# The Risk and Return Relationship

**Introduction**

This article looks at the definitions of risk and return and how they interconnect in the investment arena. It then introduces basic portfolio theory. There are some calculations involved but we hope everyone will be able to follow along. If not, leave a comment and we’ll help you out.

**What is Risk?**

Risk and return are interlinked, and so it is not surprising that in most financial textbooks the definition of risk is directly associated to the expected return on an investment.

For the purposes of this article we could define risk as follows:

**The risk of an investment is the risk that the actual return we receive on that investment will be different to the return that we expected.**

While there are different sub-sets of risk the common factor between most of those sub-sets is that they are all measured by calculating the standard deviation of the expected return on the investment. A high standard deviation indicates a high degree of risk.

We will explain this further down in this article by a worked example.

**What is return?**

The return on an investment is the gain or loss on that investment over a defined period. The gain or loss on the investment usually comprises two elements: income received on the investment and a capital gain or capital loss on the capital value of the investment.

When an investor is considering whether to make an investment that investor usually anticipates or expects a certain level of return on that investment. If the investor makes the investment the risk the investor implicitly accepts is that the return the investor receives may not match the expected return.

**The fundamental link between Risk and Return – Perception is Everything**

The link between risk and return is one of the fundamental cornerstones of Financial Theory.

The greater the amount of risk an investor is willing to take, the greater the potential return. This is just another way of saying that investors need to be compensated for taking on additional risk.

Consider the following: A government bond - a US Treasury Bond is a good example (or GORTT bond if in TTD) - is considered a safer, less risky investment than a company bond. Accordingly, because the risk of investing in a corporate bond is higher than the risk of investing in a government bond, investors will require a higher expected rate of return to induce them to invest in the corporate bond rather than the government bond.

The extra inducement is required because the investor perceives the corporate bond to be “riskier” than the government bond.

**Taking this a step further – are there any Risk-Free Investments?**

The short answer is that all investments carry a degree of risk. However, some investments are close to being considered risk-free. In the United States, an example of a risk-free investment would be United States Treasury Bills. These are securities that are backed by the “full faith and credit” of the United States Government. It is the return on United States Treasury bills, bonds and notes that are often used as a measure of the risk-free rate and yield curve in the United States.

**Risk-Free Return**

The risk-free return is the return required by an investor to compensate that investor for investing in a risk-free investment. The risk-free return compensates investors for the effect of inflation, and for foregoing consumption (because the investor is making an investment he cannot use the funds invested for general consumption purposes such as buying a new car or going on a luxury holiday).

As we have noted above, the return on treasury bills is often used as a measure of the risk-free rate in the United States.

**The Risk Premium**

Following on from the definition provided above, risk simply implies that the future actual returns received on an investment may vary from the returns that the investor originally expected. If an investor undertakes a risky investment, then that investor will require a return that is greater than the risk-free rate to compensate them for the additional risk they incur on that risky investment. The riskier the investment, the greater the compensation the investor will require.

The difference between the return the investor expects, and the risk-free return is referred to as the risk premium.

**A Worked Example**

Bob currently has all his savings deposited in a bank current account earning no interest. He is considering buying shares in a company quoted on the Trinidad and Tobago Stock Exchange and is trying to determine whether the shares are a viable investment. Each share will cost TT$100 and is expected to pay a dividend of TT$5 in the year. At the end of the year Bob’s stockbrokers expect each share to be worth TT$117.

In considering making the investment Bob should consider:

The expected return on the shares.

What risk premium he requires to compensate him for undertaking a risky investment.

**The Expected Return on the Shares**

The return on an investment in shares comes in the form of dividends received and capital gains (or losses) on the market value of the share.

Annually we could express this expected return with the following formula:

In Bob’s case, we can see that his expected annual return comprises of a dividend yield of 5% and a capital gain of 17%. In a real-world situation Bob doesn’t know what dividend will be paid in the year, or what the share price will be in one year’s time. This is the risk Bob faces. The returns he expects to receive may not match reality.

**Revisiting the definition of risk**

The definition of risk that is commonly found in finance textbooks is based on statistical analysis designed to measure the variability of the actual return from the expected return. The statistical measure of variability most commonly found in textbooks is the variance from expected return and the standard deviation (the square root of the variance).

We will explain this by looking at two possible investments.

**Worked example – risk and return with a slice of statistical analysis**

Varun is considering an investment in either Aston Limited or Zetec Limited.

An investment in Aston Limited has the following range of expected returns and associated probabilities of those returns occurring:

An investment in Zetec Limited has the following range of expected returns and associated probabilities of those returns occurring:

Varun wants to know what the expected return on each of the investments is and which of the investments exposes him to the greatest amount of risk.

**Solution Step One – Work out the Expected Return**

The first step is to work out the expected returns for the Aston Limited and Zetec Limited investments. The expected returns of each of the investments is calculated by multiplying the probability of each of the possible returns by the return expected and summing the results.

**For Aston Limited the expected return is:**

(0.1 x 30%) + (0.8 x 20%) + (0.1 x 10%) = 20%

**For Zetec Limited the expected return is:**

(0.1 x 33%) + (0.7 x 21%) + (0.2 x 10%) = 20%

We can see that despite having a different range of expected outcomes and probabilities the return on Aston Limited and Zetec Limited are equal. If we considered expected return only Varun would be indifferent between making an investment in Aston Limited or Zetec Limited.

**Solution Step Two – Work out the Variance of Returns**

Risk is measured by the variance of the expected returns of both Aston Limited and Zetec Limited.

The variance of return is calculated as the weighted sum of the squared deviations from the expected return. These are added and the square root of the sum gives us a measure of how risky each of the investment is.

**The results in summary**

In summary form, the result of our analysis of Aston Limited and Zetec Limited is as follows:

Given that the expected return is the same for both Aston Limited and Zetec Limited, Varun should opt for an investment in Aston Limited because it has the lowest risk and the same level of return as Zetec Limited.

**Taking this a step further**

In real-life we are seldom faced with a decision to invest in one of two different stocks as is the case with Varun. Usually we are considering a multitude of investments in the context of adding those investments to a portfolio of assets that we already own.

**When we start to look at portfolios with more than one investment the math gets more complicated because we should consider how each of the individual investment relates to the existing investments we already own. When exposed to the same external factors some investments in our portfolio go up while others will go down. This is the realm of advanced portfolio theory.**

When we add portfolio theory into the mix we should consider how the individual returns of the investments in our portfolio co-relate or co-vary. If two or more investments move in the same direction (for example they both go up) when exposed to the same factors they are said to be positively correlated, while if they move in different directions they are negatively correlated.

It follows from this that it is possible to add a new investment into an existing portfolio and reduce the overall risk of the overall portfolio of investments.

Portfolio theory is a subject we can return to in a future article!

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*This article is a publication of a member company of the Securities Dealers Association of Trinidad and Tobago. Nothing within it is intended to constitute legal, tax, securities or investment advice nor an opinion regarding the appropriateness of any investment, nor a solicitation of any type. Its contents are intended for general information purposes only and should not be acted upon without obtaining specific legal, tax and investment advice from a licenced professional concerning your own situation and any specific investment questions you may have. The views and opinions expressed in this article are those of the member company and do not necessarily reflect the official policy or position of SDATT.*