How Does Leverage Fit Into an Asset Allocation based on Risk Tolerance?
Leverage is a tool for investment and asset allocation that is offered to by banks and brokers to their customers. However, there is very little written about how to use it appropriately. And even less is known about how this fits in with our risk tolerance. Most of the time, the maximum amount of leverage based on the collateral provided (whether in cash or stocks and bonds) is offered, and taken, without much consideration of how much is best and how best to utilise it.
While we too have written a few blogposts on using leverage, see:
- How to Invest with Leverage Safely
- Should I borrow to invest? And how much leverage can I take on?
- An Easy Approach to Leveraged Risk Parity Investing
We have always approached the topic, like most others, from a supply-side perspective. Supply-side meaning how much leverage can be obtained from banks and brokers, what are the considerations to using leverage, and how to invest using leverage. But almost nothing is usually discussed from the demand-side perspective. Demand-side, meaning what is the best amount of leverage to use.
It should not be a surprise that the optimal amount of leverage to use should be based on our risk tolerance. And how to deploy leverage as part of our asset allocation is based on our risk tolerance, expected risk and returns, and on the level of interest rates. And these are all part of the framework around expected utility, risk tolerance, returns on stock and bond portfolios and interest rates which we explore in our earlier posts:
- How Risk Tolerance and Asset Allocation Go Together
- Asset Allocation for Cash, Bonds and Stocks Based on Risk Tolerance
So, the expected utility and risk tolerance framework seems like a natural way of looking at the subject of leverage and asset allocation. And this is where we start. Let’s see what we can figure out about how leverage fits into an asset allocation based on risk tolerance!
The Final Piece in Asset Allocation – Leverage
Asset Allocation Between Cash, Bonds and Stocks Using the Capital Market Line
In our previous blogpost Asset Allocation for Cash, Bonds and Stocks Based on Risk Tolerance, we look at the historical returns and volatility of various stock and bond portfolios to work out the efficient frontier which we can invest in. The efficient frontier shows the combinations of stocks and bonds which have the highest returns for any level of risk.
The Efficient Frontier of All the Combinations of Stocks and Bonds
If we now assume that we can also choose to invest in cash or deposits at the prevailing interest rate, we find that the optimal portfolio is the one on the tangency point of a line from the interest rate on the vertical axis, to the efficient frontier. This is the Capital Market Line, and the optimal portfolio is the one with the highest Sharpe Ratio.
Finding the Optimal Stock and Bonds Portfolio Using the Capital Market Line (Interest Rate = 5%)
Now, if we also assume that we not only can just deposit cash to earn the interest rate, but also borrow at the market interest rate to invest more in stocks and bonds, we can extend the Capital Market Line beyond the tangency point. We show this below for a case with an interest rate of 3%:
Finding the Optimal Allocation of Cash, Borrowing, Bonds and Stocks using teh Capital Market Line (Interest Rate = 3%)
Notice that with the extended Capital Market Line, we can invest in portfolios of stocks and bonds with borrowing or deposits. And all these combinations lie above the original efficient frontier. This means we can get higher returns for the same levels of risk or volatility as those points on the efficient frontier! And this is how leveraged investing can potentially offer better risk-adjusted returns.
So, how do we decide which combination of cash, leverage, stocks and bonds on the Capital Market Line is the best asset allocation for us? This is where risk tolerance comes in. For every asset allocation on the Capital Market Line, we have the expected return and volatility. We can assume the returns follow some sort of distribution around the expected return:
Example of the Distribution of Returns of each Asset Allocation
Usually, it is quite naive to assume that asset returns follow a true normal distribution, but historically, we observe the worst drawdowns of each asset allocation and work that into the distribution as well. Each possible outcome in the distribution of returns has an associated probability, so we can put all this into the isoelastic or Constant Relative Risk Aversion (CRRA) utility function:
And finally, weight all possible outcomes by their probability to obtain the expected utility of investing in each asset allocation. the asset allocation with the highest expected utility is this the best for us, given our risk tolerance.
When Should We Start Using Leverage in Investing?
Previously, we showed a rather incomplete table of the asset allocations for different levels of risk tolerance depending on interest rates. incomplete, in the sense that we were restricting ourselves to the case where no leverage was used, and hence the allocations between cash, bonds and stocks added up to 100%. However, if we allowed asset allocations along the Capital Market Line, including the use of leverage, we cn get a more complete picture as follows:
Asset Allocation to Cash, Borrowing, Bonds and Stocks
| Risk Tolerance | Interest Rate = 5.0% | Interest Rate = 4.5% | Interest Rate = 4.0% |
|---|---|---|---|
| High | 0% Cash 10% Bonds 90% Stocks | 30% Borrowing 26% Bonds 104% Stocks | 60% Borrowing 48% Bonds 112% Stocks |
| Normal | 60% Cash 4% Bonds 36% Stocks | 40% Cash 12% Bonds 48% Stocks | 10% Cash 27% Bonds 63% Stocks |
| Moderately Risk Averse | 90% Cash 1% Bonds 9% Stocks | 70% Cash 6% Bonds 24% Stocks | 50% Cash 15% Bonds 35% Stocks |
| Very Risk Averse | 96% Cash 0.4% Bonds 3.6% Stocks | 90% Cash 2% Bonds 8% Stocks | 70% Cash 9% Bonds 21% Stocks |
| Risk Tolerance | Interest Rate = 3.5% | Interest Rate = 3.25% | Interest Rate = 3.0% |
|---|---|---|---|
| High | 100% Borrowing 80% Bonds 120% Stocks | 140% Borrowing 120% Bonds 120% Stocks | 200% Borrowing 180% Bonds 120% Stocks |
| Normal | 0% Cash 40% Bonds 60% Stocks | 50% Borrowing 75% Bonds 75% Stocks | 100% Borrowing 120% Bonds 80% Stocks |
| Moderately Risk Averse | 20% Cash 32% Bonds 48% Stocks | 0% Cash 50% Bonds 50% Stocks | 50% Borrowing 90% Bonds 60% Stocks |
| Very Risk Averse | 50% Cash 20% Bonds 30% Stocks | 30% Cash 35% Bonds 35% Stocks | 0% Cash 60% Bonds 40% Stocks |
In the tables above, our ingoing assumption is that 100% of the portfolio value will be investment capital. So, if the entry reads as an asset allocation of 100% to borrowing, it means that the investor borrows another $100 on top of the $100 of investment capital, giving a gross asset value split between stocks and bonds of $200.
Using our risk tolerance and the level of interest rates to determine whether we should borrow to invest, and how much to borrow, tells us a few interesting things:
- Leverage works only if interest rates for borrowing fall below 4.5% for this with very high risk tolerance. For the rest of us with normal risk tolerance, it really only works if interest rates fall below 3.25%. For those who are moderately risk averse, the threshold interest rate is even lower at 3%
- In unleveraged investment portfolios, as interest rates fall, the asset allocation to both bonds and stocks goes up, and the asset allocation to cash falls
- However, in leveraged investment portfolios, as interest rates fall, the amount of borrowing goes higher, and the asset allocation to bonds goes up as well. Asset allocation to stocks goes up a little and stays constant at around 120% regardless of how low interest rates go!
What these observations seem to imply is that when we do leverage our investment portfolios, we are mainly borrowing to invest more into bonds. Alternatively, we can view this as borrowing against the value of bonds in the portfolio. This makes sense at several levels:
- Bonds are less risky than stocks, and hence for the purposes of borrowing, banks and brokers are often willing to lend more against their value than for stocks. In other words, having more bonds as collateral allows a higher level of leverage
- When we looked at different approaches to leveraged investments, the risk parity portfolio has highlighted as one feasible approach. This recommends increasing the allocation of bonds in the portfolio through leverage, to bring the risk of bonds on par with stocks in the portfolio. Hence the name risk parity. Pure risk parity investing can recommend asset allocation of 80% in bonds and 20% in stocks. Here, however, the recommendations based on risk tolerance are similar but not the same, being in the range of 50%-60% in bonds and 40%-50% in stocks instead
Nevertheless, the similarity of the asset allocations for investing with leverage between a pure risk parity approach and a risk tolerance-based approach give some comfort that “all roads lead to Rome” in investing. However, as pointed out previously, this also has to do with the relative risk and returns of bonds and stocks over the past quarter century, and not some deeper law of asset returns or investing.
Finally, as we do not expect to see interest rates fall to 2% or even 1% anytime soon in the current economic cycle, we have omitted reporting the results for very low borrowing rates. However, it is interesting also to note that at such low interest rates, the amount borrowed, even for the highest level of risk tolerance, do not go beyond 500%. Not surprisingly, this is also in line with most lending guidelines at banks and brokers. That is, not to extend loans beyond 4X-5X of the collateral value. Which is also what we obtain when we approach this question of leverage investments from a risk management perspective.
At the same time, even when interest rates are so low, the asset allocation to stocks also does not go past 150% of investment capital. Most of the leverage goes towards increasing the allocation to bonds instead, just like in the risk parity portfolio.
Conclusions
The use of leverage in investing, especially for stocks, is something which is not usually encouraged. However, when used according to our risk tolerance, leverage can expand the range of portfolio asset allocations to our benefit.
While the tools for using leverage are often accessible to investors, how much leverage to use is something which is more often than not left to the investor’s judgement. And there is little research on how this judgement should be made! Here, we have tried to use a simple model of risk tolerance to explore for whom it might be useful to use leverage in investing, and at what level of interest rates.
And we find that:
- For investors of normal risk tolerance, it does not make sense to use leverage unless interest rates are below 3.25%. Given the stage we are at in the interest rate cycle, it appears premature to use leverage now
- When leverage is used, it should go mainly towards increasing the allocation to bonds in the portfolio, instead of stocks. This is in line with the usual risk parity portfolios, whereby increasing the proportion of bonds to match the volatility of the stocks allocation can lead to better investment outcomes
