Question

How much money is needed up/front to pay 100 per week for 20 years, if interest is 7.5% c.c

Answer 1

compounded continuously eh, and these are payments, so its an annuity type setup

Answer 2

yeah there is a integral equation for it as well

Answer 3

they provide one for you?

Answer 4

i was thinking of trying to develop an approximation

Answer 5

well im not sure if its the same thing it says for Present value formula

Answer 6

\[B_n=B_ok^n-P\frac{1-k^n}{1-k}\]solving for B_o and since we would want Bn = 0

\[0=B_ok^n-100\frac{1-k^n}{1-k}\]
\[B_o=\frac{1}{k^n}100\frac{1-k^n}{1-k}\]

Answer 7

\[\int\limits_{0}^{t1}ce^(-rt)dt =c/r(1-e^(-rt))\]

Answer 8

the issue might be in determing a suitable k and n value

Answer 9

thats what they put on the homework page im not sure what it is though

Answer 10

if we assume compounded daily as a good approximation; and then compounded hourly, we could see how they compare and interpolate from there

Answer 11

\[k=1+\frac{r}{c}~:~n=20c\]
\[k_1=1+\frac{.075}{365}~:~n=20(365)\]
\[B_o=\frac{100}{(1+\frac{.075}{365})^{20(365)}}\frac{1-(1+\frac{.075}{365})^{20(365)}}{1-(1+\frac{.075}{365})}\]

Answer 12

okay i think i understand what does k represent

Answer 13

k is a clean up variable, its just easier to write, it represents that "1+interest" that gets compounded

Answer 14

for daily compounding, the wolf gives me: 378,059.921635

lets see how well that does for an hourly period

Answer 15

just times 365 by 24 seems reasonable to me

Answer 16

i think thy just want me to set up the equation its multiple choice i you want i can write them out

Answer 17

i was just trying out a different idea is all :) its prolly simpler to integrate the given stuff from 0 to 20

Answer 18

same integration idea may be... but this one lil off from the principal we got by above formulae..

http://www.wolframalpha.com/input/?i=x*%28e%5E%28.075%2F52*52*20%29%29+-100%281-e%5E%28.075%2F52*52*20%29%29+%2F+%281-e%5E%28.075%2F52%29%29%29+%3D+0

Answer 19

oh ok i just had wanted to know what my integral i thought it was 20 multiply by the number of months

Answer 20

hmm, this is a weekly setup too, 20*52 to start with, then incremental by weeks

F = Pe^rt, so finding the present value of one payment would amount to:

F/e^(rt) for every given payment

Answer 21

20*52 = 1040 periods

\[\frac{100}{e^{1040(.075)}}+\frac{100}{e^{1039(.075)}}+...\]
is what i believe the integral they gave you is aluding to

Answer 22

what are the options by the way?

Answer 23

well if the integral is from 0 to 1040 there is only two options left and its either integral 1000e^(-0.00144(1040) or just none

Answer 24

im thinking that the 1040 is off, and that it needs to be 0 to 20

Answer 25

it just increments in weekly payments

Answer 26

i made a mistake yeah the -0.00144t

Answer 27

1/52 would be an increment

Answer 28

would that be my t

Answer 29

52 weeks in a year, so 1/52 of a year increments

present value of: 100 at 20 - 0/52 years
+ present value of : 100 at 20 - 1/52 years
+ present value of : 100 at 20 - 2/52 years
+ present value of : 100 at 20 - 3/52 years
....

Answer 30

ok i see i was a bit confuse...so for the set up the integral goes from 0 to 1040 since is weekly and if it was monthly i would just multiply the number of years to months

Answer 31

i believe so, but im not confident that i understand what you posted as the "integral" :) im working this up in excel at the moment; they simply want a starting balance that will cover all the payments right?

Answer 32

im getting a present value of about: 5250.16

Answer 33

..but i forgot to include the interest rate

Answer 34

53901.83

Answer 35

thats within about 100 dollars of ganeshes results

Answer 36

yeah i think thats correct for the interest rate did you put it as -0.075

Answer 37

yes

Answer 38

ok haha this took longer than i thought ...thank you

Answer 39

i had to take a longer route since im not sure what the integral equation they provided meant :/

Answer 40

yeah sorry.. and thanks again

Answer 41

good luck ;)