# Asset Allocation for Cash, Bonds and Stocks Based on Risk Tolerance

In our previous post ** How Risk Tolerance and Assets Allocation Go Together**, we investigate how using our own risk tolerance, expressed with

**with an**

*expected utility***, can help us determine the asset allocation which is suitable for us. Confining ourselves to being invested in a mixture of stocks and bonds, or in cash (fixed deposits), we arrive at the following recommendations:**

*isolelastic utility function with Constant Relative Risk Aversion (CRRA)*###### Asset Allocation based on Risk Tolerance

Utility Function | Risk Tolerance | Asset Allocation when Interest Rates = 5% | Asset Allocation when Interest Rates = 4% |
---|---|---|---|

Risk Aversion = 1 | High Risk Tolerance | 90% Stocks 10% Bonds | 90% Stocks 10% Bonds |

Risk Aversion = 2 | Normal Risk Tolerance | 60% Stocks 40% Bonds | 60% Stocks 40% Bonds |

Risk Aversion = 3 | Moderately Risk Averse | 100% Fixed Deposits | 40% Stocks 60% Bonds |

Risk Aversion = 4 | Very Risk Averse | 100% Fixed Deposits | 30% Stocks 70% Bonds |

There are two things of interest here, which we seldom see in financial advice given on asset allocation:

- Even for people with high risk tolerance, it is better for them to invest a little in bonds so as to get a better risk-return tradeoff, rather than to be 100% in stocks
- For the moderately and very risk averse people, when interest rates fall, it will be beneficial for them to be invested in stocks and bonds rather than remaining in cash or fixed deposits

Hence, asset allocation is meant to be dynamic, and changes with the level of interest rates, rather than just being static for all time!

But this is still an incomplete model for asset allocation, primarily because of 2 reason:

- We seldom only invest in stocks and bonds on the one hand, and cash/fixed deposits on the other. It is more likely that we split our asset between cash, bonds and stocks. Therefore, this needs to be built into our model for asset allocation
- The choice of asset allocation between stocks and bonds may be optimal for the level of risk tolerance, but it is not optimal for the level of interest rates. Why? Because the portfolios chosen are not necessarily the ones with the best risk-return trade off, or
. Surely when we allocate our assets, we prefer to do so to give us the biggest bang for our buck!*Sharpe Ratio*

So let’s see how we can build a more complete model for asset allocation for cash, bonds and stocks based on risk tolerance! Let’s solve our financial 3 body/asset problem!

#### What’s an Optimal Portfolio?

Let’s start by discussing what an optimal portfolio is. When we looked at the utility function in our previous post ** How Risk Tolerance and Assets Allocation Go Together**, we note that it is a curved, concave function which rises with our wealth, but at a slower and slower rate. Because of its shape, the util;ity function tells us that we will be better off if our wealth is higher, and if it did not vary so much. In other words, we prefer to be wealthier, and not to be often surprised by volatility of wealth.

The Sharpe Ratio, which accounts for both the return and the volatility of a portfolio, helps us look for the mix of stocks and bonds which is best. The definition of the Sharpe Ratio is as follows:

Now, when we look at the various combinations of stock and bonds portfolios, we see that they form a curved line in the returns-volatility space, or what we call the ** efficient frontier**:

###### The Efficient Frontier of All the Combinations of Stocks and Bonds

For every combination of stocks and bonds in a portfolio, we can compute the Sharpe Ratio. The portfolio combination with the highest Sharpe Ratio is the most optimal. Here are the Sharpe ratios of all the portfolios above when the risk free interest rate is 5%, 4%, and 3% respectively:

###### Sharpe Ratios of Different Asset Allocations and Interest Rates

Asset Allocation | Interest Rate = 5% | Interest Rate = 4% | interest Rate = 3% |
---|---|---|---|

100% Stocks 0% Bonds | 20.47% | 26.81% | 33.14% |

90% Stocks 10% Bonds | 20.49% | 27.51% | 34.53% |

80% Stocks 20% Bonds | 20.16% | 28.00% | 35.84% |

70% Stocks 30% Bonds | 19.31% | 28.17% | 37.02% |

60% Stocks 40% Bonds | 17.71% | 27.83% | 37.96% |

50% Stocks 50% Bonds | 15.14% | 26.88% | 38.62% |

40% Stocks 60% Bonds | 11.05% | 24.86% | 38.67% |

30% Stocks 70% Bonds | 4.61% | 21.09% | 37.56% |

20% Stocks 80% Bonds | -5.31% | 14.37% | 34.06% |

10% Stocks 90% Bonds | -19.50% | 3.44% | 26.38% |

0% Stocks 100% Bonds | -35.71% | -11.08% | 13.55% |

From the table above, we note that the optimal portfolio combination of stocks and bonds changes when the interest rates change. For example, when the interest rates are 5%, the optimal portfolio is one of 90% Stocks / 10% Bonds. When the interest rates are 4%, the optimal portfolio combination is 70% Stocks / 30% Bonds.

But this is at odds with some of our earlier recommendations for asset allocation. 90% Stocks / 10% Bonds is the recommended asset allocation for people with high risk tolerance when interest rates are 5%, and this is indeed the optimal portfolio, but the recommendation for people with normal risk tolerance, the 60% Stocks / 40% Bonds portfolio is suboptimal. Likewise, when interest rates are 4%, while people across the range of risk tolerances are all advised to invest in stocks and bonds, the recommended allocations are all suboptimal!

This anomaly arises because we considered cash separately from stocks and bonds in the asset allocation, and hence cannot account for the changes in interest rates. So how do we consider cash together with stocks and bonds in an asset allocation?

#### Asset Allocation for Cash, Bonds and Stocks

To determine the asset allocation suited to our risk tolerance between cash, bonds and stocks, we turn to ** Modern Portfolio Theory**. Although “modern portfolio theory” it is hardly modern, having been around for some 70 years! From the research of

**we are able to construct the efficient frontier of all combinations of stocks and bonds as we do above.**

*Markowitz,***tells us that we can find the optimal portfolio of risky assets from the combinations of stocks and bonds at the point where a tangent line originating from the risk free rate touches the efficient frontier (see below):**

*Sharpe’s**Capital Asset Pricing Model (CAPM)*###### Finding the Optimal Stock and Bonds Portfolio Using the Capital Market Line (Interest Rate = 5%)

From the diagram of the efficient frontier and capital market line above, we can see that the point of tangency has a slope which is equal to the Sharpe Ratio we discuss earlier! Hence, this combination of stocks and bonds at the tangency point is the optimal portfolio which gives the highest expected return for the risk taken.

Finally, we can add cash holdings earning the risk free rate of interest to the asset mix by deciding how much of our assets we hold in cash and how much in the risky portfolio if stocks and bonds. In the diagram above, this choice of cash vs stocks and bonds is represented by every point on the tangency line. For example, if we decide to hold 100% in cash, it corresponds to the point on the vertical axis, earning a return of 5%. If we decide to hold 100% in risky assets, it is the end point touching the efficient frontier, a portfolio of 90% run stocks and 10% in bonds. And every possible combination can be chosen, depending on our risk tolerance. This, in a nutshell, is the ** two fund separation theorem** from the research of

**.**

*James Tobin*Now, combining the results of portfolio construction from modern portfolio theory with utility theory and risk tolerance, we can come up with the recommendations for asset allocations based on risk tolerance and the level of interest rates:

###### Asset Allocation Based on Risk Tolerance and Interest Rates

Risk Tolerance | Interest Rate = 5.0% | Interest Rate = 4.5% | Interest Rate = 4.0% |
---|---|---|---|

Very High | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets |

Normal | 60% Cash 40% Risky Assets | 40% Cash 60% Risky Assets | 10% Cash 90% Risky Assets |

Moderately Risk Averse | 90% Cash 10% Risky Assets | 70% Cash 30% Risky Assets | 50% Cash 50% Risky Assets |

Very Risk Averse | 96% Cash 4% Risky Assets | 90% Cash 10% Risky Assets | 70% Cash 30% Risky Assets |

Optimal RiskyAssets Portfolio | 90% Stocks 10% Bonds | 80% Stocks 20% Bonds | 70% Stocks 30% Bonds |

Risk Tolerance | Interest Rate = 3.5% | Interest Rate = 3.25% | Interest Rate = 3.0% |
---|---|---|---|

Very High | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets |

Normal | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets |

Moderately Risk Averse | 20% Cash 80% Risky Assets | 0% Cash 100% Risky Assets | 0% Cash 100% Risky Assets |

Very Risk Averse | 50% Cash 50% Risky Assets | 30% Cash 70% Risky Assets | 0% Cash 100% Risky Assets |

Optimal Risky Assets Portfolio | 60% Stocks 40% Bonds | 50% Stocks 50% Bonds | 40% Stocks 60% Bonds |

As we notice previously, the optimal risky assets portfolio changes with interest rates. This is a feature of the CAPM and the Sharpe Ratio, since its computation involves the risk free rate. We show this below:

###### Optimal Risky Assets Portfolio Changes With the Risk Free Interest Rate

What makes this somewhat disconcerting is that even while we see that the recommended asset allocation tilts more and more towards risky assets as interest rates go down, the optimal risky portfolio itself tilts towards bonds and away from stocks at the same time! Is this contradictory? Actually, no. This becomes clearer when we work out the actual set allocation to cash, bonds and stocks:

###### Asset Allocation to Cash, Bonds and Stocks

Risk Tolerance | Interest Rate = 5.0% | Interest Rate = 4.5% | Interest Rate = 4.0% |
---|---|---|---|

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% |
---|---|---|---|

Normal | 0% Cash 40% Bonds 60% Stocks | Leverage | Leverage |

Moderately Risk Averse | 20% Cash 32% Bonds 48% Stocks | 0% Cash 50% Bonds 50% Stocks | Leverage |

Very Risk Averse | 50% Cash 20% Bonds 30% Stocks | 30% Cash 35% Bonds 35% Stocks | 0% Cash 60% Bonds 40% Stocks |

When we recast the asset allocations into the weights for cash, bonds and stocks in this way, it becomes clear that as interest rates go down, for every level of risk tolerance, there will be a shift away from cash, and into both stocks and bonds. Hence, at this point in time, when global central banks have started on their easing cycles, bring interest rates down from 5% to 4%, and towards 3%, the optimal asset allocations across all levels of risk tolerance should tilt towards risky assets.

Finally, note that we have omitted the asset allocation towards cash, bonds and stocks for the “Very High Risk Tolerance” people, as well as for “Normal Risk Tolerance” in some cases. This is because the current model restricts them to being unleveraged. When leverage is allowed, the model actually recommends leverage to bring the allocations to stocks and bonds above 100%! But this is for another blogpost in the future. Keep an eye out for it!

#### Conclusions

In our previous post ** How Risk Tolerance and Assets Allocation Go Together**, we show that using both the investor’s risk tolerance, and the risk-reward available through a portfolio of stocks and bonds can let us understand how investors should allocate their assets between either cash (in the form of fixed deposits), and a portfolio of risky assets like stocks and bonds. In particular, at low interest rates (i.e. 3.5% and lower), even the very risk averse investors (with very low tolerance for risk) will still find it worthwhile to start putting their assets into stocks and bonds.

Here, we take this model further, to integrate the asset allocations between cash, bonds and stocks at the same time. Using modern portfolio theory, we show that at interest rates of 3% and below, even the very risk averse investors can be fully invested in a 60:40 portfolio stocks and bonds with only enough cash to meet emergency purposes. This is quite different from how we see risk averse investors behaving, and also quite different from how financial advisors may advise them. In particular, this seems to indicate that there is too little risk taking happening at the very risk averse end of the spectrum (and perhaps a little too much at the high risk tolerance end!).

Additionally, we see that for the investors with high risk tolerance and normal risk tolerance, when interest rates start going below 3.5% or so, they might consider leveraging the investment portfolio for additional returns i.e. holding negative cash instead. this is something we will explore in our next post.