Visualizing Risk

Risk a concept that encompasses both the consequences of actions and the probability of such consequences eventuating. It is an important concept in personal finance and heavily influences investment and insurance decisions. The better we are at identifying risk the more aligned our investment and insurance decisions will be with our risk tolerance and capacity.

Evaluating risk using probability of outcomes

Let’s assess a simple game that costs $1 to play. The game involves rolling a fair 6-sided dice (each side is numbered either 1, 2, 3, 4, 5 or 6). If you roll a 5 or 6 you will be paid $2.50. If you roll any other number, you will not win anything. Should you play this game?

The possible outcomes of this game are as follows:

Number rolled	Cost to play	Revenue	Profit
1	$1	$0	-$1
2	$1	$0	-$1
3	$1	$0	-$1
4	$1	$0	-$1
5	$1	$2.50	$1.50
6	$1	$2.50	$1.50

Expected value
In the field of probability and statistics, the expected value is calculated by multiplying the value of each outcome with its corresponding probability and summing all of these values together. In the above example the expected value is as follows:

In this case the expected value is -$0.17 which implies that if you play the game many times, you will lose money. This is an indicator that this game is not a good investment.

When does an investment become worth it?
Solving for EV = 0 will allow you to calculate the ‘breakeven’ point which indicates that in the long run an investor is likely to make a profit of $0. In the previous example, the breakeven point is where the revenue gained from rolling a 5 or 6 is equal to $3.00.

Risk tolerance and capacity
When it comes to investing, expected value needs to be combined with risk tolerance when determining if an investment is right for you. While expected value gives you an indication of the long-term profitability of the investment, you first need to be able to survive the financial journey. Allow me to change the numbers in our previous example to illustrate this point.

Number rolled	Cost to play	Revenue	Profit	Probability
1	$5,000	$0	-$5,000	0.167	-$833.33
2	$5,000	$0	-$5,000	0.167	-$833.33
3	$5,000	$0	-$5,000	0.167	-$833.33
4	$5,000	$0	-$5,000	0.167	-$833.33
5	$5,000	$0	-$5,000	0.167	-$833.33
6	$5,000	$100,000	$95,000	0.167	$15,833.33
				Expected value	$11,666.67

Alright, now we’re talking, by paying a mere $5,000 we can earn a revenue of $100,000 for a profit of $95,000. Easy money baby! In addition to this, the expected value of this investment opportunity is $11,666.67 which is over double that of the initial investment cost. Based on these metrics, the investment is a go.

This is where an understanding of individual risk tolerance becomes very important. Looking from a mathematical point of view, a player/investor in this game will have a 5/6 (83.33%) chance that they will lose their $5,000 investment in any given round. This means that it is likely they will need to play numerous rounds before winning the $100,000 pay off. In a financial risk tolerance sense, they will need enough money to facilitate repeatedly playing until they can hit the pay off. If an investor is unable to tolerate the negative consequences, then even though the game has a positive expected value it is not worth playing.

For example, if an investor has only $10,000 then they can only play 2 rounds. The probability that they will win the $100,000 pay off at least once in 2 rounds is as follows:

Now is paying up to $10,000 for a 30.56% chance of a profit of between $90,000 and $190,000 a worthwhile investment for this investor? Well, that is up to them to decide if this level of risk is acceptable. Personally, because I lean towards risk aversity I would want greater certainty – and I would be willing to accept lower returns in order to access this certainty. If this were a real-world example, I would look to bring enough capital to play the game at least 10 times, where the probability of winning at least once moves to:

Reality vs. mathematics

Mathematical scenarios like the ones above are easy to describe and calculate outcomes. In the real-world outcomes and pay offs are far more difficult to ascertain. However, there is still tremendous value in applying the same thought process of looking at scenarios and probabilities; especially when it comes to understanding your risk tolerance and capacity. I believe that understanding, and being comfortable with the downside risk is just as, if not more, important than estimating the upside. To do this, the components of investment costs and performance may need to be looked at separately.

A hypothetical investment property investment
The above example can be likened to buying an investment property, where the rounds are similar to years that an investor will need fund their investment before capital price appreciation creates an acceptable payoff.

Let’s say you decide to buy a $600,000 investment property with 20% down payment and take on an interest only mortgage with a 2.5% interest rate. To make the example simple, I’ll assume that no rent is collected. Now, assuming the property value remains steady for the first 4 years but then increases 47% during the 5^th year (equivalent to 8% p.a.) then the first 5 years of your investment journey look something like this:

Year	Investment costs	Interest costs	Value of property
0	$162,000	$0	$600,000
1	$2,000	$12,000	$600,000
2	$2,000	$12,000	$600,000
3	$2,000	$12,000	$600,000
4	$2,000	$12,000	$600,000
5	$2,000	$12,000	$881,597

After 5 years, you would have spent a total of $232,000, and have a mortgage of $480,000 on a $881,597 property. If we assume that closing costs from the sale of the house are 5% of the sale price this yields a profit of $125,517. This can be used as a representation of the expected value for this investment opportunity.

However, as discussed earlier, the varying costs of the investment need to be modelled in order to get an idea if the investment risk is worth taking. If we vary the interest rate from 0.50% to 8.00% the interest cost component varies between $2,400 p.a. and $38,400 p.a. If we fix growth at 8% p.a. and investment/maintenance related costs at $2,000 p.a.; the total costs and profits are as follows:

Interest rate	0.50%	8.00%
Yearly total investment cost	$4,400	$38,400
Value of property after 5 years	$765,769	$765,769
Profit	$173,517	-$6,483

Now, the questions that need to be asked are:

1. How high can interest rates go before the you are unable to fund the investment?
2. What is the probability of interest rates rising/falling?

If this exercise is repeated numerous times in conjunction with asset value growth, then we arrive at a plot like the one shown below:

In the case of property, price growth tends to be negatively correlated with interest rates, hence the negatively sloping ellipses. Also, the chance of outcomes is uncertain so they will be a function of the investor’s predictions – hopefully these are based on sound reasoning! The green shaded section represents the outcomes that can be funded by the investor regardless of the profitability or expected value of the investment.

Interpreting the distribution of outcomes
In my opinion, for an investment to be worthwhile, from a risk perspective, it needs to have the following features:

1. Positive net expected value that compensates the investor adequately for the risk that is taken.
2. It must be within the investor’s capability to fund the investment costs

By looking at the distributions of outcomes an investor can gain an understanding of how suitable an investment is for them. Figure 1 shows that the investor is capable of funding a wide range of interest rate movements, however roughly half of the expected outcomes are not profitable – this should be a key point of consideration.

Continuing with the theme of leveraged investments; if an investor’s distribution looks something like this:

Then their ability to withstand interest rate rises is limited. Even though a large proportion of their expected outcomes is profitable it still falls outside their funding capability so it can be said that this investment is particularly risky. In such cases, an investor needs to look at increasing their funding capability.

Conclusion

In personal finance, risk represents the consequences and probability of different outcomes. I believe the key to making educated investment decisions is to understand the risks associated with any investment opportunity presented to you. Estimating expected values and understanding your own risk tolerance, as it pertains to your overall financial situation and goals, is an important part of this process. As with all investing activities, even though outcomes and probabilities are uncertain, having a process to identify a range of consequences will generally result in a better alignment to your individual risk tolerance that simply guessing.

• What is the range of performance outcomes that might be expected from this investment?
• What is the probability of achieving various outcomes within this range?
• What sort of funding costs can I tolerate while waiting for my investment to perform?
• If cost increases arise, how will they be dealt with?

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