Question

Please show all your work step by step and all the formula that is used. This is my 3rd time posting as people do not explain what they are doing. Therefore, do not answer if you can't explain.

D. What is the present value (PV) of a 12-years lease arrangement with an interest rate of 7.5% that requires annual payments of $4250. Per year with first payment being due now?

E. A recent college graduate hopes to have $200000. Saved in their retirement account 25 years from now by contributing $150 per month in a 401(k) plan. The goal is to earn 10% annually on the monthly contribution. Will they have $200000. At the end of the 25 years?

Answer 1

1.

Present value=4250+4250/1.075+4250/1.075^2.....

This is a geometric progression with first term as 4250, common ratio as 1/1.075 and number of terms as 12 ( as it is a 12 year lease)

We know the sum of a geometric progression with first term as a, common ratio as r and number of terms as n=a*(1-r^n)/(1-r)

Hence, the above Present value=4250*(1-(1/1.075)^12)/(1-(1/1.075))=35340.55

2.

Here the first saving will be invested for 12*25=300 months, second for 299 months and so on.

The rate of interest compounded monthly=12*((1+10%)^(1/12)-1=9.569%

Future value=150*(1+9.569%/12)^300+150*(1+9.569%/12)^299+150*(1+9.569%/12)^298.........

This is a geometric progression with first term as 150*(1+9.569%/12), common ratio as 1/(1+9.569%/12) and number of terms as 300 ( =12*25)

We know the sum of a geometric progression with first term as a, common ratio as r and number of terms as n=a*(1-r^n)/(1-r)

Hence, the above Future value=(150*(1+9.569%/12)^300)*(1-(1/(1+9.569%/12))^300)/(1-(1/(1+9.569%/12)))=186474.9

So, they won't have $200000 at the end of the 25 years