Solving mathematical economics problem

Question

kindly I'm stuck in this problem instead of many attempts through net present value and other discounted cash flow methods, some one could give me a detailed information and answers on this problem : (a) Suppose your closest friend borrow you $350 000 for a year and inflation is expected to be 12%.if you set the price for the loan to be 20% interest and your (marginal) tax rate is one third. What is your real earning at the end of the year, and by what percent has your purchasing power changed? (b) suppose again you had charged no interest at all on the loan in part (a), and the expected inflation declines to 5%,what is the effect of this on your purchasing power and your end of year earning?

Answer 1

I don't know much about economics, so take this with a grain of salt!

I assume only the interest from the loan gets taxed. I also assume that backpayment of the loan and the interest are done at the same time after one year.

(a) After one year, you have your $\$350,000$ loan paid back plus $\frac{20}{100}\times\$350,000 = \$70,000$ as interest. The interest gets taxed, however, so you loose $\frac13\times\$70,000=\$23,333$ to taxes. So after taxes you have $\$350,000+\$70,000-\$23,333=\$396,667$ in cash. That makes what I would consider your 'real earning' to be $\$396,667-\$350,000=\$46,667$. However, due to inflation, the buying power of each of those Dollars is less than it was a year before. The $12\%$ inflation means that now you need to spend $\$1.12$ to buy what you could by for $\$1.00$ a year ago. So your cash now is worth is much as $\frac{\$396,667}{1.12} = \$354.167$ was a year ago. So your buying power, compared to one year ago, has increased by a factor of

$$\frac{\$354.167}{\$350,000}=1.0119\ldots,$$

so just a wee bit more than one percent. That's not surpising: Your $20\%$ interest was only $13.3\%$ after taxes, and that got almost eaten up by the inflation.

(b) If you get no interest on the loan, you haven't earned anything, so you don't need to pay taxes on it. So it is the same as if it had been in your home all the time. As before, it is worth less after that one year, due to the assumed inflation rate it is now worth $\frac{\$350,000}{1.05}=\$333,333$. Your buying power has thus increased by a factor of

$$\frac{\$333.333}{\$350,000}=0.952\ldots,$$

or rather decreased by almost $5\%$.