Money now has more value than money later. This time value of money is one of the first things anyone learns when studying finance and is the underpinnings of our entire global economy.
The idea has been since around since at least the start of the thirteenth century when the Italian mathematician known as Fibonacci wrote about it in his famous treatise Liber Abaci. The increased commerce between European, Asian and Middle Eastern traders helped Fibonacci develop his theories based on previous works by Hindu and Arabic mathematicians. By applying relatively new mathematical tools like fractions, he was able to answer tough financial questions like calculating return on investment.
The theory developed over time to better represent modern economics, eventually naming the calculation called Net Present Value (NPV). This gives a “present value” to future in and outflows of money and compares it to a “discount rate” or return on a risk-free investment. To do that this calculation is used:
NPV = number of periods in the investment x {(1 – (1 + discount rate)-Time) / discount rate} − Initial Investment *
While this helps better understand an investment’s current value, it does not give the growth percentage that we are so accustomed to seeing. To do this, we can use the same formula for NPV, but this time we have to set the present value to 0 and solve for the discount rate, or as some call it the required rate of return. Required, of course, because if the project can’t return more than a relatively risk-free interest, the juice becomes no longer worth the squeeze.
This calculation creates a well-known metric called the Internal Rate of Return. It is taught in every business school and serves as the main way that investments are compared and chosen. Boiling every investment down into a growth percentage is a way to use math to uncomplicate the world. We rely on mathematical principles for their ability to be precise and give us concrete answers to our problems.
In a perfect world a simple spreadsheet would be able to produce this kind of graph for a set of cash flows. The reality is not so simple. The problem is, math itself is not always that straight forward.
Because there are multiple variables in the NPV equation, it is not possible to determine IRR directly. Instead, the good ol’ guess and check method must be used. You see, determining an investment’s return is a bit like shooting at a moving target. Any changes made to either the cash flows or the time periods of the investment will impact the return percentage.
The guess and check method will get you close to the right answer for one cash flow but when multiple investments are put together, things start to break down. Now our variable no longer represent single numbers, but a matrix of them.
“Imagine ten buildings with cash flows for ten periods, that is a ten by ten matrix,” John Cona, founder of F9Analytics, told me in a series of phone conversations. During his career as a real estate analyst and later investment officer he couldn’t help but think, (like all of us that have had to calculate IRR), that there must be a better way to “attack” these calculations. Rather than just accepting the impossibilities of calculating multiple IRRs simultaneously, he set out to see if he could find a way to do it.
Disclaimer: This section has a lot of hard math. If you don’t really care about the process behind solving for IRR and instead would rather just learn about how it can be used, feel free to skip it. And if you need help understanding some of these terms, check out the glossary.
Not previously solved in Mathematics – with the exception of the “Approximated” Moore-Penrose Pseudo-Inverse.
The Ci is the “Cona Inverse” of a Rectangular Matrix.
The evidence of such can be easily verified – does it produce the “Identity”?
| A | 1 | 2 |
| 4 | 2 | |
| -1 | 1 | |
| 1 | 1 |
| Ci | 0.2000 | 0.2000 | -0.4000 | -0.4000 |
| -0.2000 | 0.2000 | 0.8000 | 0.2000 |
Cona’s mathematical journey was paved by literally thousands of years of human innovation. Matrices were first recorded in the second century BCE China. The modern understanding of how to use matrices in linear algebra comes from the “Prince of Mathematics” and rigorist Carl Friedrich Gauss. In order for matrices to be used in place of variables in an algebraic equation, he had to figure out how to do basic calculations with them like multiplication and division. This process gets more complicated when the matrices get bigger or become non-square. To do this Gauss borrowed and updated the method from the Chinese and it was later named the called Gaussian elimination. It reduces a table of numbers into a row echelon form which generally looks like this:
He then ruther refined the process into a row echelon form, not to be confused with Gauss-Jordan(or reduced row-echelon) , that creates an identity matrix that looks like this:
Putting the matrix in this form allows matrixes to be substituted for variables in an equation. It can then be marked in an equation by a symbol with an arrow over it like this:
While this allows basic algebra to be applied to linear system to optimize for nominal revenue and expenses, it still doesn’t directly help with more complex systems that are nonlinear like IRR that uses exponents, but it does underpin almost every current linear and non-linear algorithm in existence that need to be solved for N, or time.
That is where Cona started his work. After humble beginnings and years of study and exploratory calculations as an analyst, investment officer, and now computer scientist, he finally developed a mathematical method called the “Cona Inverse Rotation” that would allow him greater visibility to the underlying problems he faced for years in finance (see F9analytics.org for more information).
This discovery and its “Scaffolding” as Cona likes to borrow from Gauss has opened up many other new and original ways to attack problems that improve upon the rigid archaic mathematical methods that are in some case between 200 and 2000 years old. As Cona likes to say, “Why would one continue to use an abacus when we clearly have a computer.” To put it in perspective how important the Inverse is to Optimization in general, it underpins almost every algorithm in the field today – that’s why an improvement was required. The Gaussian methods which were borrowed from 2000 years ago are inherently rigid and fail miserably when N is very large – someone needed to put in the rigor.
The mathematical discovery that Cona made has set the stage for a new level of understanding about internal rates of return. He has begun applying his tool in the industry that exposed the problem to him in the first place.
Commercial real estate value is tied directly to the cash flow of its leases. And what is a lease but a predetermined set of inputs and outputs? The problem gets even trickier because lease structures can be complicated by tiered broker commissions or tenant improvement reimbursements. Plus, even just one large building could have dozens of leases, starting and ending at different times. This equates to giant, asymmetric matrices that need to be used to solve for IRR.
Historically, determining a building’s IRR has been done by calculating each individual lease IRR and averaging. This method can be inaccurate and expensive. Finding an average return on multiple IRRs with different start and stop periods can be misleading. On a portfolio that is taking in hundreds of millions of dollars in revenue, the difference can add up to millions of dollars.
Let’s say you have a 5 year lease that starts at $24 per square foot for a 5,000 square foot building. There is a lease commission of 5% and a tenant improvement allowance of $60 per square foot. This is actually a rather straightforward lease but it creates a pretty complex IRR calculation especially when you want to factor in an estimated 2% inflation rate. F9 has a calculator tool that can easily calculate the IRR (it is 8% as you can see from the graphic).
Notice the options to accommodate an even more complex lease arrangement with multiple escalations and rent-free periods.
There is another interesting aspect of this development (that is if you think things like financial calculations are interesting like I do), is that it can bring a whole new level of sophistication to lease planning. As I said earlier, using Cona’s Algorithm allows for the IRR calculation to be solved for any variable. That means that if a landlord needs to know how much that they can spend on a property upfront to keep a desired rate of return, the Cona Inverse can easily accomplish that.
So using the previous example, let’s say we are targeting an IRR of an 8% and we want to know how much we can budget for tenant improvements. Again, this answer is only one click away.
This example also uses the property’s purchase price and inflation to make the calculation even more robust (and complicated).
F9 Analytics is the company that John started to give others the power of his mathematical trick. He sees the future of real estate analytics as a much more sophisticated world. “Wouldn’t it be great if you could see the change in your building’s IRR in real time? Or if you could determine not only the current return rate, but also whether it was rising or falling? Soon, this will be the standard way that all rates of return are understood,” he told me. His platform gives access to a calculator that allows users to run the calculation any way that they see fit. He admitted that there are likely uses for the process that he has not even identified yet.
What amazes me is that, for how important of an equation IRR is, very few people actually calculate it. Instead, we leave it to online calculators and excel formulas without verifying the process. The reason might be that we view math as binary, it is either right or wrong. It takes someone willing to dive deep enough into the details to understand that there is actually some variance to certain calculations.
Almost every business decision in the modern world comes down to return on investment. It is not an exaggeration to say that the time value of money calculation has built the modern world, or at least helped determine what gets built and what doesn’t. For an equation as important as the rate of return it is paramount that those of us using it understand what it can and can’t do. Fortunately, there are people like John Cona and tools like his metrics inverse that can help those of us without the time or facilities to understand the particulars of complex mathematical calculations.
Now that my eyes have been opened to the truth about the IRR calculation it has me wondering what other seemingly infallible calculations are actually much more nuanced than we are led to believe.
Special thanks to geophysicist extraordinaire and lifetime problem solver Ernesto Vicente for helping create this glossary and explain these concepts to a math novice like myself.
*Net Present Value (NPV) = Σ [C t /(1 + r) t] – C o
Ct = net cash inflow during the period t
Co= total initial investment costs
r = discount rate, and
t = number of time periods
Internal rate of return (IRR) is a metric used in capital budgeting measuring the profitability of potential investments. Internal rate of return is a discount rate that makes the net present value (NPV) of all cash flows from a particular project equal to zero. IRR calculations rely on the same formula as NPV does. [https://www.investopedia.com/terms/i/irr.asp]
Generalized Matrix Inverse is a method of computing what is called a pseudoinverse of a singular matrix by C.R. Rao and S.K. Mitra (1966 and 1971), after initial work by Moore (1920) and Penrose (1955), and applied it to solve normal equations with a singular matrix in the least squares theory and to express the variances of estimators. The pseudoinverse defined by Rao (1962) did not satisfy all the restrictions imposed by Moore and Penrose. It was therefore different from the Moore-Penrose inverse, but was useful in providing a general theory of least squares estimation without any restriction on the rank of the observational equations. In later papers, Rao (1965 to 1970) showed that an inverse with a much weaker definition than that of Moore and Penrose is sufficient in dealing with problems of linear equations. Such an inverse was called a generalized inverse (g inverse).
