When constructing a portfolio of investments, asset allocation, or proportion, of stocks and bonds is widely acknowledged as one of the most critical decisions affecting the portfolio’s outcome over time.

Different proportions of stocks and bonds have different patterns of risk and return, and allocation plays an integral part in the overall success of the portfolio.

The challenge, of course, is deciding what allocation suits the investor.

Typically, the finance industry will provide a series of questions seeking to establish an asset allocation based on the investor’s holding period and risk tolerance.

These questionnaires pose fundamental key problems. The first, that despite what your financial advisor might say, their questionnaire is at best a mixture of art and science. The second is that on receiving the allocation ‘mix’, the method for selecting it for the investor is missing, if at all available.

The mix does little to advise the investor of their next steps, highlight the benefits or even the costs (in terms of risk) of adopting this target proportion of stocks over bonds.

Recently, Javier Estrada of IESE Business School in Barcelona, Spain, sought to develop a gain-pain index tool that overcomes these shortcomings.

### The best way to construct a portfolio

There are many approaches to producing a portfolio of stocks and bonds, but arguably, there is only one correct approach.

Portfolios are typically constructed over time through various acquisitions of asset classes. But this untargeted approach to allocation is unlikely to be sufficiently diversified according to the investor’s approach to risk.

A more targeted approach can be taken by first determining the proportions of each asset class within a portfolio then selecting assets to meet it.

This approach is essential as investors are more likely to stick with portfolios that they understand than sell-off during a downturn in the market and taking losses.

As different proportions of stocks and bonds have different patterns of risk and return the allocation.

### Introducing the gain-pain index (GPI)

The motivation behind the index is to account for risk variables that the investor considers necessary while highlighting the relevant tradeoffs in such an allocation mix.

The ‘gain’ terms of the index is given by the expected return. The ‘pain’ terms is given by two variables that summarize risk (or pain)… volatilty and the expected loss

Javier Estrada, The Gain-Pain Index

The expected loss is determined by the probability of suffering a loss and the magnitude of a loss. Finally, the index also accounts for an investor’s risk aversion.

By combining these variables and accounting for an investor’s holding period and risk aversion, the investor can derive a target allocation of stocks and bonds that is specific and more readily understandable to them as individuals.

### Steps to determining an optimal portfolio

Estrada sets out several steps, where each can be followed using historical index returns or using simulations.

- Determine an investor’s approach to risk (risk aversion coefficient)
- Determine the holding period for the portfolio
- Calculate the historical (or simulated) returns for all relevant asset classes
- Calculate the mean return (the gain), volatility and expected loss (the pain)
- Determine the GPI based on those asset allocations and select the one with the highest.

#### What is a risk aversion cofficent?

To determine risk aversion, a measure is made of the marginal reward an investor needs to take on more risk. The risk aversion coefficient is positive for risk-averse investors. It is zero for risk-neutral investors and negative for risk-seeking investors.

*For example*

A person is given a choice between two bets, one with a guaranteed pay-off and one with a risky pay-off, each with the same average values.

- A person receives £50
- A flip of a coin determines if the person receives £100

The expected pay-off for both scenarios is £50, e.g. certainty 100%*£50=£50 and 50%*£100=£50

If a person might accept £40 instead of taking a gamble, then they are considered risk-averse. If a person would accept the bet if the guaranteed payment were more than £50, then the person is considered risk loving (or seeking). An indifferent person has no preference for either the bet or a sure payment, a risk aversion figure of 1.

### Target allocations over 20 years

Using an allocation of US Stocks/Bonds, a period of 20 years and a neutral investor (neither risk-averse nor risk-loving), Estrada computes the following curve. Determining that through maximising GPI, an allocation of 95% stocks is an optimal portfolio.

Each curve below represents a different risk aversion coefficient to show the optimal portfolio percentage of stocks over different time periods. The shallower the curve, the greater the risk aversion or high the holding in bonds.

The red line in this chart presents a neutral investor, with the orange line representing a typical 60:40 stock vs bond split.

From Estrada’s analysis, it’s possible to demonstrate (in the US at least) that for time periods of greater than 10 years, a 60:40 split would suit more risk-averse investors better than they realise.

It is clear that given the higher the investor’s risk aversion (denoted by the shallower curves on the graph above) for any given holding period, the proportion of stocks is lower in the optimal portfolio. The opposite is also true for any given holding period.

### What about other countries?

Across all countries, Estrada calculates that for all holding periods and risk aversion coefficients, the average allocation to stocks is 70.1%.

However, across individual markets, the allocation various significantly, with markets such as Australia, Denmark and the UK having higher percentages of stocks on average compared to markets such as Spain, Sweden and Switzerland, which are more conservative.

### How does risk aversion change over time?

Estrada acknowledges that a person’s approach to risk changes as we approach the end of working life, estimating in a separate piece of research that the risk aversion coefficient roughly doubles during the last 25 years of work.

### The impact of the Gain-Pain Index

TThe index intended to consider an investor’s time for investing and their approach to risk in modelling optimal portfolios.

Using historical data or simulation, an investor can determine their optimal allocation of stocks versus bonds, understanding all the constituent parts in the process.