When I was in business school, all first years were required to take a Decision Sciences course as part of the core curriculum. It’s been awhile since I took that class, but I happened to recall it while writing about Dollar Cost Averaging vs Lump Sum Investing.
Decision Sciences refers generally to the mathematical models used to make decisions when there is uncertainty. One of the decision science tools we learned was Decision Tree Analysis for Decision-Making (as opposed to machine learning, which it can also be used for).
I think it’s a useful framework to have in your back pocket, and one that we don’t always employ when trying to make objective decisions that can have a significant numerical outcome.
Making deliberate and informed decisions is crucial in both personal and professional contexts. Whether you are an investor trying to maximize returns, a physician determining treatment plans, or a business manager planning for future demand, having a structured methodology is key. This is where tools like decision trees, probability, and expected outcomes play a significant role.
Since an MBA now costs $160,000, this half-credit class is worth roughly $4,000, of which the takeaways I am giving to you for free!
Decision trees are a visual and analytical tool used to map out and explore the full range of options in a decision-making process. The decision tree itself is a tree-like model of decisions, with branches representing the different choices and their potential consequences:
To create a decision tree, start by outlining the main decision or question. From there, draw branches that represent each choice available. Subsequent choices and chance events that follow from these initial options will have their own branches, forming a tree structure.
Here’s an example of a decision tree to evaluate buying a home vs investing in equity markets:
At the far left, we have a choice of buying a home or investing, represented by branches spawning from that first box. On each path, we come across a node (diamond) which represents an event — the performance of the real estate or equities markets. More branches spawn from that node, with each branch representing the outcome, which for each case is basically good (rising value), neutral (flat), or bad (declining value).
Now, let’s add some numbers to these paths and outcomes.
Probabilities quantify the likelihood of a particular outcome and are pivotal in evaluating decision trees that involve uncertainty. Probabilities are typically expressed as numbers between 0 and 1 (or 100%), where 0 indicates impossibility and 1 represents certainty. They help to weigh different outcomes within the decision tree structure.
For each chance node on the tree, we need to assign probabilities to all the branches that stem from it. These should be based on statistical data, historical observation, or your own well-informed estimates. Probabilities must total to 1 for each set of branches departing from a common node.
In this example, based on current market conditions, we’ll estimate the probabilities of each outcome as such:
By adding the probability of each branch occurring, we are accounting for uncertainty in the decision tree model.
The only thing we’re missing now in our model is the financials. What does it mean when we say “strong” or “weak” market? We need to attach a value to each branch in order to quantify that outcome and compare with the other branches.
In our example, let’s assume that in our initial decision, we would either buy a home in an all-cash transaction for $500,000 or invest that same amount as a lump sum in the equity markets.
From past historical housing market values[1], we’ll estimate home value changes over 10 years to be:
From past historical market performance, we estimate that in 10 years, the S&P returns equate to
Expected value quantifies what's likely to occur on average over time if a decision scenario were to be repeated. It's calculated by summing up the or payoff of each outcome and its associated probability. The formula is:
Expected Value (EV) = Σ (x*p) = x1*p1 + x2*p2 + x3*p3 ...
where p = probability and e = value
In decision making, the expected value helps compare the average long-term benefits or costs of different paths and aids in selecting the most advantageous one. Even if each individual outcome is uncertain, the expected value of a decision gives a weighted average of the possible outcomes if the situation occurs multiple times.
In our example, the expected value of each decision is:
Based on these expected values, the decision to invest will produce higher value in 10 years ($829,000) than the decision to purchase a home ($473,500),
This example was a fairly simple decision tree with just one decision and one stage of outcomes. Decision trees can get more complex as you add subsequent decisions and stages.
For example, what if we expect the rental market to grow in demand? Perhaps we want to evaluate the choice to rent out the home for five years:
The decision tree then looks like this:
The expected value of renting our home is is $276,000 as shown here:
Now, we have to add this value to our first real estate decision, where a strong real estate market would now have a value of 140% x $500,000 + $276,000 = $976,000.
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We can see that even accounting for the choice to rent out the home for 5 years, the decision to invest is still the higher value choice.
Decision trees can get much more elaborate than this, where we kept the problem simple by ignoring the time value of money, property tax expenses, and more. The more variables and decision nodes you add, the larger and more complex the decision tree analysis becomes.
Decision trees can applied to a broad array of industries and use cases. In addition to investment decisions for wealth building, you can utilize decision trees for evaluating whether to start a business, launch another product line, or opening up a second location. Decision trees can also be used for justifying budget for validating product-market fit, where the information gained from that decision can inform or influence the probability and values for subsequent decisions and branches. But there are limitations to using decision trees, too.
Decision trees are about maximizing the benefits while navigating the inherent uncertainties in any decision-making process. With these tools, decision-makers can better assess risks, rewards, and the average results of their choices, leading to optimal decisions in uncertain worlds.
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Citations
1 https://dqydj.com/historical-home-prices/
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