Question

An annuity immediate pays $500 per month for the first three years. After that the annuity payments increase by $50 per month for five years and then remain level for an additional six years. At a nominal rate of annual interest of 12% convertible monthly what is the present value of this annuity?

Answer 1

The annuity calculations can be divided into three sub-parts:

Part 1: Here, annuity cash flow(CF=$500) is constant every month for three years. Interest rate is compounded monthly, so effective rate is 12%/12 =1%

Present value of annuity, compounded monthly is:

CF*[1-(1+monthly interest rate)^(-3*12)]/monthly interest rate

Part 2:

Annuity increases by $50 every month for five years. So, we calculate present value of each monthly cash flow for 5 years i.e. 60 months. And then add up all these values and discount again for 36months to find present value at time t=0.

Part 3:

The annuity cash flow becomes stable at $3500, which was the cash flow in the previous month.

Applying the same formula as in part 1, we calculate the present value at the beginning of 97th month and then discount it further to arrive at present value at time t=0.

The total present value is sum of values in part 1, 2 and 3.

Qo! 04/22 - 2:34 Present value of annuity - Saved, Ą ☺ ☺ @ : fx = C5 *((1-((1+ C10 )^(- C8 )))/ - C10 ) A в с DE F р Cash flows Part 1: Periodic cashflow every month for 3 year = 500 8 36 9 Period of compoundi Nominal rate of interest per annum, 10 1% Present value of annuity: 15053.75252 15 16 Part 2: Period Cashflow Present value to time t=0, 550 380.6026953 600 411.0920202 cra Ann D EL Y co . § -

-O! O 4GR2 2:34 Present value of annuity - Saved, Ą : fx = $B18 /((1+ $C$10 )^( $218 ^ + $C$8 )) BC E F 36 8 9 Period of compoundi Nominal rate of interest per annum, 10 11 Present value of annuity: 15053.75252 13 14 15 16 Part 2: Period Cashflow Present value to time t=0, 17 nmin 0 554 380.602695 600 411.092020 650 440.9402857 700 470.1571972 750 498.7523308 800 526.7351349 850 554.1149315 900 580.9009183 25 D EL Y como o 9 su

OO! 0.46 % - 2:34 fx = $B77 /((1+ $C$10 )^( $A77 ^ + $C$8 )) A D E B C 2450 1161.617245 2500 1173.587841 2550 1185.207522 2600 1196.482065 2650 1207.417164 2700 1218.018435 2750 1228.291418 2800 1238.241573 2850 1247.874287 2900 1257.19487 2950 1266.208558 3000 1274.920515 3050 1283.335832 3100 1291.459529 3150 1299.296556 3200 1306.851792 3250 1314.130051 3300 1321.136076 3350 1327.874544 3400 1334.350066 3451340.567189 350V, 1346.530397 58965.2076 in una 1 0 60 60 79 80 Part 3: Annuity remains constant at $3500 for next 6 vears EL Y 9 sa

0 4 RA - 2:35 Quo! fx =SUM( C18:077 ) D E F. 76 77 1 78 A 5 9 60 B C 3450 1340.567189 35001346.530395 58965.20767 Part 3: Annuity remains constant at $3500 for next 6 years 80 81 Period of compounding: 72 Rate of interest for discounting: 0.01 83 84 85 86 Present value of annuity during this 6 years time period: 1 179026.3702 87 88 89 90 Present value of this annuity at time t=0 is: 68875.55686 Total Present Value: 142894.517 D EL Y fa

OO! 046R4 2:35 fx = B77 *((1-((1+ C83 )^(- 082 ) ^ ))/ 083 ) 78 A B C D E F. 59 3450 1340.567189 77 60 3500 1346.530395 58965.20761 79 80 Part 3: Annuity remains constant at $3500 for next 6 years 81 Period of 72 82 compounding: Rate of interest for discounting: 0.01 84 85 86 Present value of annuity during this years time period: 179026.3707 88 89 90 Present value of this annuity at time t=0 is: 68875.55686 Total Present Value: 142894.517 D EL Y fa

Quo! O4GARA 2:35 fx = C86 /((1+ C83 )^(36+60)) ^ 78 A B C D E F. 59 3450 1340.567189 77 60 3500 1346.530395 58965.20761 79 80 Part 3: Annuity remains constant at $3500 for next 6 years 81 Period of 72 82 compounding: Rate of interest for discounting: 0.01 83 84 85 86 Present value of annuity during this 6 years time period: am 179026.3702 87 88 89 90 Present value of thi annuity at time t= 0 68875.5568 is: Total Present Value: 142894.517 D EL Y fa