Capitalized Worth (or value)

[Tallulah]

Here is the question:
You decide to open an individual retirement account at your local bank that pays 8%/year compounded annually. At the end of each of the next 40 years, you will deposti $4000 into the account. Three years after your last deposit, you will begin making annual withdrawals. What annual amount will you be able to withdraw if you want the weithdrawals to last...
a) 20 years
b) 30 years
c) forever

For the first part of the question I assume I can determine the payments at 20 and 30 years since I have information for FV, PV, i, and n. For a) I calculated ($30,509.38) and b) I calculated ($12,133.86). To calculate the annual payment for an infinite series of payments, I assume I have to think of this as a capitalized worth problem and tabulated the cash flows for each year and assumed interest was made on the balance at the end of period. At year 43, I found that the balance of the account would be $1,414,812.98. I then used the CW (or perpetuity) formula=A/i to calculate the payment (A). I calculated ($113,185.04) for A (I did not assume inflation).

After looking at my anwers I think I went wrong somewhere because the infinite set of payments is much higher than payments made for 20 or 30 years. Can anyone give me a hint as to what I did wrong here? (I tried to upload my table, but I dont know what file extension this site accepts. I tried .xls and .doc)

 

[Denis]

Can't tell what you're doing wrong, but you're certainly doing SOMETHING wrong :shock:
Here's what "it all looks like" assuming the 20 years case:

YEAR       PAYMENT       INTEREST          BALANCE
 1        4,000.00            .00         4,000.00
 2        4,000.00         320.00         8,320.00
....
 39       4,000.00      70,501.10       955,764.88
 40       4,000.00      76,461.19     1,036,226.07[1]
 41            .00      82,898.09     1,119,124.16
 42            .00      89,529.93     1,208,654.09[2]
 1     -123,104.09[3]   96,692.33     1,182,242.33
 2     -123,104.09      94,579.39     1,153,717.63
....
 19    -123,104.09      17,562.17       113,985.27
 20    -123,104.09       9,118.82              .00

The balance at end of 40 years ([1] above) is calculated this way:
4000(1.08^40 - 1) / .08 = 1036226.07

The balance at end of 42 years ([2] above) is calculated this way:
1036226.07 * 1.08^2 = 1208654.09
NOTE that you need the balance after 42 years, not 43 years.

The payment amount during the 20 years ([3] above) this way:
1208654.09 * .08 / (1 - 1/1.08^20) = 123104.09

I'll assume you're able to understand above calculations; standard formulas used.
If not, then you're really not quite ready for this.

As far as the perpetuity goes, the payments are simply the balance at end of 42 years
multiplied by the rate .08, thus 1208654.09 * .08 = 96,692.33
Instead of calling it "interest only", financial advisers like to sound important and use
the term "perpetuity", which creates admiration in the eyes of the poor customer,
thus the fee increased to match proportionally the intensity of the admiration.

Notice that the principal amount of 1,208,654.09 remains untouched; in other words,
it'll still be there when you upset the pail (I don't like "kick the bucket") and turned
over to your wife: so make sure your tea is not poisoned!

 

[Tallulah]

Hi Denis:

Thank you for your help on this. I found what I did incorrectly. Instead of calculating the balance at 42 years, I used 43 years. That was one reason why I was getting such a huge number, especially for the perpetuity. I also didnt consider that the account holder was not making $4000 payments after year 40. It all makes perfect sense now.