FINANCIAL CALCULATIONS

Common cash flow patterns What is the present value of an annuity of A = 100 lasting for 15 periods, if the discount rate is 12%? (681) P = 100*(1/0.12 – 1/0.12/1.1215) = 681 How long should an annuity of A = 50 be in order for its present value to be at least 100, if the discount rate is 18%? (2.7 ~ 3) 100/50 = 1/0.18 – 1/0.18/1.18N  0.64 = 1.18-N  N = -log0.64 / log1.18 = 2.7 ~ 3

Common cash flow patterns Which option would you prefer: $10,000 today or $1,000 each year for 15 years, if the discount rate is 10%? (first > $7,606) P = 1000*(1/0.1 – 1/0.1/1.115) = 7606 < 10000 At least how much should the amount A of an annuity lasting for 10 periods be in order for the present value of the annuity to be at least 80, if the discount rate is 15%? (16) A = 80/(1/0.15 – 1/0.15/1.1510) ≈ 16

Common cash flow patterns What is the present value of a perpetuity of A = 100, if the discount rate is 20%? (500) P = 100/0.2 = 500 How much should the annual amount A of a perpetuity be in order for its present value to be at least 250, if the discount rate is 15%? (37.5) A = 250*15% = 37.5 What discount rate makes the present value of a perpetuity of A = 25 equal to 100? (25%) r = 25/100 = 25%

Common cash flow patterns Growing annuity & perpetuity calculations: Assume that g = 3% and r = 10% What is the present value if F1 = 100 and the series lasts for 5 periods? (400) P = 100*(1 – (1.03/1.1)5)/(0.1 – 0.03) ≈ 400 How much should F1 be in order for the present value to be 320? (g.a.: 80; g.p.: 22.4) G.a.: F1 = 320/(1 – (1.03/1.1)5)*(0.1 – 0.03) ≈ 80 G.p.: F1 = 320*(0.1 – 0.03) = 22.4

Common cash flow patterns If F1 = 100, at least how many periods should the series last for in order for the present value to be larger than 500? (6.55 ~ 7) 500/100 = (1 – (1.03/1.1)N)/(0.1 – 0.03)  0.65 = (1.03/1.1)N  N = log0.65 / log(1.03/1.1) = 6.55 ~ 7 Run the same, but for g = 10% (455; 70.4; 5.5 ~ 6) P = 100*5/1.1 = 455; F1 = 320*1.1/5 = 70.4; N = 500/100*1.1 = 5.5 The pattern is a growing perpetuity. If F1 = 100 and r = 10%, then what g, or if g = 3%, then what r makes the present value equal to 1250? (g = 2%; r = 11%) g = -(100/1250 – 0.1) = 2%; r = 100/1250 + 0.03 = 11%

Profitability index (PI) (II.) Example: consider the following projects: Ranking according to PI: D > B > C > A > E Assume that the capital constraint is 150 – then D, B, A projects would be undertaken Because F0 altogether is 80 + 20 + 50 = 150 After D and B, insufficient allotment remains for C E is by no means to be undertaken, because PIE < 1 (NPVE < 0) F0 PV PI NPV A -50 60 1,20 +10 B -20 30 1,50 C -110 150 1,36 +40 D -80 210 2,63 +130 E -70 50 0,71

Annual equivalent (AE) (II.) Illustration: We can choose from two types of air-conditioners to be installed in our office. Type A can be purchased for $10,000 and operated annually for $1,500 with expected life-span of 7 years. Type B can be purchased for $14,000 and operated annually for $1,100 with expected life- span of 10 years. The discount rate is 9%. Which type should we choose, assuming that we would stick with the choice until infinity? (-$3,487 < -$3,281, so B)

Annual equivalent (AE) (II.) Type A NPV of one cycle = -10,000 – 1,500*(1/0.09 – 1/0.09/1.097) = -17,549 Annual equivalent = -17,549/(1/0.09 – 1/0.09/1.097) = -3,487 Type B NPV of one cycle = -14,000 – 1,100*(1/0.09 – 1/0.09/1.0910) = -21,059 Annual equivalent = -21,059/(1/0.09 – 1/0.09/1.0910) = -3,281

Rearrangement of formulas Recognize that: