Utility Equation & Risk Appetite

All of us (at least most of us) definitely invest our hard earned money into some instruments. We make a choice from a variety of instruments like Fixed Deposits, Mutual Funds, Shares, Bonds, Post Office Savings Schemes, etc. Generally, our portfolio will consist of multiple instruments. This should be the case as well as this would provide the needed diversification.

However, there comes a question as to what should be our preferred investment instrument. This question can be answered by our Risk Appetite.

Risk Appetite defines our nature towards Risk.

Broadly, Risk Appetite can be categorised into whether we are

  1. Risk Averse
  2. Risk Neutral
  3. Risk Seeker

Risk Averse people always avoid Risk.

Risk Neutral people are neutral to Risk. That is they consider taking risk sometimes and avoiding it at other times.

Risk Seekers always take a risk in hope for higher Returns.

Variance and Standard Deviation

in the Investment World, the level of Risk posed by an instrument is calculated in terms of the Variance or Standard Deviation of the Returns from the Instrument.

Standard Deviation is nothing but the Square Root of the Variance.

For example, a Fixed Deposit would give a return of 7.10% over a period of 12 months (as per latest rates from HDFC Bank Ltd). This implies that the Standard Deviation of the returns from such a Fixed Deposit would be 0. Thus, we can say that the Fixed Deposit instrument is Risk Free or has Zero Risk.

Though, we need being careful here due to the fact that HDFC Bank Ltd is a Private Sector Bank. Thus, we cannot altogether say that the Fixed Deposit in HDFC Bank Ltd is actually Risk Free.

Normally, we consider Instruments like National Savings Certificates (NSC) or Public Provident Fund, issued by the Government, to be Risk Free.

Nevertheless, going by the Standard Deviation consideration, suppose a Mutual Fund gives the following returns over the last 9 months.

17.64%, 16.32%, 15.92%, 26.44%, 13.54%, 7.36%, 6.16%, 7.89%, 7.49%

We can calculate that the Standard Deviation of these returns is 6.68%. Thus, this is definitely a riskier instrument as compared to a Fixed Deposit.

How to decide where to Invest?

So, we have a proposition where we are dealing with instruments having different levels of risk. Do we have a method to determine what will be best for us? We definitely do have a method.

The method is to determine the Utility of the investment for us. The instrument, which would have higher utility for us, should be preferred.

How do we calculate Utility?

The formula for Utility is as follows:

Utility = Expected Returns – (1/2) * (Coefficient of Risk Aversion) * (Variance)

Coefficient of Risk Aversion is normally denoted by A.

If A = 0, we say that the Investor is Risk Neutral.

If A > 0, we say that the Investor is Risk Averse.

If A < 0, we say that the Investor is Risk Seeker.

Example

Let us consider 2 investment choices.

  1. In the first choice, we say that the Investor wants to invest in Fixed Deposit at a rate of 7.10% per annum. In this case, the Variance (and Standard Deviation) can be considered as 0.
  2. In the second choice, the Investor wants to invest in a Mutual Fund which may give a return of 50% with a probability of 70% OR a return of -20% with a probability of 30%.

Let us calculate the Expected Return and Variance in both cases.

Case 1: Fixed Deposit

Expected Return = 7.10%

Variance = 0%

Case 2: Mutual Fund

Expected Return = 50% * 70% + (-20%) * 30% = 35% – 6% = 29%

Variance = 70% * (50% – 29%)^2 + 30% * (-20% – 29%)^2 = 10.29%

.

So, clearly the Investment in Mutual Fund provides a much higher Return at a much higher level of Risk.

.

Now, let us see the Utility of these 2 Investments for Investors with different levels Risk Appetite.

First, we consider a Investor, who is Risk Neutral. For these Investors, A = 0.

Thus,

Utility from Fixed Deposit = 7.10% – (1/2) * (0) * (0) = 7.10%

Utility from Mutual Fund = 29% – (1/2) * (0) * (10.29%) = 29%

Clearly, for a Risk Neutral Investor, the Mutual Fund Investment has higher Utility.

.

Next, let us see the case of a Risk Averse Investor with A = 3.

Thus,

Utility from Fixed Deposit = 7.10% – (1/2) * (3) * (0) = 7.10%

Utility from Mutual Fund = 29% – (1/2) * (3) * (10.29%) = 13.565%

In this case also, the Mutual Fund Investment has more utility for this Investor.

.

Next, let us see the case of a Risk Averse Investor with A = 5.

Thus,

Utility from Fixed Deposit = 7.10% – (1/2) * (5) * (0) = 7.10%

Utility from Mutual Fund = 29% – (1/2) * (5) * (10.29%) = 3.275%

In this case, the Fixed Deposit Investment has more utility for this Investor.

 

Advertisements

This site uses Akismet to reduce spam. Learn how your comment data is processed.