It’s possible to use the brokerage money to invest with some interest i.e. margin trading. Some brokerage offers really low rate, like Robinhood’s 5.75%. With the annualized return of VOO being 14.56%, it may seem like a no brainer to use margin to buy VOO, but if factoring the risk, is it really worth it?
Max margin to avoid a margin call
Margin call can happen if your equity fall under the maintenance margin requirement set by your brokerage, which is typically 25% of the account value.
Let
- total investment = $I$
- margin used (i.e. loan) = $M$
- equity = $E=I-M$
- maintenance margin ratio = $m=\frac{E}{I}=\frac{I-M}{I}=1-\frac{M}{I}$
Suppose we use our endowment $E_0$ and margin $M_0$ to buy an asset with a return $r$, the margin ratio $m=1-\frac{M}{I}=1-\frac{M_0}{(1+r)(E_0+M_0)}$
$$1-\dfrac{M_0}{(1+r)(E_0+M_0)}\geq m_{\min}$$
$$M_0 \leq \frac{(1-m_{\min})(1+r)}{1 - (1-m_{\min})(1+r)}E_0$$
$$M_{0\max} = \frac{(1-m_{\min})(1+r)}{1 - (1-m_{\min})(1+r)}E_0$$
If the min margin ratio $m_{\min}=0.25$, and we want to sustain the worst annual return in S&P500’s history, -37% if 2008, without triggering a margin call
$$M_{0\max} = \frac{(1-0.25)(1-0.37)}{1 - (1-0.25)(1-0.37)}E_0=0.89E_0$$
Meaning if you have 100k initially, you can safely borrow 89k.
If we’re only concern about smaller crashes like VOO -18.1% in 2022,
$$M_{0\max} = \frac{(1-0.25)(1-0.181)}{1 - (1-0.25)(1-0.181)}E_0=1.59E_0$$
Meaning if you have 100k initially, you can safely borrow 159k.
Max margin for an average risk-averse person
However, an average individual is not risk-neutral but risk-averse. Since VOO can be +30% in one year but -30% in another, but the interest rate is a lot more stable does it still make sense for a rational but risk-averse investor to trade with margin?
Assuming the interest rate of the margin $r_b$ never changes. We use leverage $L$ (total investment / equity, i.e. you have L-1 of equity in margin) to buy a security with a expected return of $r$ and a standard deviation of $\sigma$. Assuming constant relative risk aversion (CRRA) with the Arrow-Pratt coefficient of relative risk aversion being $\rho$. The certainty equivalent $\text{CE}$ (effective risk-free return) of gross return is
$$\text{CE}(L)\approx L(1+r) - (L-1)(1+r_b) - \frac{1}{2}\dfrac{\rho}{(1+r)}(L\sigma)^2$$
(see appendix for proof)
To find the most optimal $L$, simply calculate the derivative of it with respect to $L$
$$\dfrac{d \text{CE}(L)}{dL}=r-r_b-\dfrac{\rho}{1+r}\sigma^2L$$
$$L^*=\dfrac{(1+r)(r-r_b)}{\rho\sigma^2}$$
The value of CRRA $\rho$ is debated, but the most widely accepted measures lie between 1 and 3.
Plugging in the numbers of VOO in the past 10 years, $\mu=13.61$%, $\sigma=15.51$%, with the margin interest rate $r_b=5.75$%, and a risk coefficient $\rho=2$, we get
$$L^*=\dfrac{1.1361(0.1361-0.0575)}{2\times0.1551^2}=1.86$$
Meaning, as an average risk-averse investor, if you have 100k in VOO without any loan/margin, you should use 86k of margin to buy VOO.
Appendix: Deriving CE of the gross return of a risky asset from CRRA utility function
The CRRA (constant relative risk aversion) utility function is
$$u(c)=\dfrac{c^{1-\rho}-1}{1-\rho}$$
where $c$ is consumption, and $\rho$ is a constant, the CRRA coefficient.
Let $c=\mu+\varepsilon$ with $\mathbb E[\varepsilon]=0$, $\mathbb E[\varepsilon^2]=\sigma^2$
Second-order Taylor at $c=\mu$:
$$u(c)=u(\mu)+u’(\mu)\varepsilon+\tfrac12u’’(\mu)\varepsilon^2+O(|\varepsilon|^3)$$
$$\mathbb E[u(c)] = u(\mu)+u’(\mu)\mathbb E[\varepsilon] +\tfrac12 u’’(\mu)\mathbb E[\varepsilon^2]+\mathbb E[R_3(\varepsilon)] = u(\mu)+\tfrac12 u’’(\mu)\sigma^2+\mathbb O(|\varepsilon|^3)$$
$$\mathbb E[u(c)] \approx u(\mu)+\tfrac12 u’’(\mu)\sigma^2$$
Let $\text{CE}=\mu+\delta$ with $|\delta|$ small. First-order Taylor of $u$ at $\mu$:
$$u(\mathrm{CE})=u(\mu+\delta)\approx u(\mu)+u’(\mu)\delta$$
By definition $\mathbb E[u(c)]=u(\mathrm{CE})$, so
$$\tfrac12 u’’(\mu)\sigma^2a\approx u’(\mu)\delta$$
$$\delta \approx \dfrac{u’’(\mu)}{2u’(\mu)}\sigma^2 = -\tfrac12A(\mu)\sigma^2$$
where $A(\mu)=-\dfrac{u’’(\mu)}{u’(\mu)}$ is the absolute risk aversion
$$\mathrm{CE}=\mu+\delta=\mu-\tfrac12A(\mu)\sigma^2$$
$$A(\mu)=-\dfrac{u’’(\mu)}{u’(\mu)}=-\dfrac{-\rho c^{-\rho-1}}{c^{-\rho}}=\dfrac{\rho}{c}$$
The risk free gross return equivalence of an asset with average gross return $\mu$ and an standard deviation of $\sigma$ is
$$\mathrm{CE}=\mu-\dfrac{\rho}{2\mu}\sigma^2$$