v2 = t v3 26/43
Example: Fishing problem You have the exclusive right to harvest fish for 3 years The initial fish stock in the lake is 10 tons If you do harvest (max once per year), you extract 70% of all the fish that are in the lake Profit 1000 e /ton Each year, the fish population grows If you harvest, the fish stock will be restored to the level at which it was before harvesting If you do not harvest, the size of the fish stock will double What is optimal harvesting policy when using a 25% discounting rate? Discount factor d = 1/1.25 = 0.8 27/43
Example: Fishing problem Build the binary lattice Period 0 1 2 3 80 40 0 10 0 20 0 14 20 28 0 40 7 10 0 14 20 7 10 0 Black (nodes): Tons of fish in lake Red (arcs): Amount of fish extracted Up: No fishing, Down: fish 7 10 28/43
Example: Fishing problem Use V (u) = max v D(u) {c v + dv (v)} Discounting yields the optimum 20.16 k e with policy (N, Y, Y) Without discounting, two solutions (N, Y, Y) and (N, N, Y) both yield 28 k e Max(0+0.8 25.2, 7+0.8 12.6)=20.16 Max(14+0.8 14, 0.8 28)=25.2 0 7 0 14 0 Max(28,0) =28 max(14,0) =14 0 28 0 14 Max(0+0.8 14, 7+0.8 7)=12.6 7 max(7,0) =7 0 7 29/43
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 30/43
Harmony theorem Lecture 1: NPV is a more sustainable principle than IRR for guiding your investment decisions If you invest in a venture as a new owner, what decision principle should you adopt to steer investment decisions in the venture to get the best possible return for your investment in the venture? IRR may lead to different decision than NPV Do current and new owners have conflicting objectives? 31/43
Harmony theorem Your friend owns a gismo There are i = 1, 2,..., n commercialization options If your friend seeks to maximize the present value of his venture, he selects the alternative that yields { P(i ) = max c i + 1 } i {1,2,...,n} 1 + r b i where b i, c i are benefit and cost of option i, respectively If you think that your investment will be used for covering the expenses, you may (wrongly) conclude that you would like to select the alternative that yields } max i {1,2,...,n} { bi c i 32/43
Harmony theorem However, to get your share of benefits, you must buy share α of the gismo (the company) at a cost αp(i ) Thus, your cost-benefit decision criterion is { } max i {1,2,...,n} αb i αp(i ) + αc i Theorem: This is maximized for i = i, meaning that your and your friend s preferred options coincide. Proof: If there were an alternative i such that αb i αp(i ) + αc i > αb i αp(i ) + αc i = 1 + r, then P(i) = c i + b i 1 + r > c i + b i 1 + r = P(i ), in contradiction with the definition of i. 33/43
Harmony theorem Theorem (Harmony theorem) Current owners of a venture should want to operate the venture to maximize the present value of its cash flow stream. Potential new owners, who must pay the full value of their prospective share of the venture, will want the company to operate in the same way, in order to maximize the return on their investment. Thus, there is no conflict between current and new owners. 34/43
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 35/43
Valuation of a firm The valuation of a firm can be based on its cash flows Different aspects, such as 1. dividends to stock holders, 2. net earning of the firm 3. cash flow that could be realized from the firm s assets lead to different valuations of firms We do not consider cash flow uncertainties explicitly These can be accounted for through larger discount rates 36/43
Dividend discount models Dividend cash flows D k in year k = 1, 2,... Present value of this cash flow V 0 = D 1 1 + r + D 2 (1 + r) 2 + + D k (1 + r) k + In the constant-growth dividend model dividends grow at a constant rate g so that V 0 = D 1 1 + r + (1 + g)d 1 (1 + r) 2 + + (1 + g)k 1 D 1 (1 + r) k + V 0 = D 1 r g = 1 + g r g D 0, r > g, which is known as the Gordon formula Example: If the firm has paid 1.5 M e of dividends, grows with 8% and is valued using a 20% discount rate, then V 0 = 1 + 0.08 0.2 0.08 1.5 106 Me = 13.5 Me 37/43
Free cash flow models Startups pay little or no dividends in order to have more capital for growth Analysis based on the free cash flow that the firm can pay without compromising growth In practice, the calculation can be ambiguous Profit in year n is Y n, of which firm invests share u [0, 1] Annual growth rate modelled as factor g(u) Profit of the firm in period n is Y n+1 = [1 + g(u)] Y n Y n = [1 + g(u)] n Y 0 Capital depreciates by a factor α annually 38/43
Free cash flow models The capital in year n is C n = (1 α) n C 0 + uy 0 Using identity n (1 α) n i (1 + g(u)) i 1 i=1 x n y n = (x y) n x n i y i 1 i=1 with x = 1 α and y = 1 + g(u), we get C n = (1 α) n (1 α) n + (1 + g(u)) n C 0 + uy 0 g(u) + α (1 + g(u)) n C n uy 0 g(u) + α = u g(u) + α Y n, because (1 α) n 0 for large n 39/43
Example: Free cash flow Income statement Before-tax cash flow Y n Depreciation αc n Taxable income Y n αc n Taxes (34%) 0.34(Y n αc n ) After-tax income 0.66(Y n αc n ) After-tax income + depreciation 0.66(Y n αc n ) + αc n Sustaining investment uy n Free cash flow 0.66(Y n αc n ) + αc n uy n 40/43
Example: Free cash flow Valuation using free cash flow yields FCF n = 0.66(Y n αc n ) + αc n uy n [ ] αu FCF n = 0.66 + 0.34 g(u) + α u (1 + g(u)) n Y 0, where we have used C n = u g(u) + α Y n Gordon formula for the constant-growth dividend model yields [ ] αu PV = 0.66 + 0.34 g(u) + α u 1 r g(u) Y 0 41/43
Example: Free cash flow How much to invest in order to maximize NPV? Turnover 10M e Capital 19.8M e Depreciation α = 0.1 Discount rate r = 0.15 Growth g(u) = 0.12(1 e 5(α u) ) Optimal u = 0.3776 can be solved, e.g., using Excel Solver u 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 g(u) -0.08-0.03 0.00 0.03 0.05 0.06 0.08 0.09 0.09 0.10 0.10 PV 29.0 34.5 39.6 44.6 49.3 53.3 56.4 58.1 58.2 56.4 52.6 42/43
Overview Capital budgeting Portfolio optimization Dynamic programming Harmony theorem Firm valuation 43/43