You probably came across many instances where risk is highlighted as a value that you need to incorporate in your projections and assessment of projects before you decide to proceed. So how do you calculate your risk tolerance and produce it as an actual percentage? Let me show you..

First let’s get a couple of concepts cleared so we can understand the approach. For the first concept let’s use this example. Let’s say that you are trying to evaluate whether buying and then selling a house is going to be a profitable project or not. For this example I will not get into interest rates, discount rates, NPV’s etc not because they are not important, but because they will muddy the waters when trying to explain risk tolerance and may confuse.

So with our simple example let’s say that the house you are about to purchase will cost you $100,000 and then you can flip it, sell it at $120,000. Therefore your profit is going to be $20,000. Clear so far?

Let’s do this using a cash flow approach, which is the correct approach for evaluating any project.

Expected Value of Project = sum of all expected cash flows

Therefore Expected Value of Project = incoming cash – outgoing cash

Therefore Expected Value of Project = 120,000 – 100,000

Therefore Expected Value of Project = 20,000.

Since the value of the project is positive then the project should be a go. You will make money if you proceed. Let’s try this again with the assumption that the sale price will be 110,000.

Expected Value of Project = sum of all expected cash flows

Therefore Expected Value of Project = incoming cash – outgoing cash

Therefore Expected Value of Project = 110,000 – 100,000

Therefore Expected Value of Project = 10,000.

Here you will make only $10,000 but hey it is still a profit. So if we assume you don’t have any guidelines insisting on a minimum profit (called hurdle rates in finance lingo), then we are still going to proceed. We are still going to make money. Let’s look at another scenario. What if the sale price is only going to be $95,000? Then we have this situation.

Expected Value of Project = sum of all expected cash flows

Therefore Expected Value of Project = incoming cash – outgoing cash

Therefore Expected Value of Project = 95,000 – 100,000

Therefore Expected Value of Project = -5,000.

Here there is going to be a loss of $5,000. Therefore the obvious decision would be not to proceed with the transaction or project. You will lose money.

So far, I think, this explanation has been extremely simple and any (most) people could follow it.

Now let’s add in our second concept: risk. Risk is really just the probability that something will happen. So we have this house, it costs $100,000. It may sell for $110,000 or it may sell for $120,000. Let’s assume that these are the only possible scenarios. Only one or the other may happen and nothing else. And let’s say that there is a 60% chance that it will sell at $110,000 and a 40% chance that it will sell at $120,000. Notice that 60% (0.6) and 40% (0.4) add up to 100% or 1. That has to be the case. The sum of the probabilities of all scenarios has to equal to 1 or 100%. There is a 100% chance that everything will happen.

Now let’s go back to evaluate the transaction or project:

Expected Value of Project = sum of all expected cash flows

Therefore Expected Value of Project = 0.6 x (110,000 – 100,000) + 0.4 x(120,000 – 100,000)

Therefore Expected Value of Project = 0.6 x 10,000 + 0.4 x 20,000

Therefore Expected Value of Project = 6,000 + 8,000

Therefore Expected Value of Project = 12,000

Again in this case since the outcome is positive you will want to proceed with this project or transaction.

Let’s add a third scenario where it is possible we could make a loss just like in our earlier example. Let’s say that there is a 50% chance that we could sell the house at $110,000, a 40% chance that we will sell at $120,000 and a 10% chance that it would sell at $95,000. Again note the sum of all probabilities has to equal 100% or 1.

Now let’s do our calculations:

Expected Value of Project = sum of all expected cash flows

Therefore Expected Value of Project =

0.5 x (110,000 – 100,000) + 0.4 x(120,000 – 100,000) + 0.1 x(95,000 – 100,000)

Therefore Expected Value of Project = 0.5 x 10,000 + 0.4 x 20,000 + 0.1 x (-5,000)

Therefore Expected Value of Project = 5,000 + 8,000 – 500

Therefore Expected Value of Project = 12,500

Again the outcome is positive and this looks like a go.

So we have 2 concepts nailed down:

- To calculate expected value you have to add up all the future cash flows
- To add an element of risk you have to identify different scenarios of cash flows and put a probability against each and then add them all, making sure all the probabilities add up to one.

Both the above concepts then roll up into only one:

Expected Value of Project = Sum of (Each cash flow scenario x probability of that scenario occurring)

Again, worth repeating, all the probabilities must add up to 1 or 100%.

Now that we have this understood, how do we calculate our risk tolerance? This is what we set out to do in this article.

Let’s see. Let’s say that we don’t have any appetite for any loss. This means that the minimum we are ready to put up with is a breakeven situation. This means that all expected money coming in must equal all expected money going out.

Expected Incoming Cash = Expected Outgoing Cash.

In this case we would break even and we don’t make or lose any money.

Let’s examine this formula. Remember your algebra. If Expected Incoming Cash = Expected Outgoing Cash, then it means that:

Expected Incoming Cash – Expected Outgoing Cash = 0.

I think this is clear. If Expected Incoming Cash = $100 and Expected Outgoing Cash = $100 then obviously $100 – $100 = 0.

Another way of saying this is that when the Expected Value of the Project = 0 we are in a breakeven situation.

Now let’s go back to our equations with two possible scenarios, one where there is a chance that we would make a profit of $20,000 and another scenario where there is a chance of losing $5,000. The max risk we would tolerate would be the probability of the loss, the risk. Remember all the probabilities must add up to 100% or 1. Let’s call this risk ρ. This is the Greek letter rho and is the one usually used to stand for risk.

Expected Value of Project = sum of all expected cash flows

Therefore Expected Value of Project = (1-ρ) (20,000) + ρ(-5,000)

I hope the above equation is not confusing after what we have already gone through. The probability of loss is ρ is the one we are looking for. If the probability of loss is ρ then the probability of the other scenario, profit has to be 1 – ρ. This is because all the probabilities must add up to 1.

We also said that the max situation we would tolerate is a breakeven situation which is when the value of all our expected outcomes = 0.

0 = (1-ρ) (20,000) + ρ(-5,000)

Now let’s do some juggling around, going back to elementary school algebra.

0 = (1-ρ) (20,000) + ρ(-5,000)

Therefore: 0 = 20,000 – 20,000ρ – 5,000ρ

Therefore: 0 = 20,000 – 25,000ρ

Therefore: 25,000ρ= 20,000

Therefore: ρ = 20,000/25,000

Therefore: ρ = 20/25

Therefore: ρ = 0.8

Now let’s check if that is accurate by inserting it into our earlier equation.

0 = (1-ρ) (20,000) + ρ(-5,000)

Therefore: 0 = (1-0.8) (20,000) + 0.8(-5,000)

Therefore: 0 = (0.2) (20,000) + 0.8(-5,000)

Therefore: 0 = (400) – (400)

Therefore: 0 = 0 which means that our calculation that ρ = 0.8 is dead accurate.

Now don’t forget we were looking for the ρ when that is equivalent to our maximum tolerable situation. This is the maximum risk we would tolerate – not the actual risk.

Therefore if the maximum risk we would tolerate is breakeven and that is equivalent to a risk of 80% or 0.8 then that means any risk below that is a good risk to take.

I hope I have clarified this very important concept. Let me know what you think. Send me a note. If you like my posts, please follow me.

Thanks for reading.

Muneer