# Rate of Return Required on Buying Stocks with Loan

A bank gives a loan $L$ for $m$ months and the monthly interest rate is $i$.
The bank requires monthly installments - which I calculate is $I = \frac{L}{m} + Li$.

I use this loan to buy stocks. If the stock price is $p(m = 0)$, then I can buy $\frac{L}{p(m = 0)}$ shares. I plan to pay the $m$ monthly installments only by selling the optimal number of stocks every $x$ months.

How can I calculate if this plan will pay off? How do I optimize $x$? What variables should I be looking at? Can profit be made from this example? How much does the stock price have to rise to break even? I match abbreviations with their referents.

If I just accept the loan of $L$ and do nothing with it except return it in $m$ installments, then my loss is only the total interest $=m \times Li$. So to break even, $m \times Li$ is what I need to gain from my stocks. But this feels too simple and naive?

Furthermore, the rate of return over the entire period = total interest/loan amount $= \frac{mi}{L}$? This feels wrong because it doesn't account for the fact that stocks will be sold every $x$ months to cover the $m$ monthly installments? I think $x\frac{ I}{p(m)}$ shares must be sold to cover $x$ monthly installments?

I'm already getting lost so maybe some numbers will help - suppose
☻ $L = 10,000$ USD,
☻ $m = 24$,
☻ $i = 0.25\%$
☻ $p(m = 0) = 80$,
☻ The stock price is expected to go up to $100$ within the 24 months.

I'm new to financial math and only know first-year undergrad math. Sorry for any faults.

• Are you actually contemplating this scheme -- that is, looking for practical advice -- or is this an academic exercise? I am sure people can help you with the former. – Drew Saunders May 31 at 2:48
• Honestly speaking, the problem here is not the break even share appreciation. The problem here is to adjust your leverage according to your expected shortfall, which is the delicate trade off between risk and safety. Buying and hold stocks purchased through leverage is a good way to blow up. – Lisa Ann May 31 at 8:16

Frankly this is barely understandable.

Assume the nominal of the bond is $L$, the monthly interest rate is $i$, then you have to compute the monthly installments $C$ over $M$ months as follows:

$$L= \sum_{j=1}^M \frac{C}{(1+i)^j}$$

Now, we know that the price of a perpetuity is :

$$\sum_{j=1}^\infty \frac{C}{(1+i)^j}=\frac{C}{i}$$

So you can compute $L$ by subtracting a discounted perpetuity at time $M$ from a perpetuity at time $0$:

$$L= \sum_{j=1}^M \frac{C}{(1+i)^j}= \frac{C}{i} - \frac{1}{(1+i)^M} \frac{C}{i}=\frac{C}{i} \left( 1 - \frac{1}{(1+i)^M} \right)$$

In any way, you want to invest this amount in some stock.

I can't really understand the rest of the question.