Using an example, I'd like to show how risk theory can help us manage our uncertain future. Risk theory is a branch of mathematics that provides a framework for making decisions with uncertain outcomes.
When it comes to retirement, we are facing the unknown: we don't know how long we'll live and what will be the rates of investment returns earned on funds used for retirement income. Also, expenses are unknown because of future inflation and the possibility of health-related and other expenses such as long-term care down the road.
Knowing the range of possible outcomes is our starting point to manage uncertainty.
Framework
The starting point of risk theory is the set of possible uncertain environments. Decisions we make influence the results we get in the environment that prevails.
Decisions + Environment = ResultsFor each decision there is a probability distribution. Making the best decision means choosing the "best" distribution of results among those available.
Example
Suppose I am retiring with a nest egg of $1 Million.
Decision
I am considering two withdrawal options:
In both cases, my annual withdrawal will increase each year by the rate of inflation, so my purchasing power is preserved.
- Spending $50,000 per year (5% of my initial capital), or
- Spending $70,000 per year (7% of my initial capital).
Environment
My environment is an uncertain future with unknown longevity, rates of investment returns and inflation. I will run out of money if the returns are too low, the inflation too high, or if I live too long.
This may happen whether I decide to spend 5% or 7%, but it is more likely if I decide to spend the higher amount of 7% annually.
Results
Let's define a simple range of possible results that can occur for my retirement:
Awful
|
I ran out of money and spent several years penniless
|
Bad
|
I ran out of money at the end of my life while curbing my spending throughout retirement
|
Good
|
I didn't run out of money but had to curb my spending
|
Excellent
|
I didn't run out of money and enjoyed a good standard of living
|
We can calculate odds for each decision that we make. Let's go with the probabilities below for our example. They're in line with these types of calculations and will be sufficient for our purposes.
Probabilities for 5% spending decision
Awful
|
Bad
|
Good
|
Excellent
|
0%
|
30%
|
70%
|
0%
|
Awful
|
Bad
|
Good
|
Excellent
|
70%
|
0%
|
0%
|
30%
|
Utility
My optimal decision means that I must choose the probability distribution that has the best overall outcome for me.
A common approach consists in assigning an "utility" to each result, and calculating the expected utility of each decision. I then choose the decision that has greatest expected utility.
I will assign a rating from 0 to 10 to the utility of each possible result. The rating is arbitrary but one with which we are familiar.
Value
|
Utility
|
Awful
|
0
|
Bad
|
3
|
Good
|
6
|
Excellent
|
10
|
And the winner is...
Now we have all the ingredients to calculate the "expected utility" of each decision.
Decision
|
Calculation
|
Utility
|
Spend 5%
|
0% * 0 + 30% * 3 + 70% * 6 + 0% * 10
|
5.1
|
Spend 7%
|
70% * 0 + 0% * 3 + 0% * 6 + 30% * 10
|
3.0
|
Not surprisingly, the 5% middle of the road approach is more sensible. The odds of "excellent" are too low to add a significant utility to this decision.
This is sensitive to the rating and the odds we assign to each result. You can see that I'll need a 50% chance of success with the 7% spending decision to match the utility of the lower spending decision. But even at 50% it is still a big gamble.
What if
What if I have lifetime income, say public pensions and an annuity that are sufficient to cover my essential expenses? Now my nest egg is only for discretionary expenses such as travel, leisure and bequest.
My results will be different: if I run out, it's not so "awful", it's just bad! My life will just have less leisure but I won't be destitute.
7% spending decision
Awful
|
Bad
|
Good
|
Excellent
|
0%
|
70%
|
0%
|
30%
|
And here's my new utility with the 7% spending decision:
0% * 0 +70% * 3 + 0% * 6 + 30% * 10 = 5.1
Now both spending decisions are just as good.
Theory and practice
This framework is a way to apply some logic to help us take decisions. It can help with any type of situation.
With retirement, we cannot know how the future will unfold, and in spite of the uncertainty we need to make decisions that involve risk. Results of our decisions are influenced not only by our decisions, but also by the range of outcomes.
With retirement, we cannot know how the future will unfold, and in spite of the uncertainty we need to make decisions that involve risk. Results of our decisions are influenced not only by our decisions, but also by the range of outcomes.
By keeping our options open and monitoring our progress, we can take corrective action mid-course to mitigate the damage if our decisions turned out to be not "bad" but "awful".
