Necessary and Sufficient

Young companies often have limited cash and thus offer prospective new employees with an equity stake (often referred to as "equity participation"). Despite the recent changes to the taxation of such schemes by the Australian government employee equity stakes still seem common in technology start-ups, at least for the more senior roles. So candidates considering such roles need to evaluate the proposed, or expected equity stake in order to make a call on the value of the included equity. Below are a series of steps that I've found useful in tackling this problem.

1. When you've reached the negotiation step of the courtship get the business model/plan from the founders. It is normal for founders to NOT want to disclose it until there is an established relationship and serious interest from both parties. However, if you reach this point and they don't offer at least the financials then you ought to be very suspicious.

2. Look at the forecast revenue figures and compare these to other companies in the same industry. What percentage of the overall market do they represent? If it represents a large percentage do you think it's realistic given the competitors in the market? If it's very low you might want to ask why the firm thinks its' product can only capture such a small percentage. Of course a smaller piece of a bigger pie is always better than the lion's share of a small pie, unless that smaller pie is growing quickly. NB: before you get the business plan you can estimate revenue crudely by: no. of customers X revenue per customer.

3. Look at the forecast profit margin (profit/revenue) and ask yourself if this is reasonable for the type of product/service being developed. The beauty (and curse) of digital assets is that they are easily copied leading to high margins once the R&D expense is out of the way. But you also need to consider the overall architecture because op-ex and cap-ex costs also come into play if you're storing lots of user generated content and chewing through bandwidth like it's going out of fashion, or your offering needs massive amounts of compute power which, in essence, translates to high incremental cap-ex costs making margin expansion dependent on Moore's law (cpu), Kryder's law (storage) and Nielsen's law (bandwidth).

4. Look at the forecast profit figures and compare them to the historical profit figures of other players in the same industry. Of course future cash flows are not equivalent to past cash flows in dollar terms but you can still get an idea without needing to dive into discounted cash flow modelling if that's not your thing.

5. If there are any other publicly-traded companies in the same space look at the P/E multiple that they are trading on. If you can't find one pick something between 13-18.

6. Using the profit figures and forward P/E calculate a tentative valuation for the company and give the resulting figure a sanity check adjusting parameters as necessary.

7. Consider your equity stake over time. Is it likely to grow as a result on acquired rights or allocations over time? If the company is planning additional rounds of capital raising then dilution will occur and your percentage will shrink, though the company valuation would most likely have increased.

8. With the equity percentage estimates above you can quantify your ownership in dollar terms by multiplying the percentage with the company valuation derived above.

Apart from the valuation of the equity stake, there are of course other important things you can elicit from the business plan which will help you come to a conclusion about whether the firm is worth betting on. Basically you need to think like an investor/VC and evaluate whether you would want to put your money into the venture. The list below has some things to consider in this phase.

• Look at the break-even time line which will be shown in a revenue vs cost graph. If it is going to take X years to turn a profit the likelihood of an exit before this time is not great so you need to plan to be around until at least this time plus a few years.

• Look at the strengths-weaknesses-opportunities-threats (SWOT) analysis and ask yourself if it is accurate. Are there big things missing from the W/T listings? Are the S/O components overstated?

• What are the barriers to entry? Is it easy for someone else to come along and do exactly the same thing? If so what makes you think that this offering is going to be the winner in that showdown?

• Look at the free cash flow as this will tell you how much additional capital the company is likely to need in future rounds - which will effect your equity stake due to dilution.

Lastly, there are the gut feel things that aren't really quantifiable but none-the-less important:

• It's common sense but worth repeating... don't do something you don't enjoy just for the money! If you are passionate about a particular area you'll find the work enjoyable and highly satisfying, but be careful - passion can blind you from reality and cloud your judgement on the business case.

• If you don't have faith in the management team then move on to other opportunities. The management team of a speculative venture is responsible for both strategy and execution, as well as raising additional rounds of capital. If management fail on any of these 3 items the business is doomed.

So all-in-all it's just like playing the stock market - you'll lose some and win some. You just need to pick more winners than losers because each investment could be 5 years of your working life. Also note that positive things can come out of failed ventures - the learning experience for one is invaluable and you'll get respect for trying from individuals who aren't completely risk-averse. I think Theodore Roosevelt said it best though:

"Far batter to dare mighty things, to win glorious triumphs even though checkered by failure, than to rank with those poor spirits who neither enjoy nor suffer much because they live in the gray twilight that knows neither victory nor defeat" .. Theodore Roosevelt (quoted on the Harvard Business School Admissions website)

As a technical person you are often in a position of having to explain intricate technical issues to folks who aren't deeply technical or knowledgeable in a particular field. Perhaps the listeners are your superiors, your investors, your work colleagues, or your customers. Whoever they are you need to be able to explain seemingly complex technical issues in a language they understand. This isn't always easy as you may not know the level of interest or understanding the listener has of the particular topic of conversation. My advice is to initially offer the "simplified, but not completely dumbed-down executive summary" response, and only offer up further technical details if requested. That way you are not insulting them by a gratuitous generalisation but you are giving them the choice of going deeper if they request it, rather than forcing an in-depth discussion on them and suffering the indignity of being cut-off before you've made your point. The Board of Directors doesn't want to hear a techo-babble sermon.

Clearly, this is just sensible communication skills, and is something that can easily be examined during interviews. For instance, a potential Product Manager might be asked:

Can you explain a how a database works to a 7-year old?

Or a maths geek with post-graduate education applying for a programming gig might be thrown a curve ball out of left field with something like this (yes I've actually seen that before, believe it or not)...

How would you explain to high school students the Taniyama-Shimura Conjecture as it applies to Fermat's Last Theorem?

So WTF is that you're asking? Well it's a particularly complex mathematical theorem that even Harvard professors have trouble explaining, and therein lies the problem. You might be clever enough to have an understanding of the subject matter but conveying it to others is particularly challenging.

But first a slight digression to get everyone on the same page...

Background Info

Perhaps one of the most famous theorems in mathematics is Fermat's Last Theorem. This was a conjecture proposed by Fermat in the 17-th century. It states that no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ (the Diophantine equations) for any integer value of n greater than two. Famously, Fermat made a margin note in one of his manuscripts saying he had a proof for it but didn't include it due to space restrictions. Without Fermat committing his proof to paper the problem stood as an unproven conjecture for some 300 years. Because of the importance of the conjecture, the simplicity of the equation, the number of famous mathematicians who had failed to find a proof for it, and the theatrical story around the "missing proof", Fermat's Last Theorem, which was actually still a conjecture at the time, became very famous in the mathematics world.

Fermat's Last conjecture was a conjecture for over 300 years, meaning it had never been proven, but the conjecture became a theorem when Mathematics Professor, Andrew Wiles, proved it in the mid 90s. But a key part of the breakthrough proof for the conjecture was a lesser known but equally important conjecture known as the Taniyama-Shimura conjecture for elliptic curves. This conjecture, which had been around since the 50s, stated that every elliptic curve was a modular form. It was later proved and became known as the Modularity Theorem.

So how exactly is it related to Fermat's theorem? Well it provided the key link that bridged 2 disparate fields of mathematics: modular forms and elliptic curves, and in doing so brought a proof for Fermat's theorem much closer than it had been before. The Taniyama-Shimura conjecture had as one of its consequences, Fermat's Last Theorem, meaning that formally proving Taniyama-Shimura would result in a proof for Fermat's theorem. Basically the plan of attack was as follows:

1. Assume that Fermat was wrong and there is in fact a solution to the equation aⁿ + bⁿ = cⁿ , n>2.
2. With the assumed solution we can generate an elliptic curve with some rather weird properties - the curve seems not to be modular, but the Taniyama-Shimura conjecture stated that all elliptic curves are modular - a clear contradiction.
3. This gave incentive for researchers to find a proof for Taniyama-Shimura because the discovery of such a proof would be the critical piece in proving Fermat's Last Theorem.

In essence, this is proof by contradiction (a simple example of this can be found here).

Explanation By Way of an Analogy

So how do you summarise the relationship between these two conjectures so that high school students - who have never studied any number theory - can understand it? Use a simplified analogy! Here's one I came up with that I hope shows how these 2 pieces are related:

Imagine you have to figure out if there exists a special one-of-a-kind metallic needle located somewhere within a massive haystack the size of Texas. We don't know for sure if the needle is in the haystack or not. Some people say that the special needle is not in the haystack at all and we'd like to confirm this before we waste time searching. We know that it's not really feasible to try and search through the haystack because it's just too damn big so we need another way to confirm that the needle does or doesn't exist. Coincidently an eccentric Japanese scientist has recently claimed that he has built the worlds' first super-range needle magnet that, he claims, can suck in a metallic needle even if it is as far away as the moon is from the earth. If we use his super-range needle magnet and we don't find anything it doesn't really tell us that the needle doesn't exist - it could mean that, or it could mean that his magnet is not as good as he claims. Given that it is relatively easy to try the whiz-bang new magnet - and most likely not find anything as most people suspect - we only need to prove that the super-range needle magnet can detect a needle at a distance of "the size of Texas" to be completely confident that the needle doesn't exist. This seems like an easier problem to tackle so you decide to focus on it.

In this analogy, finding the needle in the haystack is analogous to proving Fermat's Last Theorem, and checking the range of the new magnet is analogous to proving Taniyama-Shimura. High school students would at least comprehend the analogy.

Problem: An old lady was going to a street market when a reckless kid on a skateboard bumped into her and made her drop a basket full of painted porcelain figurines breaking all of them into many small pieces. The kid's father saw the whole thing and offered to pay for all the damaged items so he asked her how many figurines she had bought. The old lady couldn't remember the exact number, but she remembered that when she had taken them out two at a time, there was one figurine left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of figurines she could have had?

Solution: First, let's formulate this using mathematical notation. Let x = the number of porcelain figurines the old lady originally had. Clearly, x is a positive integer and we also know that:

where % is the modulus operator. These equations can also be written as congruence relationships:

If you haven't already figured it out, this is a problem that derives from the Chinese Remainder Theorem. This theorem deals with solving simultaneous modular arithmetic equations.

The least common multiple of 2,3,4,5,6 is 60, thus the first 5 equations above can be reduced to a single equation:

So clearly the solution to the simultaneous equations is a number that is a multiple of 7 and gives a remainder of 1 when divided by 60. We can therefore examine numbers of the form: 60m+1 where m=0,1,2,3,... Here is the first 10 numbers in this sequence:

1, 61, 121, 181, 241, 301, 361, 421, 481, 541, ...

Starting with the lowest number and dividing each by 7 we can quickly determine that 301 is the answer since it is the first number divisible by 7.