Utility Theory: The Investment Decision

Now that we’ve looked at Morgan’s dietary and budget situation it’s time to move onto Morgan’s optimal investment decision. In order to understand this, there are a few new concepts to introduce. Specifically risk and return.

Risk and return form the bedrock of the investment decision. Both risk and return can be decomposed into various metrics for acute, detailed analysis. For our purposes though we will measure risk as standard deviation and return as expected return. Both of these measures will be derived from past data – and I’m sure you’ve heard the phrase “past returns are not guarantees of future performance” or something of the sort – well the same can be said here. It is not bad practice to use past returns for analysis, it’s just important to keep in mind they are backward looking.

Standard deviation is defined as the average amount by which a series of returns deviates from the the average of the series. So if the average monthly return of an investment for the past 12 months was 2% with a monthly standard deviation of 5%, you could expect the return each month to fluctuate between 7% (2% + 5%) and -3% (2% – 5%). Expected return, for our illustration, is simply the average monthly return of 2%.

So each month we expect the investment to make a return of 2%, keeping in mind that it could go as high as 7% and as low as -3% for any given month.

Risk Aversion

Traditional finance speaks to risk and an individuals aversion towards it. There are three types of risk oriented investors. Risk averse, risk seeking and risk neutral. To illustrate consider two investment options:

Option A: Expected return of 10% with a standard deviation of 5%

Option B: Expected return of 11% with a standard deviation of 9%

A risk averse investor would choose option A because the extra 4% of risk, quantified as standard deviation, is more than double the extra 1% expected return between options A and B. To a risk averse investor the extra return is not worth the extra risk. The risk seeking investor would go for option B because to them a 9% standard deviation may justify the 11% return. A risk neutral investor would probably also go for option B since they are more concerned with the return side of things rather than the risk side. Although this is a very generic example it gets the point across. In reality there will be varying degrees of risk aversion and is not a one size fits all.

Traditional finance assumes most individuals are risk averse and bases its theories around this assumption. For our example, we will consider Morgan as being a risk averse investor.

Measuring expected utility of an investment

Enter the risk-free return. Throughout finance there is a concept of a risk-free return. This return is usually the return of a specific government bond, in most cases a United States government bond, and is considered risk-free because governments have a solid ability (especially the United States) of paying back the money they borrow plus interest. It doesn’t always have to be a government bond and depending on the analysis or financial securities involved it could be a money market rate for example. In our example though we will assume it is a US T-bill (which is the classic risk-free asset in traditional finance).

We use the risk-free rate in the investment decision by comparing the expected utility (covered shortly) of an investment to the risk-free rate. If the expected utility of an investment is greater than the risk-free rate, then the individual will investment in that risky asset rather than the T-bill. That begs the question though, how do you determine the expected utility of an investment?

We look at the following formula:

Expected utility = E(r) – (Λ)(σ2)

where;

E(r) = expected return.

Λ = Risk aversion coefficient, which is normally between 0 and 6, with 6 being highly risk averse.

σ2 = Variance, which is just the square of the standard deviation of the investment.

Great. Now let’s consider Morgan who is a somewhat risk-averse investor with a risk aversion coefficient of 4. For option A and B above Morgan’s expected utility will be 9.5% (0.1 – (4)(0.0025)) and 9.38% (0.11 – (4)(0.0081)) respectively. So Morgan will choose option A over option B since it has the higher expected utility, and will choose option A as long as the prevailing risk-free rate is below 9.5%.

Risk Indifference curve

Now that we know how to measure the expected utility of an investment, we can build a risk-indifference curve for Morgan. The concept of a risk indifference curve is as follows: as long as the expected utility of an investment is equal to the prevailing risk-free rate, the investor will be indifferent between investing in the risk-free asset or the the asset that carries some risk. As an example let us look at the expected utility calculation for a risk-free asset (T-bill) and a risky asset that results in the investor being indifferent.

Expected util
Note: In the calculation, the standard deviation for the risk-free asset is zero. This is because we know for certain that the return will be 4% so there is zero deviation and therefore zero variance. For the risky asset with a standard deviation of 10%, the variance is 10% x 10% = 1% = 0.01.

Because Morgan’s expected utility of both assets are equal, Morgan is indifferent as to which asset to invest into. This concept is represented as a risk-indifference curve shown below.

Expected util2

Risk indiff
The take-away here is that for each level of expected return and corresponding standard deviation (risk of the investment) the expected utility equals 4%. So Morgan could invest anywhere along this curve with the same expected utility.

This concept is analogous to Morgan’s indifference curve with regards to the combinations of burgers and salads, where any combination along the indifference curve provides Morgan with the same amount of utility.

An important concept here for a risk-averse individual is that the shape of the curve shows that a risk-averse investor needs a greater expected return for each additional unit of risk, in order to remain indifferent between investing in the risk-free asset or a risky asset.

The Efficient Frontier

We are in the same situation after we identified Morgan’s indifference curve with regards to the burgers and salads, and the sweet spot for the best combination was found by applying the budget constraint. In this way Morgan is satisfied with the amount of utility received as well as the full use of the budget.

Now that we’ve identified Morgan’s risk-indifference curve, it begs the question, what should Morgan invest into? This is where the investable universe comes in. Morgan will be exposed to countless investment options to satisfy the investment decision. And all of these options are represented by the rawest of metrics, namely expected return and standard deviation. Let’s assume Morgan is exposed to the following investment options.

Options
This universe is represented by the graph below.

Universe

Note: This is the universe for available risky investments and does not include the risk-free asset.

Now that we have Morgan’s investment universe, we need to identify which options are superior relative to one another. Note the following: Option B is superior to option C because it has the same expected return for a lower amount of risk. Option D is superior to option E for the same reason, while option F is superior to option E because it has a higher expected return for the same amount of risk. Thus, the efficient frontier of Morgan’s investable universe is represented by the curve below.

Efficient
Combining Morgan’s expected utility curve and the efficient frontier we are able to determine that investment option F is the optimal investment choice for Morgan. This is because it satisfies the expected utility as well as the investment into the optimal investment choice. This can be seen below.

Optimal

Next: Bounded Rationality

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