My Math Forum  

Go Back   My Math Forum > Science Forums > Economics

Economics Economics Forum - Financial Mathematics, Econometrics, Operations Research, Mathematical Finance, Computational Finance


Reply
 
LinkBack Thread Tools Display Modes
January 30th, 2014, 04:03 PM   #1
Newbie
 
Joined: Jan 2014

Posts: 4
Thanks: 0

Finance Problem

Hi there
I have difficulty solving this question. Any help appreciated

Question:
Once you retire you will be withdrawing $48,000 at the beginning of every year for another 30 years. To make your retirement plans and for the coming 30 years, you will start depositing at the end of every year over 24 years a certain amount of money in your bank account. Since you are expecting that your salary will grow by 2% every year for the coming 24 years then you expect your yearly savings to grow by the same rate. If your bank is paying you 3.25% Annual Percentage Rate compounded weekly. What should be your first deposit so that you can achieve your retirement plan?
Marcipanas is offline  
 
January 31st, 2014, 07:49 PM   #2
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 13,302
Thanks: 935

Re: Finance Problem

Hello. Have a look at the thread by "Noritaka" which is almost identical to yours.

If you need more help, come back and ask.
Denis is offline  
February 2nd, 2014, 01:14 PM   #3
Newbie
 
Joined: Jan 2014

Posts: 4
Thanks: 0

Re: Finance Problem

Ok so i get that its three step process going backwards.

First I need to know effective annual interest rate

k = (1 + 0.0325/52)^52 - 1 = 0.03302

Then we need to know how much money I have to have to allow to take 48,000 per year for 30 years.

PV1 = 48000 / 0.03302 * ( 1 - 1/1.03302^30 ) = 905,132.061 ( so after 30 year I have to have this amount in my bank to be able to take 48,000 dollars a year for 30 years right? )


Then because I was depositing money only for 24 years for 6 years money was growing on interest

PV2 = 905,132.061 / (1 + 0.003302)^6 = 744,834.131 ( SO after depositing for 24 I need to have this amount in my banck if I want to retire after 6 years and take 48000 per year for 30 years)


And now Lastly is the part I do not get.

What formula do I use here?

Growing annuity? ? ?

is it this one? But this one is present value but i need future value right?

PV0 = PMT/(k-g) * ( 1 - ((1+g)/(1+k))^n

Lets solve for PMT as I need to know PMT

PMT = PV0 * (k - g)/ (1 - ((1+g)/(1+k))^n) = 744834.131 * (0.03302 - 0.02) / (1-((1-0.02)/(1+0.003302))^24) = 36,951.5616 ( SO this means that I have to start by depositing 36,951 first time to achieve it. But the answer is not correct as there are
a) 18,495.93 b) 20,590.39 c) 21,270.29 d) 442,895.39 e) non of above

So please help me if you can I am pretty sure I am wrong with last formula as I dont need PV I need FV right? Or am I totally lost in finance?
Marcipanas is offline  
February 2nd, 2014, 07:08 PM   #4
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 13,302
Thanks: 935

Re: Finance Problem

Quote:
Originally Posted by Marcipanas
Then because I was depositing money only for 24 years for 6 years money was growing on interest

PV2 = 905,132.061 / (1 + 0.003302)^6 = 744,834.131 ( SO after depositing for 24 I need to have
this amount in my banck if I want to retire after 6 years and take 48000 per year for 30 years)
That's wrong. There is no 6 years involved.
Deposits are made for 24 years, then the 48,000 starts right away (end of 24th year)
or 1 year later (end of 25th year).

Here's the calculations:

GIVENS: r = .0325, j = .02, p = 48000, n = 24, m = 30

Step 1; convert rate:
i = (1 + r/52)^52 - 1 : .0330234....

Step 2; get required value v at end of n years:
v = p[1 - 1/(1 + i)^m] / i : 905092.957848....

Step 3; get 1st deposit d (which increases by j annually) which will accumulate to v :
d = v(i - j) / [(1 + i)^n - (1 + j)^n] : 20590.39 .....which is choice b)
This assumes that the 1st 48000 starts at end of 25th year.

IF 1st 48000 starts at end of 24th year,
(as your problem states "$48,000 at the beginning of every year"; but does not say which year)
then the formula at Step 2 needs to be slightly revised to:

Step 2; get required value v at end of n years:
v = {p[1 - 1/(1 + i)^m] / i} * (1 + i) : which gives 934982.209896....
Step 3 remains same:
d = v(i - j) / [(1 + i)^n - (1 + j)^n] : 21270.29 .....which is choice c)

Looks like your teacher wants you to decide if there is a 1 year "waiting" period or not!
Denis is offline  
February 2nd, 2014, 10:04 PM   #5
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 13,302
Thanks: 935

Re: Finance Problem

In case you insist that there is a 6 years delay:

GIVENS: r = .0325, j = .02, p = 48000, n = 24, m = 30, w = 6 (delay)

Step 1; convert rate:
i = (1 + r/52)^52 - 1 : .0330234....

Step 2; get required value v at end of n years:
v = p[1 - 1/(1 + i)^m] / i : 905092.957848.... (same as before)

Step 3; get 1st deposit d (which increases by j annually for n years, then interest only for w years)
which will accumulate to v (notice change to formula):
d = v(i - j) / {[(1 + i)^n - (1 + j)^n](1 + i)^w} : 16942.5428...
This assumes that the 1st 48000 starts at end of 31st year.

IF 1st 48000 starts at end of 30th year,
(as your problem states "$48,000 at the beginning of every year"; but does not say which year)
then the formula at Step 2 needs to be slightly revised to:

Step 2; get required value v at end of n years:
v = {p[1 - 1/(1 + i)^m] / i} * (1 + i) : which gives 934982.209896....(same as before)

Step 3; get 1st deposit d (which increases by j annually for n years, then interest only for w years)
which will accumulate to v:
d = v(i - j) / {[(1 + i)^n - (1 + j)^n](1 + i)^w} : 17502.0432...

So we have 16942.54 or 17502.04: both NOT in your answer choices...convinced?!
Denis is offline  
February 3rd, 2014, 08:15 AM   #6
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Finance Problem

Quote:
Originally Posted by Marcipanas
a) 18,495.93 b) 20,590.39 c) 21,270.29 d) 442,895.39 e) non of above
I get a number rather close to c). I used a spreadsheet with
=B2*(1+B$1)+C3
in B3 (dragged to B26),
=(B26-48000)*(1+B$1)
in B27 (dragged to B56),
=C3*1.02
in C4 (dragged to C26),
= (1 + 0.0325/52)^52 - 1
in B1, and the answer to the problem in C3. This should leave B56 at 0.
CRGreathouse is offline  
February 3rd, 2014, 11:32 AM   #7
Newbie
 
Joined: Jan 2014

Posts: 4
Thanks: 0

Re: Finance Problem

Thank you guys.
I see my mistakes as english is not my mother tongue I got messed up with extra 6 years and I forgot to use Annuity due formula and took simple annuity.
Marcipanas is offline  
February 3rd, 2014, 11:34 AM   #8
Newbie
 
Joined: Jan 2014

Posts: 4
Thanks: 0

Re: Finance Problem

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by Marcipanas
a) 18,495.93 b) 20,590.39 c) 21,270.29 d) 442,895.39 e) non of above
I get a number rather close to c). I used a spreadsheet with
=B2*(1+B$1)+C3
in B3 (dragged to B26),
=(B26-48000)*(1+B$1)
in B27 (dragged to B56),
=C3*1.02
in C4 (dragged to C26),
= (1 + 0.0325/52)^52 - 1
in B1, and the answer to the problem in C3. This should leave B56 at 0.
Our teacher said to round intermediate calculations to 5 digits after point ( y.xxxxx) . And excel takes every number after point.
Marcipanas is offline  
February 3rd, 2014, 01:52 PM   #9
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Finance Problem

Quote:
Originally Posted by Marcipanas
Our teacher said to round intermediate calculations to 5 digits after point ( y.xxxxx) . And excel takes every number after point.
You could force it to round however you like, of course.
CRGreathouse is offline  
February 3rd, 2014, 08:07 PM   #10
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 13,302
Thanks: 935

Re: Finance Problem

Quote:
Originally Posted by Marcipanas
I forgot to use Annuity due formula and took simple annuity.
Would have saved lots of time if that had been specified...
The problem can be "SIMPLY" worded this way:

An annuity due (or annuity immediate) allowing $48,000 annual for 30 years, starting in 24 years
is to be set up with 24 annual deposits starting in 1 year, the deposits increasing by 2% annually.
The annual savings rate for the full period is 3.25% compounded weekly.

That's it! No need for word-happy teachers to build a story confusing the issue

Solved as following:

GIVENS: r = .0325, j = .02, p = 48000, n = 24, m = 30

Step 1; convert rate:
i = (1 + r/52)^52 - 1 : .033023405816

Step 2; get required value v at end of n years:
v = {p[1 - 1/(1 + i)^m] / i} * (1 + i) : which gives 934982.209896983414

Step 3; get 1st deposit d (which increases by j annually for n years) that accumulates to v:
d = v(i - j) / [(1 + i)^n - (1 + j)^n] : 21269.129877079599
The "choice" given is 21,270.29 ; I'm using greater accuracy as you can see...

The whole thing will look like this (bank statement format):
Code:
YEAR    TRANSACTION       INTEREST         BALANCE
  1       21,269.13            .00       21,269.13  
  2       21,694.51         702.38       43,666.02
...
 24       33,539.28      28,817.07      934,982.21
 24      -48,000.00            .00      886,982.21
 25      -48,000.00      29,291.17      868,273.38
...
 52      -48,000.00       3,019.84       46,465.55
 53      -48,000.00       1,534.45             .00
Notice that 1st deposit of 21269.13 * 1.02^23 = 33539.28, the last deposit.
Denis is offline  
Reply

  My Math Forum > Science Forums > Economics

Tags
finance, problem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Need help with finance math problem Mikev078 Economics 5 December 7th, 2012 12:45 PM
Simple Mathematical Finance problem - please help... jamie89 Calculus 1 November 3rd, 2009 01:47 PM
Finance problem mr.confused Algebra 3 September 28th, 2008 09:53 PM
Simple Mathematical Finance problem - please help... jamie89 Economics 0 December 31st, 1969 04:00 PM
finance help moralvirus Algebra 3 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.