Question

You invest $1000 per month for 9 months starting 4 months from now (i.e., from month 4 to month 12), and then you increase your investment by $100 per month for 2 years (i.e., from month 13 to month 36). If the rate of return is 60% per year, what is the equivalent annual worth ($ per year) of this cash flow in years 1 to 3?

Answer 1

The cash flow contains 2 parts-

(1) Fixed amount of $1,000 per month during 9 months from months 4 to 12

(2) Additional amount increased by $100 every month (arithmetic gradient) from months 13 to 36 (24 payments)

Value (as at end of month 3) of fixed uniform series (item 1 above)= $7,107.82 Calculated as the present value of annuity as follows:

с D А в 1 Present Value of Annuity Payments at the end of each period 2 - 12 3 Present value of annuity is calculated using the formula PVA=P*{[1-(1+r)^-n]/r]} 4 Where P= Periodical payments, 5 n= Number of payments. 6 r= Rate of interest per period in decimals All amounts in $ Given that 9 Periodical payments (P)= 1000 No. of years (N)= 0.75 10 Rate of interest (% p.a)= R= 60 No. of payments a year (t)= 11 12 Calculations: 13 Formula Cell reference Value 14 Interest rate per period in decimals (r )= R/(100*t) D10/(100*G10) 0.05 15 No. of periods (n)= N*t G9*G10 16 (1+r)= 1+G14 1.05 17 (1+r)^-n= G164-G15 0.64460892 18 [1-(1+r)^-n]= 1-G17 0.35539108 19 [1-(1+r)^-n]/r= G18/G14 7.10782168 20 Present value of annuity PV= P*{[1-(1+r)^-n]/r]} D9*G19 7107.82168 21 Rounded to $7,107.82 9

Value of this amount as at the beginning (month 0) = $7,107.82/(1+60%/4)= $6,180.72

Uniform series of payments required for the gradient (increase in amount)(item 2 above)=$579.79

Details of calculation as follows:

E B C D 1 Uniform series of payments required for the gradient (increase in amount)= 2 =G* {(1/i )-n/((1+i)^n)-1} 3 Where G= gradient (increase in periodical amount), 4 i= interest rate and n= number of periods of payment. 5 Given, 6 Gradient G= $ 100.00 7 Period (number of months) n= 24 8 Interest rate per month 60%/4 15.00% 9 Calculations: Formula Cell reference Values 1/i 1/E8 6.66666667 11 (1+i)^n (1+E8)^E7 28.6251762 (1+i)^n -1 E11-1 27.6251762 13 n/(1+i)^n-1 E7/E12 0.86877274 (1/i)-n/(1+i)^n-1 E10-E13 5.79789393 15 Uniform series payments G*((1/1)-(n/(1+i)^n-1)) E6*E14 $ 579.79 10 12

Value of this uniform series equivalent, as at the end of month 12= $8,000.31 Calculated as the present value of annuity as follows:

21 A B C D E F G 1 Present Value of Annuity Payments at the end of each period 2 3 Present value of annuity is calculated using the formula PVA= P*{[1-(1+r)^-n]/r]} 4 Where P= Periodical payments, 5 n= Number of payments. 6 r= Rate of interest per period in decimals 7 All amounts in $ 8 Given that 9 Periodical payments (P)= 579.79 No. of years (N)= 10 Rate of interest (% p.a)= R= 60 No. of payments a year (t)= 12 11 12 Calculations: 13 Formula Cell reference Value 14 Interest rate per period in decimals (r )= R/(100*t) D10/(100*410) 0.05 15 No. of periods (n)= G9*G10 24 16 (1+r)= 1+G14 1.05 (1+r)^-n= G164-G15 0.31006791 18 [1-(1+r)^-n]= 1-G17 0.68993209 19 [1-(1+r)^-n]/r= G18/614 13.7986418 20 Present value of annuity PV= P*{[1-(1+r)^-n]/r]} D9*G19 8000.31453 21 Rounded to | $8,000.31 N*t 17 22

Present Value of this amount as at the beginning (month 0)= $8,000.31/(1+60%) = $ 5,000.20

Therefore, total present value of the cash flows as at the beginning= $6,180.72 + $ 5,000.20 = $ 11,180.92

Equivalent annual worth of the cash flow= $8,875.40 calculated as uniform annuity payment from this present value as follows:

A B C D Ε 1 Annuity Payment Payments at the end of each period 2 3 Amount of periodical payments is calculated using the formula PMT= (PV*r)/(1-(1+r)^-n] 4 Where PMT= Periodical payments required, 5 PV= Present value in lumpsum 6 n= Number of payments. 7 r= Rate of interest per period in decimals All amounts in $ 9 Given that 10 Present value (PV) 11,180.92 No. of years (N)= 3 11 Rate of interest (% p.a)=R= 60 No. of payments a year (t)= 12 13 Calculations: Formula Cell reference Value 15 Interest rate per period in decimals (r )= R/(100*t) D11/(100*611) 0.6 16 No. of periods (n)= N*t G10*G11 17 (1+r)= 1+G15 1.6 18 (1+r)^-n= G174-G16 0.244140625 19 [1-(1+r)^-n]= 1-G18 0.755859375 20 PV*r= D10*G15 6708.552 21 Equivalent Annual worth (PV*r)/(1-(1+r)^-n] G20/619 8875.396961 22 Rounded to | $8,875.40 14 3