Annuity: $50K in bank, $200K in shares, retiring in 15 yrs

smudge

New member
Joined
Mar 5, 2009
Messages
8
Hi I'm having a problem solving this question. My attempted workings are below. Thank you.

Today is your 50th birthday, and you anticipate that you will continue working until your 65th birthday. You currently have $50,000 in a bank account and $200,000 in shares. You plan to add to these savings by the following annual increments:

i. depositing $2,000 a year for 10 years, and then $4,000 per year, into your bank account, the first deposit to be made today, and the last on your 65th birthday;

ii. adding $6000 to your share portfolio today, and increasing this amount by 4% per annum with the last addition on your 65th birthday.

The bank account is expected to earn 5% per annum, and the share portfolio 12% per annum.

On your retirement aged 65, you intend to deposit all your savings into an investment account that will earn 10% per annum.

a. What is the value of your savings on your 65th birthday (after you have made your annual deposit)?

b. If you expect to live for 20 years after your retire, how much can you withdraw each year after retirement (20 equal withdrawals beginning one year after you retire) to end up with a zero balance on your anticipated demise?

My workings thus far
Part A
i) FVAdue= $2000(1.05/0.05)^10 –1 * 1.05= $26414
then FVAdue= $2000(1.05/0.05)^15 –1 * 1.05 = $45315

ii) Stuck on this part. Cannot work out the cash flows.
$6000 today meaning FVAdue?

a)Value of savings on 65th bday?
$50000(1.05)^15= $103,946
$200000(1.12)^15= $1,094,714
2000(1.05/0.05)^10 –1 * 1.05= $26,414
2000(1.05/0.05)^15 –1 * 1.05 = $45,315
Stuck on ii) $$$$$$

Part B
Yearly withdraws
=Savings Amount / (1-(1.10/0.10)^-20 = ?????
 
smudge said:
> a)Value of savings on 65th bday?
> $50000(1.05)^15= $103,946
> $200000(1.12)^15= $1,094,714
Both correct!

> 2000(1.05/0.05)^10 –1 * 1.05= $26,414
> 2000(1.05/0.05)^15 –1 * 1.05 = $45,315
1st one should be 2000(1.05/0.05)^5 –1 * 1.05= $11,604
2nd one correct!

> Stuck on ii) $$$$$$
With this one, the $6000 increases by 4% each year, hence flows are:
6000 + 6000(1.04) + 6000(1.04^2) + .....+ 6000(1.04^14)
There is a formula for this kind of FV (I forgot what it is...YOU find it!);
anyway, the FV will be ~308,499 (computer program)

So at retirement:
[103946 + 11604 + 45315] + [1094714 + 308499] = 1,564,078
20 years @ 10% means an annual withdrawal of 183,716
 
Hi Denis
Thanks for your reply! :)
I arrived at a different answer to the FV for share portfolio.
My FV were $322484.81
Cash flows FV
0 6000.00 6000.00
1 6000 6240.00 6988.80
2 6000 6489.60 8140.55
3 6000 6749.18 9482.12
4 6000 7019.15 11044.77
5 6000 7299.92 12864.95
6 6000 7591.91 14985.09
7 6000 7895.59 17454.64
8 6000 8211.41 20331.16
9 6000 8539.87 23681.73
10 6000 8881.47 27584.48
11 6000 9236.72 32130.41
12 6000 9606.19 37425.50
13 6000 9990.44 43593.22
14 6000 10390.06 50777.38

322484.81
Therefore:
$103946
$1094714
$11604
$45315
$322485
Savings amount =$1,578,064.00

Yearly withdrawals $1,578,064.00 / PVA = $185,218.78
 
No; 322485 is not correct.

Cash flows with FV's for each are (closest dollar):
1: 6000 : FV = 6000(1.12)^15 = 42841
2: 6240 : FV = 6240(1.12)^14 = 30496
...
14: 9990 : FV = 9990(1.12)^2 = 12532
15: 10390 : FV = 10390(1.12^1) = 11637
The 15 FV's add up to 308499
Don't forget that the 1st flow of 6000 is immediate;
so last flow of 10390 is at beginning of year 15, therefore earns interest during year 15.

A "statement" showing transactions looks like:
Code:
Jan 1/01        6000                    6000
Jan 1/02        6240       720         12960
Jan 1/03        6490      1555         21005
...
Jan 1/14        9990     24286        236657
Jan 1/15       10390     28398        275445
Jan 1/16                 33054        308499
 
In imitation of Jeremy Clarkson of the car show Top Gear whenever he feels that his choice of car is better that those of his colleagues, I feel I must point out that you both got the wrong car...er...I mean assessment.

My assessment:

\(\displaystyle \left( {50,000 + 2,000} \right)\left( {1.05} \right)^{15} + 2,000 \cdot s_{\left. {\overline {\, {15} \,}}\! \right| .05} + 2,000\cdot s_{\left. {\overline {\, 5 \,}}\! \right| .05} +\)

\(\displaystyle \left( {200,000 + 6,000} \right)\left( {1.12} \right)^{15} + 6,000\left( {1.04} \right)\frac{{\left( {1.12} \right)^{15} - \left( {1.04} \right)^{15} }}{{0.12 - 0.04}}\)

\(\displaystyle = R \cdot a_{\left. {\overline {\, {20} \,}}\! \right| .10}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle R \approx \$ 185,155.3343...\)

Batteries sold separately and this member assumes no responsibility for any loss or damages arising from use of his semi-drunk assessment.
 
Ya Jonah, but you're using one more annual contribution than I am; another 15 following the initials.
I can now see that you're probably right, 50 to 65 is initial plus 15;
I didn't pay much attention to that, since Smudge had used 14 following the initials in his 1st post;
so I assumed that was the intention.
 
Hi
Thanks for reply
I dont understand where the mistake is. Can you please clarify.
Cheers
 
Contribitions from 50th to 65th means 16 contributions; I (and you) assumed none at age 65;
Jumpin' Jonah assumed (correctly, from the wording) that contributions included one at age 65.

Your problem SHOULD probably be worded more clearly: using 50 to 65 makes one jump at
the conclusion that 15 contributions are involved...and YOURS DOES involve 15 contributions!
Tell your teacher to smarten up :wink:
 
These are my workings:

i) FVAdue= 2000(1.05)^6 –1 * 1.05= $14284
0.05
then 2000(1.05)^16 –1 * 1.05 = $49680
0.05
ii)
Cash flows FV
0 6000.00 6000.00
1 6000 6240.00 34155.05
2 6000 6489.60 31715.40
3 6000 6749.18 29450.02
4 6000 7019.15 27346.45
5 6000 7299.92 25393.13
6 6000 7591.91 23579.33
7 6000 7895.59 21895.09
8 6000 8211.41 20331.16
9 6000 8539.87 18878.93
10 6000 8881.47 17530.44
11 6000 9236.72 16278.26
12 6000 9606.19 15115.53
13 6000 9990.44 14035.85
14 6000 10390.06 13033.29
15 6000 10805.66 12102.34
=$314,738
Value of Savings
$14,284
$49,680
$103,946
$1,094,713
$314,738
=$1,577,361

Withdrawals
$1,577,361/(1-(1.10)^-20 = $185276
0.10
 
Well whatever, Smudge. Still not CLEAR :shock:
50: 52000 (50000 + 2000)
51: 2000
52: 2000
...
59: 2000 : now 10 contributions of 2000 have been made, the 1st included in the opening 52000.
60: 4000
61: 4000
62: 4000
63: 4000
64: 4000
65: 4000 : now 6 contributions of 4000 have been made.

66: 1st withdrawal (1 year after last contribution of 4000)

IS THAT THE INTENT ???
 
Smudge, you're almost there. To paraphrase from two excellent authors, Dr. Zima and Dr. Brown, the most efficient way to solve an annuity problem is to make a time diagram, determine the type of annuity, and then apply the proper formula(s). Now I strongly suggest you make your own time diagram in the form of a line graph and make the necessary marks on it until you see what seems clear to Denis and me. You'll get it eventually.
 
jonah said:
\(\displaystyle \left( {50,000 + 2,000} \right)\left( {1.05} \right)^{15} + 2,000 \cdot s_{\left. {\overline {\, {15} \,}}\! \right| .05} + 2,000\cdot s_{\left. {\overline {\, 5 \,}}\! \right| .05} +\)
Jonah, methinks there's an extra tequila shot in there: your 2nd 2000 flow should be for 6 years, not 5.

That'll result in an ending balance of 1,578,884.300330305115...
for 20 withdrawals of 185,455.157476061132...
 
Noted Sir Denis.
But methinks I’ll have to stick with my initial assessment.
By the way, I’ve been meaning to ask you what calculator or software are you using? My Excel could only give me $185,455.157476061…
 
jonah said:
> But methinks I’ll have to stick with my initial assessment.

...with your initial "erroneous" assessment :roll:

> By the way, I’ve been meaning to ask you what calculator or software are you using?
> My Excel could only give me $185,455.157476061…

I use programming language Ubasic; look it up: you'll love it.
Handles stuff like factorial(1000) : 2568 digits !
 
First you think I’m probably right. Now you think my initial assessment is erroneous.
You might be seeing things that are not there.
Methinks you took an extra strong cup of coffee there yourself.
Maybe you should cut down a little. Decaf perhaps Sir Denis?
smudge said:
depositing $2,000 a year for 10 years, the first deposit to be made today
However way you see it, 50 to 60 is initial plus 10, making for 11 $2,000 and 5 $4,000 on the 61st, 62nd, 63rd, 64th, and 65th birthday.
 
jonah said:
> First you think I’m probably right. Now you think my initial assessment is erroneous.

I said you were right in assuming a deposit on 65th birthday, hence from/including 50 to 65 = 16 deposits.
That was BEFORE your erroneous assessment :)

>
smudge said:
depositing $2,000 a year for 10 years, the first deposit to be made today
> However way you see it, 50 to 60 is initial plus 10, making for 11 $2,000 and 5 $4,000 on the 61st,
> 62nd, 63rd, 64th, and 65th birthday.

CLEARLY(!) means the first deposit made today is the first of the 10, so 9 left (beginning of period):
2000: today=1, 51=2, 52=3, 53=4, 54=5, 55=6, 56=7, 57=8, 58=9, 59=10
4000: 60=1, 61=2, 62=3, 63=4, 64=5, 65=6

No need to apologize, Sir Jonah :wink:
 
First, let me express a stolen humorous outrage phrase from a young friend of mine.

You WILL rue this day, Sir Denis!

RUE IT I SAY!

START RUEING! RUIING! RUING... blast it, how do you spell that?

I’ve always wanted to use that ever since I came across it. With that out of my system, let’s move on.

Judging from you last post, it would seem that you’re literally equating the phrase “depositing $2,000 a year for 10 years” with 10 years = 10 $2,000 deposits. I suppose that is a valid interpretation if 10 years have actually elapsed since the first deposit.
Methinks however that my interpretation is more plausible somehow. (Seems like a throwback to that old argument on which date is the real beginning of the 3rd millenium controversy – 1/1/2000 or 1/1/2001)
Using your own date example on your 2nd post on this thread, the first $2,000 deposit is made on January 1, 2001 and the 10th deposit is on January 1, 2010. Accordingly, the time from January 1, 2001 to January 1, 2010 is equal to 9 years only. Thus, 10 $2,000 deposits in 9 years.
Since only 9 years have elapsed since the first deposit, this is hardly in keeping with the spirit of the condition of “$2,000 a year for 10 years, with the first deposit to be made “today” January 1, 2001”.
My interpretation is this: an initial deposit of $2,000 and a 10-year period standard annuity-immediate (or end of period deposits – January 1 in this case) valuation to comply with the condition set forth.


If you have children with you, secure their seatbelts first before you secure yourself. Emergency exits are provided here, here, there. Beverages will be served shortly.
 
Sir Jonah, you talk more plus faster than a mother-in-law. :)
I hereby pretend I agree with you...only if you buy the next round :idea:
 
No need to pretend, Sir Denis.
Remembering Einstein’s theory of relativity, I realized that the problem statement is not explicit enough to allow for an absolute interpretation. I guess a $2,000 10 year annuity-due valuation and 5 $4,000 on the 60th, 61st, 62nd, 63rd, 64th, birthday with a final deposit on the 65th birthday is just as plausible as an initial deposit of $2,000 and a 10-year period standard annuity-immediate (or end of period deposits – January 1 in this case) valuation and 5 $4,000 on the 61st, 62nd, 63rd, 64th, and 65th birthday. Nothing like rest and some brandy to clear and calm your mind. Until then, I look forward to our next joust, Sir Denis.
 
So I'm the hero?
Let's have Smudge play the guitar, Sir Jonah sing "Happy Trails To You"
while Sir Denis rides into the sunset with Ample Annie...
 
Top