Deriving the Return on a Leveraged Stock Investment

# Decomposition of stock returns with leverage


# Decomposition of stock returns with leverage












I came across this question in Kopp's portfolio management and risk, where we are told, that a stock is bought at S(0) with leverage - a mixture of $wS(0)$ cash and $(1-w)S(0)$ loan and told to prove that the return on the leveraged position $R_{l}$ is $r + \frac{1}{w}(R_{S} - r)$ where $R_s$ is the return on the stock, $w\in(1,0)$ and where $r$ is the risk free rate.

This implies that $R_{S} = wR_{l} + (1-w)r$ - so the stock returns are decomposed into the return on the leveraged position and another component.

- I am unable to derive the relationship. My initial thought was to look at the initial cost of setting up the portfolio, which is $S(0)$, and note that at any subsequent point the portfolio value is a combination of the final stock price $S(1)$ (say) and the loan $-(1+r)(1-w)S(0)$. However this does not simplify to give what I need as in particular I don't have an expression for $S(1)$.

- I can't fully understand the expression we are meant to prove - $R_{S} = wR_{l} + (1-w)r$. Does this mean if we use leverage to buy a stock the stock return is simply a proportion of the leveraged position and some asset growing at a risk free rate? But both these terms are deterministic? Have I misunderstood what they mean by leverage?

## Answer by nbbo2 (score 1, accepted)

https://quant.stackexchange.com/a/80403

First consider unlevered investment. We buy one share for $S_0$ and a year later the price is $S_1$. It is straightforward to find an expression for $S_1$ namely $$S_1 = S_0 (1+R_S)$$

since we know the return on the stock is $R_S$.

Now the levered case. We still buy one share. But this time we only pay $w S_0$ and take out a loan for the remainder namely $(1-w)S_0$. What is the value of the loan 1 year from now? We will have to reimburse $(1+r)(1-w)S_0$ since the rate on the loan is $r$.

What is the profit on the levered operation? The gain on the stock is $S_1-w S_0$ but we have to subtract the loan reimbursement so the profit is $\pi=S_1-w S_0-(1+r)(1-w)S_0$ or substituting $S_1$ as before and cleaning things up a little $$\pi= S_0[(1+R_S)-w-(1+r)(1-w)]$$ $$\pi= S_0[R_S-r+rw]$$

The rate of return is this profit $\pi$ divided by the initial investment $w S_0$. So finally $$R_L = \frac{\pi}{w S_0}=\frac{R_S-r}{w}+r$$

QED

As to the interpretation, note that if $r$ is small compared to $R_S$ this becomes approximately $$R_L \approx \frac{R_S}{w}$$ and this is often used as a rule of thumb, for example people say if you invest putting up only half in cash your return will be approximately doubled (on both the up and down side of course :-) ). But the above exact expression shows this is not quite true, some "adjustments" involving $r$ are necessary.