Consulting Math Techniques Flashcards
Multi-stage growth problems
Growth problems: Multi-stage problems *Always* ask whether an exact answer is required first. If estimating. use the compound growth formula: PV * (1 + growth rate) ^ (# of periods). For example, 50m with 2 years of growth at 10%: $50,000,000 * (1 + 0.10) ^ (2 years). Then, the key insight is to prove to yourself that (1.10)^2 is very, very close to adding the growth rates together and then adding 1: Multiplying for exact: 1.10 * 1.10 = 1.21 Adding for estimation: 1 + 0.10 + 0.10 = 1.20 Obviously, the latter route is much easier and for the purposes of estimating gives you a solid footing to work from. Now return to the problem with the estimated approach, the formula is now $50,000,000 * 1.2. This could be simplified to $50 * 1.2 * M, or $60M, very close to accurate.
Gut check for growth rate
Rule of 72. If you divide 72 by the growth rate, you’ll get the number of periods in which your present value will double. For example, if you have a 6% growth rate, take 72 / 6 which is 12 and you now know that if something is growing at 6% it will double in 12 years. This is useful for two reasons: You can use it to gut check an answer quickly. Given the prior problem we did with the $250M growing at 3%, we could use rule of 72 to determine that the market should double in 24 years. Given we were only asked to calculate 7 years, we would know that if our estimate was anywhere near $500M, something went wrong. Taking it a step further we know 7 / 24 is slightly less than a ⅓ and $250 / 3 is just above $80M, we know that if our answer is above $250M + $80M, which is $330M, something has gone wrong! The other obvious use is finding out how long it will take something to double by simply dividing the growth rate into 72. And as we saw above, once you have that period, you can make rough assumptions that in half the time it would grow roughly 50%, or in a quarter of the time it would grow 25% and so on. Again, these are rough estimates but they can be very useful for ball parking.
Rate Change formula
Rate change = 1 + rateQuantity + ratePrice + (rateQuantity * ratePrice) Let’s look at why this is true: Say you’ve got a scenario where a firm selling widgets grows sales (the number of units sold) by 50% but in the process drops price 45% in year one. If they had a target to grow overall revenue by 5%, would they achieve it? Revenue0 = Quantity0 * Price0 Revenue1 = (Quantity0 * (1 + 0.5)) * (Price0 * (1 + -0.45)) Revenue1 = Revenue0 * (1 + 0.5) * (1 + -0.45) Revenue1 / Revenue0 = (1 + 0.5) * (1 + -0.45) Rate change = (1 + 0.5) * (1 + -0.45) Rate change = 1 + 0.5 + -0.45 + (0.5 * -0.45) Rate change = 1 + 0.05 + (-0.225) Rate change = 0.825
Market Math - finding shortcut multipliers
Market math: Finding shortcut multipliers Market math is tricky but the best way to approach it is 1) quickly check for shortcuts and 2) if no shortcuts are present, smartly round the numbers and proceed with the calculation. Shortcuts: start by looking for easy shortcuts where the % of the known market is the inverse of a multiplier that is fairly easy to work with. Say something like 5%, which is the same as 1/20 or multiplying times 20. That turns the problem into a fairly simple multiplication problem. What are the easy multipliers to look. These are a good start: If you know 5% of the market is X, you can multiply X against 20 If you know 10% of the market is X, you can multiply X against 10 If you know 25% of the market is X, you can multiply X against 4 If you know 50% of the market is X, you can multiply X against 2 If you know 33% of the market is X, you can multiply X against 3 If you know 66% of the market is X, you can multiply X against 1.5
Market math - messy multipliers
Market math: Estimating with messy percentages If no shortcuts are present, you’ll have to deal with messy percentages which you’ll want to round up or down to an easier number you can work with. Since interviewers won’t expect an exact answer, you can reach an acceptable answer as long as you don’t round too aggressively. For ex: If 39% of the market is $700M, what is the overall market size? Start by finding an appropriate rounding you can use. In this case, 40% is a good option. Next, you should pick a “gut check” percentage that you can do very quickly to anchor your own expectations. For example, in this case, you could choose 50% and quickly calculate that if $700M were 50% of the market, the whole market would be $1.4B. Since the your real percentage is 40%, you know that your answer should be greater than $1.4B. In an interview, this logic is something you could voice over, so your interviewer can follow along with you. Now start working on your actual estimation. If calculating the market size with the estimation is still tricky, you can use a handy tool: break up the percentage you’re working with into 10% or 5% increments. Since you know 40% of the market is $700M, this means that you also know 10% of the market is $700M divided by 4 (since you’re just dividing both by 4). That means that 10% of the market is $175M. Now, you can easily multiply that times 10 to get to 100% of the market, which is $1.75B Technique recap: Check for a shortcut percentage Round the percentage Calculate a 10% share Multiply by ten
