Money Excercise Qs

Take Your Pants Off

John plans to deposit 1000 every month in an account with an AER of 10.9%. How much will he be in the account after 3 years, continuously compounded.
Now I know this can be treated as an ordinary annuity question and you get the answer to be 42030, but why can you not use the compounding continuously formula, is it because that formula only works for years ?
Pe^(i)(n)

Another question I'm finding difficult is:
Say if you borrowed 10000 today, and must repay the loan in weekly installements over next 7 years. If quoted interest is 5%, compounded monthly, what is the value of each weekly payment you make.
First I got the apr for the monthly, worked out to be 0.795856 and then I am trying to convert it to the periodic rate per week, but how would I set up the equation for that
I/m=(1+0.795856)^(1/?)-1
I tried using 52 (as 52 weeks a year) but it does not work out.

Yakuza

Hi, that P * e^(i * n) is for a single sum invested at i (continuously compounded) for n years (doesn't have to be whole).
Also, you need to be careful as to what "i" you use.
If the AER is 10.9%, then the equivalent continuously compounded interest rate (δ (delta) in the actuarial exams) is defined as ln(1+i) where i is the annual effective rate.
The situation you describe is where 1000 is invested at the end of the month (so in effect the last investment gets no interest) for 36 months, so you end up with ...
1000 invested for 35 months + 1000 invested for 34 months +...1000 invested for 0 months.
Expressing the months as years, this turns into a nice little GP (Geometric Progression) sum (similar to the accumulation formula) a * (r^n-1)/(r-1), where a = 1000, r = exp(δ/12) and n = 36. This will give you the 42,030.85 you got when you used the formula based on annual interest rates.

On to your second part - you were almost there (assuming the question meant 5% interest PER MONTH (ouch!))
You correctly worked out the annual equivalent of it to be 79%, the weekly equivalent of that is (1.795856)^(1/52)-1 or .011323 (1.1323%).
Now you just have to equate 10000 with the present value of an annuity at rate 1.1323% for 364 weeks (7*52).
Let P be the payment, so:
10000 = P (v +v^2+...+v^364) (v = 1/1+i (where i in this case is 0.011323))
Pulling a v out, we get :
10000 = Pv (1 +v + ...+ v^363). The bit in the parentheses is another GP, with a = 1 and r = v and 364 terms, so the sum therefore is (1-v^364)/(1-v) (or equivalently (v^364-1)/(v-1) but that involves dividing two negative numbers and just a little more room for error)
So now we have :
10000 = Pv * (1-v^364) / (1-v), solving for P we get approx. 115.14 a week.

Just goes to show you how savage a rate of 5% per month is; you borrow 10k over 7 years and you end up paying the guts of 42 grand!

Take Your Pants Off

Yakuza wrote: »

Hi, that P * e^(i * n) is for a single sum invested at i (continuously compounded) for n years (doesn't have to be whole).
Also, you need to be careful as to what "i" you use.
If the AER is 10.9%, then the equivalent continuously compounded interest rate (δ (delta) in the actuarial exams) is defined as ln(1+i) where i is the annual effective rate.
The situation you describe is where 1000 is invested at the end of the month (so in effect the last investment gets no interest) for 36 months, so you end up with ...
1000 invested for 35 months + 1000 invested for 34 months +...1000 invested for 0 months.
Expressing the months as years, this turns into a nice little GP (Geometric Progression) sum (similar to the accumulation formula) a * (r^n-1)/(r-1), where a = 1000, r = exp(δ/12) and n = 36. This will give you the 42,030.85 you got when you used the formula based on annual interest rates.

On to your second part - you were almost there (assuming the question meant 5% interest PER MONTH (ouch!))
You correctly worked out the annual equivalent of it to be 79%, the weekly equivalent of that is (1.795856)^(1/52)-1 or .011323 (1.1323%).
Now you just have to equate 10000 with the present value of an annuity at rate 1.1323% for 364 weeks (7*52).
Let P be the payment, so:
10000 = P (v +v^2+...+v^364) (v = 1/1+i (where i in this case is 0.011323))
Pulling a v out, we get :
10000 = Pv (1 +v + ...+ v^363). The bit in the parentheses is another GP, with a = 1 and r = v and 364 terms, so the sum therefore is (1-v^364)/(1-v) (or equivalently (v^364-1)/(v-1) but that involves dividing two negative numbers and just a little more room for error)
So now we have :
10000 = Pv * (1-v^364) / (1-v), solving for P we get approx. 115.14 a week.

Just goes to show you how savage a rate of 5% per month is; you borrow 10k over 7 years and you end up paying the guts of 42 grand!

THANK YOU!!