Annuties (Again) (1 Viewer)

[Paj20]

Kylie wants to take a world trip in 5 years time. She estimates that she will need $25,000 for the trip. The best investment that Kylie can find pays 9.2% p.a interest, compounded quartely.

Kylie plans to save for the trip by depositing $100 p/w into an annuity. Calculate if this will be enough for Kylie to achieve her savings goal.

I got $20,654 which is no but the book says yes??

 

[pLuvia]

Firstly she deposits 100 per week, so change the interest rate to a weekly effective rate which is 0.001750..., let that be i⁵²/52

Hence

$100((1+i⁵²/52)^(52*5)-1)/(i⁵²/52)
= $32891.69 (2dp)

Hence yes

 

[Paj20]

Firstly she deposits 100 per week, so change the interest rate to a weekly effective rate which is 0.001750..., let that be i⁵²/52

Hence

$100((1+i⁵²/52)^(52*5)-1)/(i⁵²/52)
= $32891.69 (2dp)

Hence yes

i dont get that... what formula r you using is it the future value 1??

and for the interest rate i got 9.2 divded by 52 to get 0.00176923.....

 

[pLuvia]

The future value formula is

((1+i)ⁿ-1)/(1-(1+i))

That's the wrong interest rate you got there. Since it says the interest rate is 9.2% p.a. compounded quarterly it means that the current interest rate is 9.2/4% effective quarterly. To work out the equivalent effective weekly interest rate

(1+0.092/4)⁴=(1+i⁵²/52)⁵²

Then you solve for i⁵²/52 which is what I stated before

 

[Paj20]

lol sorry still dont get how you got the rate

I just usually divide 9.2 by 4

 

[pLuvia]

But then the interest rate you are using for the annuity will be effective quarterly and not effective weekly. You have to convert to a effective weekly interest rate then you can do the question. Because the payments are weekly payments you can't use the interest rate given to you because it compounded quarterly, so you need a weekly interest rate

Ok since you are trying to find an equivalent nominal weekly rate to the nominal quarterly rate given to you. It should be obvious lets say if you were compounding $1 for a year at the interest given to you, the future value of it would be (for the quarterly rate case) $1(1+i⁴/4)⁴. And for the weekly rate case, $1(1+i⁵²/52)⁵²

Since both these should be the same amount

Hence
$1(1+i⁴/4)⁴=$1(1+i⁵²/52)⁵²


Then you solve for i⁵²/52

You're textbook should show you how to convert from two different interest rates where it compounds at different frequencies

The formula should be

(1+iⁿ/n)ⁿ=(1+i^m/m)^m

Where iⁿ/n and i^m/m are the nominal interest rates compounded n-thly or m-thly respectively

 

[Paj20]

lol yeh thanks i get it now..

But for this 1 it says 'At the end of each year for 15 years, Kiohero invests $2500 into an account earning 8%p.a compounded interest. Calculate the Future Value of:

(A) The First Payment:

What does it mean by the first payment???

 

[pLuvia]

What are the next parts to the question?

It might be just the future value of the first payment which is 2500*(1.08)^15

 

[Paj20]

lol nevermind i get it now

But lol again, for this 1: Find the amount of each monthly instalment if $50,00 is borrowed at 1.2% per month over 7 years?

Why isnt the AxR OVER (1+r)^n -1 formula used to find the answer cause i use it and its wrong

 

[pLuvia]

Present value formula is
[1-(1+i)^-n]/(i) where i is the effective interest rate, n is the number of payments

You need to find the present value at time 0 and make that equal to $50000

Let X be the monthly instalments
50000=X(1-(1.012)^-84)/(0.012)
X=948.08

But you can also do it from first principles

Let X be the monthly instalments in dollars
Let v be (1+i)^-1 where i is the effective interest rate

So basically you need to discount every payment to time 0 where the 50000 was borrowed so
$50000=Xv+Xv²+Xv³+...+Xv⁸³+Xv⁸⁴
The left hand side is a geometric series so:

$50000=Xv(1-v⁸⁴)/(1-v)
X=948.08

I'm not sure what formula that is, I didn't study general maths sorry