Question

Linear Gradient

4.- If you make a deposit of 5,000 dollars at the end of this year and you decrease this amount by 200 dollars each year, and the interest rate is 9%. a) what is the present value of this cash flows b) what is the balance after the last deposit.

Please step by step.

Thank you!

Answer 1

If the amount of deposit is reduced by $200 each year, the last deposit will be $200 at the end of the 25th year.

Now we will calculate the present value of these deposits, using 9% interest rate to discount the cash flows

Present value = 5000/1.09 + 4800/(1.09^2) + 4600/(1.09^3) + 4400/(1.09^4) + 4200/(1.09^5) + 4000/(1.09^6) + 3800/(1.09^7) + 3600/(1.09^8) + 3400/(1.09^9) + 3200/(1.09^10) + 3000/(1.09^11) + 2800/(1.09^12) + 2600/(1.09^13) + 2400/(1.09^14) + 2200/(1.09^15) + 2000/(1.09^16) + 1800/(1.09^17) + 1600/(1.09^18) + 1400/(1.09^19) + 1200/(1.09^20) + 1000/(1.09^21) + 800/(1.09^22) + 600/(1.09^23) + 400/(1.09^24) + 200/(1.09^25)

= 4,587.16 + 4,040.06 + 3,552.04 + 3,117.07 + 2,729.71 + 2,385.07 + 2,078.73 + 1,806.72 +1,565.45 + 1,351.71 + 1,162.60 + 995.50 + 848.06 + 718.19 + 603.98 + 503.74 + 415.93 + 339.19 + 272.20 + 214.12 + 163.70 + 120.15 + 82.67 + 50.56 + 23.19 = $33,727.5

The balance after the last deposit, assuming deposits grew 9% per annum is

5000*(1.09^24) + 4800*(1.09^23) + 4600*(1.09^22) + 4400*(1.09^21) + 4200*(1.09^20) + 4000*(1.09^19) + 3800*(1.09^18) + 3600*(1.09^17) + 3400*(1.09^16) + 3200*(1.09^15) + 3000*(1.09^14) + 2800*(1.09^13) + 2600*(1.09^12) + 2400*(1.09^11) + 2200*(1.09^10) + 2000*(1.09^9) + 1800*(1.09^8) + 1600*(1.09^7) + 1400*(1.09^6) + 1200*(1.09^5) + 1000*(1.09^4) + 800*(1.09^3) + 600*(1.09^2) + 400*(1.09^1) + 200

= 39,555.41 + 34,837.80 + 30,629.56 + 26,878.75 +23,538.52 + 20,566.65 + 17,925.06 + 15,579.48 + 13499.04 + 11,655.94 + 10,025.18 + 8584.25 + 7312.93 + 6193.02 + 5208.20 + 4343.79 + 3586.61 + 2924.86 + 2347.94 + 1846.35 + 1411.58 + 1036.02 + 712.86 + 463+ 200 = $290,835.8