## Thursday, August 21, 2014

### Paying off the mortgage with funds in the stock market.

Question:   A person buys a \$800,000 house with a 7-year 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 pre-tax 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 = (1-t) x PTP

So PTP = ATP/ (1-t)

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.