Question: A person buys a $800,000 house with a 7year
ARM. The interest rate on the ARM is
3.0%. The down payment is $300,000. The person is planning to pay off the ARM in
exactly seven years when he retires.
The person has a lump sum of $200,000 invested in common
stock. He sells his entire investment
only at the end of seven years.
What rate of return on common stock is needed to pay off the
entire $500,000 loan in seven years if capital gains are untaxed?
What is the required rate of return if the capital gains tax
rate is 20%?
What is the required rate of return if the capital gains tax
rate is 40%?
Discuss implications for tax policy and include a discussion
of the merits of taxing capital gains versus taxing corporate income.
Answer: The investor needs $500,000 total after seven
years ($200,000 in his down payment and a $300,000 gain after tax.
When the tax rate is zero after tax income is equal to
pretax income. The person needs to obtain $500,000 total to
pay off his loan.
Denote FV the final value of the investment and IV the
initial value of the investment.
Without taxes
FV=IV*(1+r)^{7}
Rearrange to get r
R=(FV/IV)^{1/7}
1
Plug in FV=$500,000 and IV=$200,000 to get
R = 0.13985
When taxes exist the investor needs an after tax profit of
$300,000 and returns must exceed 13.985%.
By how much?
Denote ATP as after tax profit and PTP as pre tax profit
ATP = (1t) x PTP
So PTP = ATP/ (1t)
At t=0.20 the required PTP to get $300,000 after taxes is
$375,000. We need a FV of $575,000.
Plugging in FV=$575,000 and IV=$200,000 we get r=0.16284%.
At t=0.40 the required PTP is $500,000. The investor needs a $700,000.
This requires a rate of return of 0.19598.
I checked my answer by plugging IV=$200,000 and the
calculated r into the FV equation subtracting out tax and making sure I had
after tax profits of $300,000 in all cases.
I am happy the checks confirmed my numbers are correct.
Impact of Taxes on
Required Return to Cover Debt


IV

$200,000

$200,000

$200,000

ATP

$300,000

$300,000

$300,000

Tax Rate

0

0.2

0.4

PTP

$300,000

$375,000

$500,000

FV consistent with PTP

$500,000

$575,000

$700,000

Return needed to pay off
debt

0.13985

0.16284

0.19598

FV Consistency Check

500000

575000

700000

ATP Consistency Check

$300,000

$375,000

$500,000

PTP Consistency Check

$300,000

$300,000

$300,000

Concluding thoughts:
Unless this borrower/investor has the acumen of Warren
Buffet he is not likely to be able to pay off the loan after 7 years if
$200,000 in the stock market in the stock market is his only source of funds.
Some people believe that corporate tax rates should go down
and that more taxes should be collected from the individual.
This argument makes sense if you believe that households are
better able to predict and manage their finances than corporations. People need to save because they need funds
when bad events occur. High personal tax
rates make people pay a bill to the government when they themselves most need
their funds.
Many analysts do not fully appreciate the fact that taxing
people when they are in dire need of funds leads to bankruptcy. I would prefer a system that taxed corporate
profits when corporate profits are high than a system that taxed personal
income when the person is experiencing an emergency that requires use of his or
her funds.
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