solomon (01/29/83)
There's been a lot of unsupported claims in this group recently on the
value of an IRA as a savings account. I decided to do a few simple
calculations to get a better feeling for the value of such an approach.
Suppose your marginal tax rate (the fraction of the last dollar you make
that goes to Uncle Sam and his state and local relatives) is t and you
can get a rate of r on investments. As several have pointed out,
you can put your IRA into an account that gets that same rate r
(subject to some restrictions). Now suppose you put $1000 in an IRA,
while I simply invest $1000 (I have the same tax rate as you).
Further, for simplicity, let's suppose the compounding period for
everything is one month and the rate r is scaled to the rate per month.
For example, we could take r=0.80/12=.066667 and t=0.40. At the start of
the period, you would have $1000 invested, while I would have 1000(1-t).
after one month, you would have 1000 principal + 1000r interest.
On the other hand, I would get 1000(1-t)r in interest, but pay 1000(1-t)rt
in taxes. Thus you would have 1000(1+r) in your account and I would have
1000(1-t)(1+r(1-t)) in mine. Similarly, after n months, your account
balence would be
n
1000 (1+r)
whereas mine would be
n
1000 (1-t) (1+r(1-t))
Now suppose we both decide to spend our money at the end of n months.
You have to pay taxes at the rate of t as well as a 10% penulty.
I've already paid the taxes, so I get the whole amount. Thus you get
n
1000 (1+r) (1-t) 0.9
Thus, the difference between what I get and what you get is
n n
1000 (1-t) [ (1+r(1-t)) - .9(1+r) ]
When n = 0, this value is 100(1-t), so I start out ahead. However,
n n
(1+r(1-t)) grows slower than (1+r) , so my advantage slowly drops and
eventually falls below zero. The break-even point is when this value zero:
n n
(1+r(1-t)) = .9(1+r) , or
n
[1+r(1-t)]
[--------] = .9
[ 1+r ]
Taking logs, we get n log([1+r(1-t)]/(1+r)) = log(.9), or
n = log(.9)/log([1+r(1-t)]/(1+r)).
Using our example of 8% interest and a marginal rate of 40%, we get
n = 39.7. In other words, it takes you just under 3 years and 4 months
to catch up and pass me. At the end of 5 years, you would have $804.48
to spend while I would have only $762.34, or $42.14 less (poor me!).
Here is the bc script "ira.bc" I used to get these results. To run it, type
"bc -l ira.bc". After the results are printed, you will still be in bc;
experiment with different values of r, t, and m (number of months)
and call f() (to calculate months to break-even point) or g() (to
see what happens over a period of m months). Type Control-D to quit.
-------------------------------------------------------------
/* bc program to compare straight investment with investment in a tax-free
* IRA.
* Usage: bc -l ira.bc or bc -l <ira.bc
*/
/* calculate number of months before the IRA account is better */
define f() {
return( l(x) / ( l( (1+r*(1-t)) / (1+r) ) ))
}
/* show what happens to an initial 1000 investment over a period of months */
define g() {
auto i,p,q
/* p = non-IRA balence; q = IRA balence after i months */
p=1000*(1-t); q=1000
for(i=0; i<=m; i++) {
i; /* time */
p; /* non-IRA balence */
q*x*(1-t); /* IRA balence after taxes and penulties */
p-q*x*(1-t); /* difference: non-IRA - IRA */
p =* 1+r*(1-t); q =* 1+r
}
}
scale = 6
x=.9 /* 1-penulty for early withdrawal */
r=.08/12 /* interest rate (per month) */
t=.4 /* marginal tax rate */
m=60 /* number of months to run g() */
x;r;t;m /* print default values */
f()
g()
-------------------------------------------------------------
Enjoy!
Marvin Solomon
Computer Sciences Department
University of Wisconsin, Madison WI
solomon@uwisc
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