Interest calculation (Interest paid after) Simple interest – FV = PV*(1+r*t) ==> PV = FV/(1+r*t) Amount of interest - I = PV*r*t Compound interest – FV = PV*(1+r)^n, certainly PV = FV * (1+r)^-n Combined interest - FV = PV*(1+r)^n*(1+r*t) , where n…. number of whole interest periods Continuous interest - FV = PV*e^(f*t) (1+re) = (1+r(m)/m)^m re=(1+r(m)/m)^m - 1 e^f f=ln(1+re) (Interest paid ahead) PV = FV – D, D=FV*d*t PV = FV*(1-d*t); d=r/(1+r) and analogous r = d/(1-d), always must be d < r Compound discount … PV = FV*(1 - d)^n, PV = FV*(1 - d/m)^(m*n) (1 - de) = (1 – d(m)/m)^m = 1/e^f (calculation for PV…), obviously it must be true de = 1 – e^-f Note 1: Discounting = using the discount factor to calculate PV in contrary to use a COMERCIAL DISCOUNT – D=FV.d.t PV=FV*(1-d*t) Note 2: You pay tax just from the obtained interest!!! (do not tax the principal) PV…. Present Value FV…. Future Value D …. Commercial discount (similar to Interest amount I) r…. Interest rate e… 2,182818…. d…. discount rate de…. Effective discount rate d(m) … nominal discount rate with m-conversions TO UNDERSTAND THE INTEREST CALCULATION IT IS NOT IMPORTANT (and useful) DO A MECHANICAL MEMORIZATION, but understand the logic of each approach!!! 1. Decide which of the four investments is the best…? (t1 = 88,000.00, t3 = 107,000.00, t5 = 129,300.00, t6 = 132,064.00) (risk = 2 % p. a., Opportunity costs 3 % p. a., Inflation = 1,5 %) r = 0,02 + 0,03 + 0,015 = 6,5 % PV(1) = 88000/1,065 = 82629,11 4. PV(2) = 107000/1,065^3 = 88579,85 3. PV(3) = 129300/1,065^5 = 94372,59 1. PV(4) = 132064/1,065^6 = 90507,97 2. With cost (- 800, - 3000, - 4000, - 7000) PV(1) = (88000-800)/1,065 = 81877,93 4. PV(2) = (107000-3000)/1,065^3 = 86096,31 2. PV(3) = (129300-4000)/1,065^5 = 91454,07 1. PV(4) = (132064-7000)/1,065^6 = 85710,63 3. 2. What is better choice of this two: pay the car in 6 months 460,000.00 or pay it in 3 days with 5 % reduction of price? A bank offers you 6 % p. a., but the fee for opening the account is 1.300,-- PV (in 3 days) rebate = 460000-0,05*460000 = 437000 PV (t=0) = 437000/(1+0.06*3/360) = 436781.6 PV (of 460000) = 460000/(1+0.06/2) = 446601.9 You prefer the first offer! PV1