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Mathematics for Asset Valuation Budi Frensidy Faculty of Economics, University of Indonesia 4th International Conference on Research and Education in Mathematics Renaissance Hotel Kuala Lumpur, Malaysia 21-23 October 2009 Budi Frensidy - FEUI 1 Introduction Present value concept is needed for: Project evaluation Asset valuation (real and financial) Recording and reporting an asset on the balance sheet We usually do the discounting one by one by using the basic equation This method is impractical as the number of periods can be infinite (~) Budi Frensidy - FEUI 2 The Purpose of the Paper We actually can use the short-cut equations if certain conditions are met namely equal cash flows or equal growth (in percentages) There are at least 13 other equations besides the basic and continuous equation that can be used for this This paper tries to enumerate and give examples of all the equations using a simple and interesting case on valuation Budi Frensidy - FEUI 3 Definition of Annuity and Perpetuity Annuity is a series of cash payments or cash receipts, generally in equal amounts, in equal periodic intervals. Installments on home loan or car loan and bond coupon are some examples Whereas perpetuity or perpetual annuity is infinite annuity which is a special type of annuity when the periods are countless like stock dividends, pension allowances, royalties, and copyrights Budi Frensidy - FEUI 4 Valuation Case (1) A young investor is contemplating investment in the following financial assets which are offered at the same price i.e. Rp 100 million. With his budget constraint, he can choose one and only one of the alternatives given. a. Zero-coupon bond with the nominal value of Rp 250 million, due in 8 years b. No-par bond that pays cash Rp 18 million every year for 10 times starting next year c. No-par bond that pays cash Rp 16 million every year for 10 times starting right now d. No-par bond with the payoff Rp 50 million every year for five times, but starting in five years Budi Frensidy - FEUI 5 Valuation Case (2) e. Discretionary fund that gives cash Rp 12.5 million every year for the whole life starting next year f. Discretionary fund that gives cash Rp 11.5 million every year for the whole life starting next week g. Hedge fund with the payoff Rp 15 million every year for the whole life starting in three years h. Hedge fund that pays cash Rp 15 million next year, then becomes Rp 15.9 million the following year and steadily rises 6% every year, and the payments are for ten times only i. Banking product with the payoff Rp 13 million next week then Rp 14.04 million next year and rises consistently 8% every year, and the payments are for ten times only j. Banking product that promises cash Rp 40 million in five years then becomes Rp 43.2 million a year after and steadily rises 8% every year, and the payments are for five times only Budi Frensidy - FEUI 6 Valuation Case (3) k. Investment trust that promises cash Rp 4 million next year, and then rises 8% every year for the whole life l. Investment trust that gives cash Rp 7 million next week, then Rp 7.35 million next year and rises on at 5% every year for the whole life m. Private equity with the payoff Rp 12 million starting in three years and rises to Rp 12.36 million the following year and at 3% every year for the whole life n. Private equity that distributes cash Rp 12 million next year and rises Rp 60,000 every year for 50 years Budi Frensidy - FEUI 7 Valuation Case (4) The security, risk, and certainty of the above financial assets are assumed the same. The investor is expected to act rationally and will base his decision only on the mathematical calculation. Using a different formula for each alternative, which asset should be chosen if the relevant discount rate is 12% p.a.? To decide which asset should be chosen, the investor must calculate the present value of all the assets. As long as the present value of the cash flows generated from an asset is more than the cost which is Rp 100 million, the asset is worth buying The problem is when there is a budget constraint, as we usually encounter, we have to rank all the choices and choose the highest value. This is what is meant by mutually exclusive projects in finance Budi Frensidy - FEUI 8 Equations and Results Present Asset Equation Use value (Rp) FV Basic equation (To calculate A 100,970,807 PV = (1 i) n PV from a single FV) 1 (1 i) n B 101,704,014 PV A Ordinary annuity i 1 1 i n 1 C 101,251,997 PV 1A Annuity due i 1 (1 i) n A D 114,545,022 PV i Deferred annuity (1 i) m -1 A E 104,166,667 PV Ordinary perpetuity i A F 107,333,333 PV A Perpetuity due i A PV i1 i G 99,649,235 m 1 Deferred perpetuity 1 g n 1 PV 1 i A H 105,848,743 Ordinary growing annuity ig 1 Budi Frensidy - FEUI 9 Equations and Results 1 g n 1 1 I 110,977,751 PV 1 i A A Growing annuity due ig 1 0 1 g n 1 1 i A J 105,663,644 ig 1 Deferred growing annuity PV 1 i m 1 A1 K 100,000,000 PV Ordinary growing perpetuity ig A1 L 112,000,000 PV A0 Growing perpetuity due ig A PV (i g )1 i M 106,292,517 m 1 Deferred growing perpetuity Variable annuity where A1 = 1 n (a1 + d/i + nd) 1 N 103,719,727 PV 1 i A nd d is the nominal difference i 1 i between periods and a1 is the first cash flow Budi Frensidy - FEUI 10 Summary Based on the above results, the investor must choose asset d which is a no-par bond with the payoff Rp 50 million every year for five times, but starting in five years. This asset gives the highest present value namely Rp 114,545,022 Present value is the basics to understand financial management, investment management, and accounting Certainly, there are some assumptions to be met in order for us to use the 13 equations. The amount of periodic cash flow must be equal, or they grow at an equal percentage or in equal difference from one period to another period. If one of these assumptions is fulfilled, the use of mathematical equations will give the same result as the present value derived from one by one calculation Budi Frensidy - FEUI 11 THANK YOU Budi Frensidy - FEUI 12

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