BBA307 Assignment 4.1 Solutions1. Draw a timeline for (1) a $100 lump sum cash flow at the end of year 2, (2) an ordinary annuity of $100 per year for 3 years and (3) an uneven cash flow stream of -$50, $100, $75, and $50 at the end of years 0 through 3. Solution A time line is a graphical representation which is used to show the timing of cash flows. The tick marks represent end of periods (often years), so time 0 is today; time 1 is the end of the first year, or 1 year from today; and so on.I% 0 1 2 year | | | lump sum 100 cash flowI% 0 1 2 3 | | | | annuity 100 100 100 I% 0 1 2 3 | | | | uneven cash flow stream-50 100 75 50 A lump sum is a single flow; for example, a $100 inflow in year 2, as shown in the top time line. An annuity is a series of equal cash flows occurring over equal intervals, as illustrated in the middle time line. An uneven cash flow stream is an irregular series of cash flows which do not constitute an annuity, as in the lower time line. -50 represents a cash outflow rather than a receipt or inflow.2. Calculate the following:a. The future value of an initial $100 after 3 years assuming annual interest of 10%. Solution:Show dollars corresponding to question mark, calculated as follows:10% 0 1 2 3 | | | |100 FV = ? After 1 year:FV1= PV + I1 = PV + PV(I) = PV(1 + I) = $100(1.10) = $110.00.Similarly:FV2= FV1 + I2 = FV1 + FV1(I) = FV1(1 + I)= $110(1.10) = $121.00 = PV(1 + i)(1 + i) = PV(1 + i)2.FV3= FV2 + I3 = FV2 + FV2(I) = FV2(1 + I)= $121(1.10)=$133.10=PV(1 + I)2(1 + I)=PV(1 + I)3.In general, we see that:FVn= PV(1 + I)N,SOFV3= $100(1.10)3 = $100(1.3310) = $133.10.Note that this equation has 4 variables: FVN, PV, I, and N. Here we know all except FVN, so we solve for FVN. We will, however, often solve for one of the other three variables. By far, the easiest way to work all time value problems is with a financial calculator. Just plug in any 3 of the four values and find the 4th.Finding future values (moving to the right along the time line) is called compounding. Note that there are 3 ways of finding FV3: using a regular calculator, financial calculator, or spreadsheets. For simple problems, we show only the regular calculator and financial calculator methods.(1)regular calculator:1.$100(1.10)(1.10)(1.10) = $133.10.2.$100(1.10)3 = $133.10.(2)financial calculator:This is especially efficient for more complex problems, including exam problems. Input the following values: N = 3, I = 10, PV = -100, PMT = 0, and solve for FV = $133.10.b. The present value of $100 to be received in 3 years if the discount rate is 10%.Solution:Finding present values, or discounting (moving to the left along the time line), is the reverse of compounding, and the basic present value equation is the reciprocal of the compounding equation:10% 0 1 2 3 | | | |PV = ? 100FVn = PV(1 + I)N transforms to:PV = = FVn = FVn(1 + i)-nthus:PV = $100 = $100(PVIFi,n) = (0.7513) = $75.13.The same methods used for finding future values are also used to find present values.Using a financial calculator input N = 3, I = 10, pmt = 0, FV = 100, and then solve for PV = $75.13.3. If a companys sales are growing at a rate of 20% per year, how long will it take for the sale to double?Solution:We have this situation in time line format:20% 0 1 2 3 3.8 4 | | | | | |-12 2Say we want to find out how long it will take us to double our money at an interest rate of 20%. We can use any numbers, say $1 and $2, with this equation: FVn = $2= $1(1 + i)n = $1(1.20)n. (1.2)n= $2/$1 = 2n LN(1.2)= LN(2) n= LN(2)/LN(1.2) n= 0.693/0.182 = 3.8.Alternatively, we could use a financial calculator. We would plug I = 20, PV = -1, PMT = 0, and FV = 2 into our calculator, and then press the N button to find the number of years it would take 1 (or any other beginning amount) to double when growth occurs at a 20% rate. The answer is 3.8 years, but some calculators will round this value up to the next highest whole number. The graph also shows w