Complicated Retirement Problem

annuity_man

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Nick and Sharon would like to retire when they are both 50 years old. Nick and Sharon are both 26 years old. Nick is currently employed, but Sharon is trying to become a lawyer. Nick currently has $30,000 in an investment account, which earns him 6% per year compounded semi-annually. Nick would like to contribute to their joint retirement fund every two months until Sharon finishes her studies when they’re both 30. Nick’s contributions would be made at the end of each two month period. From that point Sharon would like to make all of the contributions to their retirement fund. She would also contribute every two months. These contributions would be made at the end of each two month period. During their retirement, they would like to withdraw $15,000 per month at the beginning of each month. They would like the amount they receive to increase by ½% each month. Nick is expected to live until he is 85. Sharon expects to live until she is 90. However, after Nick’s death she’s content with receiving only 50% of what she received while Nick was alive. These payments would be made at the beginning of each month and would continue to grow by ½% each month. During retirement they expect to earn 4% compounded quarterly.
a) What is the bi-monthly contribution both Nick and Sharon would end up making to their joint retirement account?
b) Nick would like to establish a scholarship for students. He would like the scholarship to pay $5,000 the first year it was paid. He would like the first scholarship to be paid 5 years after he retires. He would also like the scholarship payment to grow by 3% per year and the payments under the scholarship to go on in perpetuity. To fund both the scholarship and their retirement what will be the amount of the contributions they have to make?

Now this is an intense problem and I've taken 3 hours to try and solve this.
For a), I'm trying to find out how much they need for when Nick and Sharon are 50. I've found an amount for the future value of a growing annuity in advance to be = $9,124,916.928. This number represents how much they both need at age 50, up to until Nick dies at age 85.

Whats next is that I'm trying to find next is how much Sharon needs from age 85 to age 90.
I'm trying to find the monthly payment of what they both received on the 419th or 420th month after they were age 50, before the death month of Nick. I'm not sure how to calculate this monthly payment, because of the growing annuity part. Once I have this amount, I can use that for the C value and do the same as I did to solve part a).

Thank you for the help.
 
1) I get more like $10.2 Million - How did you get $1 Million shorter?

2) There is no information concerning the earnings of the ongoing retirement contributions. Are we to assume they go into the "Investment Account"?

3) If you handled the first inscreasing annuity, what's the impediment to the second? Just cut the payment at that time in half and start over. Of course, you'll also need to discount to the retirement date.

4) "content"? Right. I'd say $72,000 per month and growing would cover most people's living expenses.
 
annuity_man said:
For a), I'm trying to find out how much they need for when Nick and Sharon are 50. I've found an amount for the future value of a growing annuity in advance to be = $9,124,916.928. This number represents how much they both need at age 50, up to until Nick dies at age 85.

Whats next is that I'm trying to find next is how much Sharon needs from age 85 to age 90.
I'm trying to find the monthly payment of what they both received on the 419th or 420th month after they were age 50, before the death month of Nick. I'm not sure how to calculate this monthly payment, because of the growing annuity part. Once I have this amount, I can use that for the C value and do the same as I did to solve part a).
I agree with your $9.125 million, this being the required amount to accomodate
increasing monthly payments for 35 years, but only for that.
The final monthly payment will be a whopping 15000(1.005)^419 = 121,247.03

Now, to find what is required to accomodate Sharon's last 5 years, you need to:
1: find amount at age 50 that will accumulate (4% cpd quarterly) to $k as at age 85
2: this $k is the required amount to permit 60 increasing monthly payments (to age 90)
3: the 1st will be 121,247.03 / 2
 
Too bad. You had a chance to defend your figures but Denis beat you to it. I was winging it in a spreadsheet, but the more careful and compact calculation produces $9,124,912.65. Good work. I still don't get why you are lost on the Age 85-90 part. It's nearly the same calculation.
 
i = periodic interest rate, j = payment increase rate, p = 1st payment

PV = p(r^n - 1) / [(1 + i)(r - 1)]
where r = (1 + j) / (1 + i)

S'that your formula too, TK? Btw, your $9,124,912.65 should be $9,124,912.787653.... :shock:

And as far as calculating individual contributions:
>Nick would like to contribute to their joint retirement fund every two months until Sharon finishes her studies
>when they’re both 30. Nick’s contributions would be made at the end of each two month period. From that point
>Sharon would like to make all of the contributions to their retirement fund. She would also contribute every two
>months. These contributions would be made at the end of each two month period.

Can't be done, right? Nick could contribute only $1; Sharon then needs to contribute enough to reach target.
 
For the age 85-90 part, I get $949,264.31.

Denis said:
PV = p(r^n - 1) / [(1 + i)(r - 1)]
where r = (1 + j) / (1 + i)

(1 + i) in
PV = p(r^n - 1) / [(1 + i)(r - 1)]
has to be a typo.
 
Agree with your $949,264.31, jonah, BUT as at age 50:
at that point, $10,074,177.10 needs to be in the account (to accomodate ages 51-85 and age 86-90);
the 1st payment of $15,000 is deducted right away (1st of month).

The PV I'm calculating with my formula is the PV to accomodate the series of increasing payments,
which is $3,810,123.26: that's the required amount in account to accomodate Sharon's 60 payments,
a month before her 1st payment. There's no typo: try it.

The $949,264.31 is the PV of the $3,810,123.26 (at age 50).

Btw, the deposit required every 2 months from age 26 to 50 works out to $31,454.473468....
Makes the "real life" aspect attempted by the "teacher" through the "story" a joke!!
They gotta save over $188,000 per year!!
 
Sorry for the delay. I finally got my spreadsheet and my formulas together.

Normally, I wouldn't use or publish a formula. I tend to lean toward basic principles unless I'm coding a program.

In this case:

\(\displaystyle i = 0.04\)

\(\displaystyle i^{(4)} = \frac{0.04}{4} = 0.01\)

\(\displaystyle i^{(12)} = (1+0.01)^{1/3}\)

\(\displaystyle v^{(12)} = \frac{1}{1+i^{(12)}} = 0.996688717477339\)

\(\displaystyle r^{(12)} = 1.005\)

c = Combined Discount and Payment Increase

\(\displaystyle c = v^{(12)}*r^{(12)} = 1.00167216106473\)

And we have:

\(\displaystyle 15,000*\frac{1-c^{420}}{1-c} = 9,124,912.79\)

This assumes there is NO payment at anyone's age 85. Payments are made on the first month of age 50 through the last month of age 84.

Initial Payment: $15,000
Final Payment: $121,247.04

Similar assumptions for age 85-89 produce:

Initial Payment: $60,623.52
Final Payment: $81.635.22

Thus, \(\displaystyle \left[60,623.52*\frac{1-c^{60}}{1-c}\right]*\left(v^{(12)}\right)^{420} = 949,264.34\)

Of course, if I made any sense at all, I would round the payments to pennies and get:

50-84: Final Payment of $121,247.13 for a total of $9,124,914.59
85-89: Initial Payment $60,623.57, Final Payment $81,365.26 for a total of $949,265.07

These are, indeed, very high-end earners and savers. Note: We haven't even talked about funding the endowment! I'm confident they can get that out of their paltry monthly living expenses unless they are trying to run a comercial airline in their 80's.
 
Mainly to keep jonah busy and off the streets(!):
how much is left in the account after the withdrawal made on the 1st of the 600th month;
assume the $30000 initial deposit is at end of month 0 (so 1st bimonthly deposit at end of 2nd month).
 
Denis said:
The PV I'm calculating with my formula is the PV to accomodate the series of increasing payments,
which is $3,810,123.26: that's the required amount in account to accomodate Sharon's 60 payments,
a month before her 1st payment.
That certainly makes more sense. Sorry about that. I thought I might have had a little too much to drink at the time. I thought I might have overlooked something. It’s a good thing I’m always hammered. Otherwise, my arthritic shoulder, my gout, my asthma and the depressingly large amount of money involved with this problem might have made me really depressed. And you thought a buck 27 was silly in that other thread.
Denis said:
how much is left in the account after the withdrawal made on the 1st of the 600th month;
I see where you’re going with this. It’s a rather juicy puzzle. Unfortunately, I don’t have much experience with growing annuities (the geometric kind – it’s a relatively new find for me). Hopefully, the principles involved in determining the outstanding liability of a growing annuity (geometric) at the beginning/end of any period without using a spreadsheet schedule are the same. I’ll check in again after I escape from my other mind numbing pressing activities/duties. In the meantime, the alcohol in my brain is wearing off and I’m beginning to lose my focus. I believe I’ll be needing my next beer bottle. Now that I’ve thought about it, maybe questioners ought to send a check for a six-pack for our troubles every time their question gets answered. Cheers.
 
Step 1: we admitted we were powerless over alcohol and that our wives were unmanageable.

Whoops: should be "our lives" !
 
“Nick is expected to live until he is 85. Sharon expects to live until she is 90. However, after Nick’s death
she’s content with receiving only 50% of what she received while Nick was alive.”

This is definitely the beer talking but I get the feeling that the preceding statements are open to different interpretations.
First, the conversions.

\(\displaystyle \left( {1 + \frac{{j_6 }}{6}} \right)^6 = \left( {1 + \frac{{.06}}{2}} \right)^2\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle j_6 \approx .0594098043\)
\(\displaystyle \Rightarrow i_6 = \frac{{j_6 }}{6} \approx .00990163405\)
+
\(\displaystyle \left( {1 + \frac{{j_{12} }}{{12}}} \right)^{12} = \left( {1 + \frac{{.04}}{4}} \right)^4\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle j_{12} \approx .03986740251\)
\(\displaystyle \Rightarrow i_{12} = \frac{{j_{12} }}{{12}} \approx .00332228354\)
+
\(\displaystyle g = .005\)
\(\displaystyle r = \frac{{1 + g}}{{1 + i_{12} }}\)

Assumption A
The initial payment for the age 85-89 (or if you prefer, on the 1st of the 421st month after age 50) is assumed to be ½ of 15,000(1 + g)^419, or $60,623.52; the assumption everyone followed, the interpretation being “what she received” is the same as what the couple received. Under this assumption, as demonstrated by earlier posts, we have:

\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\)
\(\displaystyle = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}}\)\(\displaystyle + \tfrac{1}{2} \cdot 15,000(1 + g)^{419} \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot (1 + i_{12} )^{ - 420}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\)
\(\displaystyle = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}}\)\(\displaystyle + 60,623.52 \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot (1 + i_{12} )^{ - 420}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 9,124,912}}{\text{.79 + 949,264}}{\text{.31}}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 10,074,177}}{\text{.10}}\)

Assumption B
Being a couple, it can be argued that Sharon’s share in their 1st of month payment (I would imagine half - while Nick was alive) is represented by \(\displaystyle \tfrac{1}{2} \cdot 15,000(1 + g)^k\) , or \(\displaystyle 7,500(1 + g)^k\) where \(\displaystyle k \in (0,1,2,...,419)\).
If we’re to interpret the statement “she’s content with receiving only 50% of what she received while Nick was alive.” to mean a continuance of this series [after all, her share last time – in keeping with this line of reasoning - when Nick was still alive is \(\displaystyle 7,500(1 + g)^{419}\) , or $60,623.52], then the initial payment for age 85-89 should be \(\displaystyle 7,500(1 + g)^{419} (1 + g) = 7,500(1 + g)^{420}\), or $60,926.64. Under this interpretation, we have:

\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\)
\(\displaystyle = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}}\)\(\displaystyle + 7,500(1 + g)^{420} \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot (1 + i_{12} )^{ - 420}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\)
\(\displaystyle = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}}\)\(\displaystyle + 60,926.64 \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot (1 + i_{12} )^{ - 420}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 9,124,912}}{\text{.79 + 954,010}}{\text{.64}}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 10,078,923}}{\text{.42}}\)

Assumption C
Again, (apologies for the repetition) being a couple, it can be argued that Sharon’s share in their 1st of month payment (I would still imagine half - while Nick was alive) is represented \(\displaystyle \tfrac{1}{2} \cdot 15,000(1 + g)^k\) , or \(\displaystyle 7,500(1 + g)^k\) where \(\displaystyle k \in (0,1,2,...,419)\).
If we’re to interpret the statement “she’s content with receiving only 50% of what she received while Nick was alive.” to mean just that, then 50% of what she received while Nick was alive is represented by 50% of
\(\displaystyle \tfrac{1}{2} \cdot 15,000(1 + g)^k\), or \(\displaystyle 3,750(1 + g)^k\) where \(\displaystyle k \in (0,1,2,...,419)\).
Under this interpretation (since 50% of what she received last time is \(\displaystyle 3,750(1 + g)^{419}\) , or $30,311.76) the initial payment for age 85-89 should be 50% of \(\displaystyle \tfrac{1}{2} \cdot 15,000(1 + g)^{419} (1 + g)\) or
\(\displaystyle \tfrac{1}{2} \cdot \tfrac{1}{2} \cdot 15,000(1 + g)^{420} = \tfrac{1}{4} \cdot 15,000(1 + g)^{420} = 3,750(1 + g)^{420}\) , or $30,463.32.
We then have:

\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\)
\(\displaystyle = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}}\) \(\displaystyle + 3,750(1 + g)^{420} \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot (1 + i_{12} )^{ - 420}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\)
\(\displaystyle = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}}\) \(\displaystyle + 30,463.32 \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot (1 + i_{12} )^{ - 420}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 9,124,912}}{\text{.79 + 477,005}}{\text{.32}}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 9,601,918}}{\text{.11}}\)

“how much is left in the account after the withdrawal made on the 1st of the 600th month”

The 1st of the 600th month (a.k.a. the beginning of the 600th month) is also the end of the 599th month.
In this case, at the end of the 288th month (age 50), we let n = 599 – 24(12) = 311. We then determine the account balance at the end of the 311th month after age 50 and subtract from this the first of month payment due on that time.
Depending upon which assumption you choose, the account balance after the withdrawal made on the 1st of the 600th month is represented by
\(\displaystyle \left[ {\ddot A(1 + i_{12} )^{311} - 15,000\frac{{(1 + g)^{311} - (1 + i)^{311} }}{{r - 1}}} \right] - 15,000(1 + g)^{311}\) where
\(\displaystyle \ddot A\) , by abuse of notation is the present value of a growing (geometric, exponential or whatever) annuity due. Thus, with
\(\displaystyle \ddot A\) = $10,074,177.10, the account balance is $11,043,966.33
\(\displaystyle \ddot A\) = $10,078,923.42, the account balance is $11,057,281.28
\(\displaystyle \ddot A\) = $9,601,918.11, the account balance is $9,719,128.70

At least $9.6 million savings account? Perhaps the couple is better off with the perpetuity option instead. Better for me anyway.
 
You tired me out just reading your stuff, jonah!

jonah said:
\(\displaystyle 30,000(1 + \tfrac{{.06}}{2})^{24*2} +\) \(\displaystyle R \cdot s_{\left. {\overline {\, {24*6} \,}}\! \right| } _{i_6 }\) \(\displaystyle {\text{ = 10,074,177}}{\text{.10}}\)
..........
\(\displaystyle \ddot A\) = $10,074,177.10, the account balance is $11,043,966.33
That to me is the one that is obscurely meant.

BUT the 10,074,177.10 must be reduced by 15,000.00, since this 1st withdrawal is to be made "beginning of period",
thus coinciding with the last bimonthly deposit.
So at this point we have 10,059,177.10, from which the 1st withdrawal will be 15,075.00 a month later.

This (according to my calculations after 2 coffees!) means 11,009,552.13 instead of your 11,043,966.33
Code:
Balance 2nd of month 288 (or 1st of month 289)                         :  10,059,177.10

10,059,177.10(1.0033228...)^312                                        =  28,312,963.41

Less: 15075(r^312 - 1)(1.0033228...)^312 / [1.0033228...(r - 1)]       = 17,303,411.28
(where r = 1.005 / 1.0033228...)

28,312,963.41 - 17,303,411.28 = 11,009,552.13
Note: 312 is from 600 - 288
 
With no "little cute story" involved:

$30,000 is deposited on Dec.31/00 in an account earning 6% per year compounded semi-annually.

Bimonthly (every 2 months) deposits of $D are made until Dec.31/24.

On this day (Dec.31/24), $15,000 is withdrawn from the account. The rate changes to 4% compounded quarterly.

Monthly withdrawals will then be made from the account until Dec.31/59, the 1st being $15,075.
These withdrawals will increase by 1/2 % each month.

The next withdrawal will be 1/2 of the Dec.31/59 withdrawal.
Withdrawals will continue until Dec.31/64, again increasing by 1/2% each month.
After the Dec.31/64 withdrawal, the account will be exactly ZERO.

Calculate the bimonthly deposit $D.

BUT teachers feel a need to "embellish", thus creating confusion :evil:
Even to the point that sometimes it takes a student longer to "decipher" the story than it does solving the problem:
very unfortunate if a test is timed...
 
My computer went down for the count and I thought that was the end of it. I got into a bad drunken habit of just unplugging it whenever it takes forever to shut down normally and I finally paid the price. Fortunately for me, my patron, the wine god Dionysus heard my prayers and decided to resuscitate my poor abused computer back to life. At any rate, I see that my friend, Sir Denis disagreed with my assessment of $11,043,966.33. Having just re-read Don Quixote lately – what with being cut off from the online community and all – and being put in an extremely humorous mood in the process, I’m somewhat inclined to challenge Sir Denis to a traditional joust or sword duel of sorts to make him “confess” that my assessment of $11,043,966.33 is the correct one. Also, with this re-reading of Don Quixote, I’m also inclined to believe that those who indulge in whatever facet of Mathematics, being the queen of the sciences – I distinctly remember her being called that – be they students, dabblers like myself, or professionals, are practically knights-errant of that ruthless and exacting queen; thus the prefix title of “Sir”. Ah! There goes that rambling drunk again, I could almost hear the silent rolling of eyes everywhere. With my amortization spreadsheet schedule confirming my theoretical calculations, I’m afraid I’ll have to tire you out once again with my poor excuse of an inductive argument, Sir Denis.
Let’s begin by letting
\(\displaystyle R_1 =\)$15,000 and \(\displaystyle \ddot A\)= $10,074,177.10.
From a pure growing annuity due perspective:

The account balance after withdrawal made on the 1st of the 290th month relative to age 26 is the same as
the account balance after withdrawal made on the 1st of the 2nd month relative to age 50. Since the 1st of the 2nd month is the end of the 1st month, this account balance is represented by
\(\displaystyle \begin{gathered} \left[ {\ddot A - R_1 \left( {1 + g} \right)^0 } \right]\left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^1 \hfill \\ = \ddot A\left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^1 \hfill \\ = \ddot A\left( {1 + i_{12} } \right)^1 - R_1 \frac{{\left( {1 + g} \right)^1 \left( {1 + i_{12} } \right)^1 }}{{r - 1}} - R_1 \left( {1 + g} \right)^1 \hfill \\ \end{gathered}\)
= (FV of \(\displaystyle \ddot A\) at end of 1st month)-(FV value of 1st payment at end of 1st month)-(2nd payment at 1st of 2nd month)
= $10,077,521.54


The account balance after withdrawal made on the 1st of the 291st month relative to age 26 is the same as
the account balance after withdrawal made on the 1st of the 3rd month relative to age 50. Since the 1st of the 3rd month is the end of the 2nd month, this account balance is represented by
\(\displaystyle \begin{gathered} \left[ {\ddot A\left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^1 } \right]\left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^2 \hfill \\ = \ddot A\left( {1 + i_{12} } \right)^2 - R_1 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^2 - R_1 \left( {1 + g} \right)^1 \left( {1 + i_{12} } \right)^1 - R_1 \left( {1 + g} \right)^2 \hfill \\ = \ddot A\left( {1 + i_{12} } \right)^2 - \left[ {R_1 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^2 + R_1 \left( {1 + g} \right)^1 \left( {1 + i_{12} } \right)^1 } \right] - R_1 \left( {1 + g} \right)^2 \hfill \\ = \ddot A\left( {1 + i_{12} } \right)^2 - R_1 \frac{{\left( {1 + g} \right)^2 - \left( {1 + i_{12} } \right)^2 }}{{r - 1}} - R_1 \left( {1 + g} \right)^2 \hfill \\ \end{gathered}\)
= (FV of \(\displaystyle \ddot A\) at end of 2nd month)-(FV value of 1st payment + 2nd payment at end of 2nd month)
-(3rd payment at 1st of 3rd month)
= $10,095,851.55

\(\displaystyle \begin{gathered} \bullet \hfill \\ \bullet \hfill \\ \bullet \hfill \\ \end{gathered}\)

The account balance after withdrawal made on the 1st of the 600th month relative to age 26 is the same as
the account balance after withdrawal made on the 1st of the 312th month relative to age 50. Since the 1st of the 312th month is the end of the 311th month, this account balance is represented by
\(\displaystyle \ddot A\left( {1 + i_{12} } \right)^{311}\)
\(\displaystyle - [R_1 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^{311} + R_1 \left( {1 + g} \right)^1 \left( {1 + i_{12} } \right)^{310} + R_1 \left( {1 + g} \right)^2 \left( {1 + i_{12} } \right)^{309} + \cdot \cdot \cdot\)
\(\displaystyle + R_1 \left( {1 + g} \right)^{310} \left( {1 + i_{12} } \right)^1 ]\) \(\displaystyle - R_1 \left( {1 + g} \right)^{311}\)
\(\displaystyle = \ddot A\left( {1 + i_{12} } \right)^{311} - R_1 \frac{{\left( {1 + g} \right)^{311} - \left( {1 + i_{12} } \right)^{311} }}{{r - 1}} - R_1 \left( {1 + g} \right)^{311}\)
= (FV of \(\displaystyle \ddot A\) at end of 311th month)
-(FV value of 1st payment + 2nd payment + 3rd payment + · · · + 311th payment at end of 311th month)
-(312th payment at 1st of 312th month)
= $11,043,966.33

\(\displaystyle \begin{gathered} \bullet \hfill \\ \bullet \hfill \\ \bullet \hfill \\ \end{gathered}\)

The account balance after withdrawal made on the 1st of the 708th month relative to age 26 is the same as
the account balance after withdrawal made on the 1st of the 420th month relative to age 50. Since the 1st of the 420th month is the end of the 419th month, this account balance is represented by
\(\displaystyle \ddot A\left( {1 + i_{12} } \right)^{419}\)
\(\displaystyle - [R_1 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^{419} + R_1 \left( {1 + g} \right)^1 \left( {1 + i_{12} } \right)^{418} + R_1 \left( {1 + g} \right)^2 \left( {1 + i_{12} } \right)^{417} +\)
\(\displaystyle \begin{gathered} \cdot \cdot \cdot + R_1 \left( {1 + g} \right)^{418} \left( {1 + i_{12} } \right)^1 ] - R_1 \left( {1 + g} \right)^{419} \hfill \\ = \ddot A\left( {1 + i_{12} } \right)^{419} - R_1 \frac{{\left( {1 + g} \right)^{419} - \left( {1 + i_{12} } \right)^{419} }}{{r - 1}} - R_1 \left( {1 + g} \right)^{419} \hfill \\ \end{gathered}\)
= (FV of \(\displaystyle \ddot A\) at end of 419th month)
-(FV value of 1st payment + 2nd payment + 3rd payment + · · · + 419th payment at end of 419th month)
-(420th payment at 1st of 420th month)
= $3,810,123.26
which is, as you pointed out in an earlier post, “the required amount in account to accommodate Sharon's 60 payments, a month before her 1st payment.” Relative to your simplified analysis on your last post, this point of time is on Nov.30/59.

\(\displaystyle \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet\)

From an ordinary growing annuity perspective:

With \(\displaystyle A =\)$10,074,177.10 - $15,000 = $10,059,177.10
And \(\displaystyle R_2 = 15,000(1 + g) = 15,075\)

The account balance after withdrawal made on the 1st of the 600th month relative to age 26 is the same as
the account balance after withdrawal made on the 1st of the 312th month relative to age 50. Since the 1st of the 312th month is the end of the 311th month, this account balance is represented by
\(\displaystyle A\left( {1 + i_{12} } \right)^{311}\)
\(\displaystyle - [R_2 \left( {1 + g} \right)^0 \left( {1 + i_{12} } \right)^{310} + R_2 \left( {1 + g} \right)^1 \left( {1 + i_{12} } \right)^{309} + R_2 \left( {1 + g} \right)^2 \left( {1 + i_{12} } \right)^{308} +\)
\(\displaystyle \cdot \cdot \cdot + R_2 \left( {1 + g} \right)^{309} \left( {1 + i_{12} } \right)^1 + R_2 \left( {1 + g} \right)^{310} \left( {1 + i_{12} } \right)^0 ]\)
\(\displaystyle = A\left( {1 + i_{12} } \right)^{311} - R_2 \frac{{\left( {1 + i_{12} } \right)^{311} - \left( {1 + g} \right)^{311} }}{{i_{12} - g}}\)
= (FV of \(\displaystyle A\)at end of 311th month)
-(FV value of 1st payment + 2nd payment + 3rd payment + · · · + 310th payment + 311th payment at end of
311th month relative to age 50)
= $11,043,966.33

Denis said:
Monthly withdrawals will then be made from the account until Dec.31/59, the 1st being $15,075.
These withdrawals will increase by 1/2 % each month.
I’d say this should be
“Monthly withdrawals will then be made from the account until Nov.30/59, the 1st being $15,075. These withdrawals will increase by 1/2 % each month.”
Denis said:
The next withdrawal will be 1/2 of the Dec.31/59 withdrawal.
I’d say this should be
“The next withdrawal will be 1/2 of the Nov.30/59 withdrawal.”
Denis said:
Withdrawals will continue until Dec.31/64, again increasing by 1/2% each month.
After the Dec.31/64 withdrawal, the account will be exactly ZERO.
I’d say this should be
“Withdrawals will continue until Nov.30/64, again increasing by 1/2% each month.
After the Nov.30/64 withdrawal, the account will be exactly ZERO.”
Denis said:
BUT teachers feel a need to "embellish", thus creating confusion :evil:
Even to the point that sometimes it takes a student longer to "decipher" the story than it does solving the problem:
very unfortunate if a test is timed...
I feel your pain Sir Knight. At least teachers these days no longer require students to use tables like what I had to go through despite the availability of scientific calculators. Or do they?
 
Well Sir Jonah, I haven't had a drink of alcohol since Apr 8/85 (over 23 years) and, in order to continue with
my hangoverless and vomitless ways, will accept your $11,043,966.33 .... even if mine is the correct one :)
 
Denis said:
.... even if mine is the correct one :)
Ah Sir Denis! I see that you are humorously inclined as well as you are mathematically inclined.
If my fellow knights-errant in this forum will humor/permit me to indulge further, I’m sure our fellow knights-errant will appreciate your sense of humor more by quoting a long passage from Don Quixote as translated by Senor Charles Jarvis.

Here it goes:

And having gone about two miles, Don Quixote discovered a company of people, who, as it afterwards appeared, were certain merchants of Toledo, going to buy silks in Murcia. There were six of them, and they came with their umbrellas, and four servants on horseback, and three muleteers on foot. Scarcely had Don Quixote espied them, when he imagined it must be some new adventure; and, to imitate, as near as possibly he could, the passages he had read (of) in his books, he fancied this to cut out on purpose for him to achieve. And so, with a graceful deportment and intrepidity, he settled himself firm in his stirrups, grasped his lance, covered his breast with his target, and posting himself in the midst of the highway, stood waiting the coming up of those knights-errant; for such he already judged them to be: and when they were come so near as to seen and heard, Don Quixote raised his voice, and with an arrogant air, cried out:
‘Let the whole world stand, if the whole world does not confess, that there is not in the whole world a damsel more beautiful than the empress of la Mancha, the peerless Dulcinea del Toboso.’
The merchants stopped at the sound of these words, and to behold the strange figure of him who pronounce them; and by one and the other they soon perceived the madness of the speaker: but they had a mind to stay and see what that confession meant, which he required of them; and one of them, who was somewhat of a wag, but withal very discreet, said to him:
‘Senor Cavalier, we do not know who this good lady you mention may be: let us but see her, and, if she is of so great beauty as you intimate, we will, with all our hears, and without any constraint, confess that truth you demand from us.’
‘Should I show her to you,’ replied Don Quixote, ‘where would be the merit in confessing a truth so notorious? The business is, that, without seeing her, you believe, confess, affirm, swear, and maintain it; and if not, I challenge you all to battle, proud and monstrous as you are: and whether you come on one by one (as the laws of chivalry require) or altogether, as is the custom and wicked practice of those of your stamp, here I wait for you, confiding in the justness of my cause.’
‘Senor Cavalier,’ replied the merchant, ‘I beseech your worship, in the name of all the princes here present, that we may not lay a burden upon our consciences, by confessing a thing we never saw nor heard, and especially what is so much to the prejudice of the Empresses and Queens of Alcarria and Estremadura, that your worship would be pleased to show us some picture of this lady, though no bigger than a barley-corn; for we shall guess at the clue by the thread: and herewith we shall rest satisfied and safe, and your worship remain contented and pleased: nay, I verily believe we are already so far inclined to your side, that, though her picture should represent her squinting with one eye, and distilling vermilion and brimstone from the other, notwithstanding all this, to oblige you, we will say whatever you please in her favour.’ :wink:
‘There distils not, base scoundrels,’ answered Don Quixote, burning with rage, ‘there distils not from her what you say, but rather ambergris and civet among cotton; neither is she crooked, nor humpbacked, but as straight as a spindle of Guadarrrama: but you shall pay for the horrid blasphemy you have uttered against so transcendent a beauty as my mistress.’
And so saying, with his lance couched, he ran at him who had spoken, with so much fury and rage, that, if good fortune had not ordered it that Rosinante stumbled and fell in the midst of his career, it had gone hard with the daring merchant. Rosinante fell, and his master lay rolling about the field a good while, and endeavouring to rise; but in vain, so encumbered was he with his lance, target, spurs, and helmet, and with the weight of his antique armour. And while he was thus struggling to get up, and could not, he continued calling out:
‘Fly not, ye dastardly rabble; stay, ye race of slaves; for it is through my horse’s fault, and not my own, that I lie here extended.’
 
DENIS AND MASON (a keep-it-simple wise elderly friend)
===============================================
DENIS: Hey Mason, just decided to take this amazing course.
MASON: Ya? What's it called?
DENIS: The Sylva Method of Mind Control.
MASON: Hmmm...what will you learn there, Denis?
DENIS: Amazing things, Mason; I'll be able to control my own mind and
make it do all sorts of things.
MASON: Well...can you give me a couple of examples?
DENIS: Sure. At the demonstration, they told me I'd be able to figure
out what my dog is thinking; also, when I get a letter in the
mail, I'll hold it at both ends with my fingertips, concentrate,
and I'll be able to tell what's in the letter...imagine that!
MASON: Gee. Tell me, Denis, how much is this course costing you?
DENIS: Only 325 bucks, Mason.
MASON: Well, why don't you buy yourself a plastic letter opener instead:
only 79 cents at K-Mart....

That's a TRUE story...I've learned to make fun of myself :idea:
 
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