# 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.

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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.

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