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January 19th, 2012, 06:01 PM  #1 
Joined: Jan 2012 Posts: 2 Thanks: 0  Very difficult question (need help)
You are a female 30 years old and planning to retire at age 65. In order to save for your retirement, you are able to invest 300 per month in an account that earns an average of 10% compounded semiannually. As females, you have an expected longevity of 75 years, you believe you will live for ten years past your retirement age and will require a sum to live on, the beginning of each year. The annual amount you require in your retirement years will grow at the expected annual inflation rate of 5% per annum. You also plan to bequeath, upon your death at 75 years of age, 10,000 to the accounting and finance department of your university. Given these palns, what is the amount you would withdrawal in the first year of your retirement. Answer (132,691.91) Can someone please help me figure out how they got this answer? Thanks in advance. 
January 19th, 2012, 09:03 PM  #2 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,566 Thanks: 108 Math Focus: The calculus  Re: Very difficult question (need help)
Every six months $1800 is deposited and interest compounded, so after 35 years, or 70 compounding periods, the account has a value V in dollars given by: Now subtract the $10,000.00 going to the university and we are left with: $1102318.89 To find the first year's withdrawal W, we set: Close, but no cigar. I post my incorrect response so perhaps someone who knows how to actually solve this can show me where I went wrong. 
January 20th, 2012, 01:27 AM  #3  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
(1 + r)^12 = 1.05^2 Then use 300 (not 1800) and 420 (not 70).  
January 20th, 2012, 09:06 AM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Code: Time Dep/Wd Interest Balance 0 .00 1 300.00 .00 300.00 2 300.00 2.45 602.45 3 300.00 4.92 907.37 ..... 419 300.00 8,680.68 1,072,157.76 420 300.00 8,754.01 1,081,211.77 1 132,841.67 948,370.10 (same time as 420th deposit) 2 139,483.75 97,207.94 906,094.29 ..... 9 196,267.65 36,468.57 195,991.85 10 206,081.03 20,089.18 10,000.00 I am interpreting your "unclear problem" this way (make sure you tell your teacher to quit making up stories that are not clear mathwise!) : 420 monthly deposits of $300 are made in an account earning 10% compounded semiannually; the deposits are made at monthend. On the same day as the last deposit is made, $W is withdrawn; withdrawals continue annually for the next 9 years, increasing by 5% annually. The rate remains at 10% compounded semiannually (note that this was not stated in problem). "EDIT" After the last withdrawal, $10,000.00 remains in the account. Calculate W If instead I started the monthly deposits 1 month sooner (beginning of month), it would make the situation worse, since the account would be higher after 420 months. Anyway, 132,891.67  132,691.91 = 199.76: such a difference is really negligible when you consider a span of 45 years. If the "scenario" is different from what I assumed, let me know. 
January 20th, 2012, 01:31 PM  #5 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,566 Thanks: 108 Math Focus: The calculus  Re: Very difficult question (need help)
To obtain the equivalent monthly rate: To find the first year's withdrawal W, we set: 
January 20th, 2012, 02:26 PM  #6  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
Using actual year, last (420th) $300 deposit is made Dec.1(year 35). Interest is earned/credited on Dec.31(year 35). The 1st withdrawal is theoretically on same date...even if Jan.1(year 36). So really we're at the 44 year point when last withdrawal is made. Death is 1 year later: so I think we have to "revisit" this, and calculate to end up with ~9,070.29 after the last withdrawal: so at death (1 year later or end of 45th year), we end up with 9070.20 * 1.1025 = 10,000.00 What you say to that, Mark? Am I clear?  
January 20th, 2012, 02:41 PM  #7 
Joined: Jan 2012 Posts: 2 Thanks: 0  Re: Very difficult question (need help)
The answer is what is stated in the problem, I have asked the professor. I am pulling my hair out because of this problem. There is some trick with that 5% inflation rate. I am getting an answer very similar to you Dennis, but apparently, that is the wrong answer.

January 20th, 2012, 07:17 PM  #8  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
are they together, or a month apart, or something else? And did you ask about the $10,000 donation: was it made 1 year after the last annual withdrawal? Problem states: "You also plan to bequeath, upon your death at 75 years of age, ...."; but the last withdrawal is at 74 years old (beginning of year). Ask him/her if this is the intended cash flow (starting with year 1 for simplicity): Jan.1/01 300.00 (deposit#1) Feb,1/01 300.00 (deposit#2) ... Nov.1/35 300.00 (deposit#419) Dec.1/35 300.00 (deposit#420) Jan.1/36  W (wd#1) Jan.1/37  (1.05)W (wd#2) Jan.1/38  (1.05^2)W (wd#3) ... Jan.1/45  (1.05^9)W (wd#10) Jan.1/46 10,000.00 .....leaves account at exactly ZERO Surely they'll tell you if that's the correct interpretation of their confusing "wording"!  
January 21st, 2012, 03:00 AM  #9  
Joined: Apr 2011 From: USA Posts: 782 Thanks: 1  Re: Very difficult question (need help)
I do not know how to work this out using the math you guys are, and I don't know how to work out the last part of it. That is, I don't know how to increase a payment by 5% a year, and at the same time still compound the 10% on the remainder. I only have my "builtin" equations and I don't know any that do that. So I will merely toss in my thoughts on what's been posted and you two can grab it from there. Quote:
Because it's not compounding, I just figured out what six $300 payments would come out to at the end of six months, starting at the beginning of the month. (The problem doesn't say beginning or end, which is a serious flaw in the problem, but since the retirement is at the beginning, I'm going to just go with that throughout. I haven't yet reworked it for end of the month.) $300 a month, not compounding but earning, at the end of six months is $1853.24. (i.e. $300 for 6 months, $300 for 5 months, etc.) So that's how much there is at the end of the first six months. That amount will now just continue to compound at 5% per six months. The next six months the same thing is going to happen. So there's another 1853.24 plus the amount compounded on the first 1853.24. This seems to me is the equivalent of making a 1853.24 payment at the end of every six months for 70 periods at 5%. No? That gave me 1,090,681.80. Quote:
Quote:
Quote:
But once that's subtracted off, I have no clue how to do the rest of the retirement part of it.  
January 21st, 2012, 06:04 AM  #10  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
 
January 21st, 2012, 06:20 AM  #11  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
I think you used .10/12 as rate: slightly too high: rate compounds semiannually, not monthly. .10/12 = .008333.... ; should be .008164... It is USUAL to make the interest rate factor match the frequency of payments.  
January 21st, 2012, 11:49 AM  #12  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
Code: j = 10% i = 12% Year Deposit Interest Balance 0 .00 .00 .00 1 1000.00 .00 1000.00 2 1100.00 120.00 2220.00 3 1210.00 266.40 3696.40 F = Future value (?) D = initial deposit (1000) j = deposit increase (.10) i = interest rate (.12) n = number of years (3) F = D*[(1 + i)^n  (1 + j)^n] / (i  j) F = 1000(1.12^3  1.10^3) / (.12  .10) = 3696.40 Basically all that simple...google will give ya lotsa sites, like: http://www.financeformulas.net/Growing ... Value.html  
January 21st, 2012, 08:33 PM  #13  
Joined: Apr 2011 From: USA Posts: 782 Thanks: 1  Re: Very difficult question (need help) Quote:
 
January 21st, 2012, 08:49 PM  #14  
Joined: Apr 2011 From: USA Posts: 782 Thanks: 1  Re: Very difficult question (need help) Quote:
This is a present value relative to the withdrawals, payments growing at 5% annually, but earning (we assume) 10% semiannually. That equation is growing to a future value, not shrinking from a present value.  
January 22nd, 2012, 07:00 AM  #15  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 3,084 Thanks: 36  Re: Very difficult question (need help) Quote:
This was the rate per the problem: "in an account that earns an average of 10% compounded semiannually". 10% cpd semiannually means 10.25% annually ; 1.05^2  1 = .1025 So the problem statement could as well be: ....10.25% compounded annually (same thing). It is standard practice to convert the effective ANNUAL rate such that it matches the payments frequency; on this case, to 12 periods annually; so: (1 + i)^12 = 1.1025 ; i = .0081648... So the "semiannual equivalent deposit" is ~1852.14, not ~1853.24. Another example:If given rate is 12% annual cpd. quarterly and payment frequency is 6 (every 2 months), then equivalent rate becomes: (1 + i)^6 = (1 + .12/4)^4 ; i = .019901...slightly lower than 2%, as expected.  

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