IRR

This is my first attempt at something that is directly subject related. Though the idea is basic maths (middle school maths to be specific), I have been having some torrid times trying to convince certain section of students about its simplicity and middle school origins. What is even more baffling is that these basic stuffs are offered as excuses for aversion to certain subjects. And my "Wtf?" moment comes when these ideas are identified as reasons why Financial Management is tough.

Internal Rate of Return

Mathematically, it is the discount rate at which Net Present Value is equal to Zero. The aggregate of present value of cash inflows equals present value of investment cash outflows. Most of the investment cash flows take place at T-0 (Time Zero).

For beginners, Present Value (PV) is nothing but discounting of future cash flows to T-0 cash flow (Discounting = Inverse of Compounding).

Conceptually, it is the rate at which the project cash flows recovers the initial investment. It can also be considered as the “Implied Reinvestment Rate” i.e. the rate at which the future cash flows are deemed to be reinvested.

The IRR is essentially used as a benchmark for Financing Decisions, including identifying sources and the suitable capital structure.

And for those who understand, we can proceed further. And for those who don’t, don’t worry, you won’t anyway, since the idea of this post isn’t explaining IRR, but how to compute IRR in a simple manner.

With an excel file in hand, or even using a scientific calculator, the task of computing “Internal Rate of Return” isn’t worth even referring to as a task! A simple “Goal Seek” act would throw the answer in a second. This write up is for the one who takes up examinations, where scientific calculators or computers are not permitted. The object is not to be find the exact IRR, but to determine a close / approximate.

There are three approaches, that I know of. Two I was taught, One I learnt. All are based on intrapolation, and nothing more. And therefore, this write up is relevant for any exercise involving intrapolation.

The first two methods are the same. One goes by the graphic method, and the second by formula. In all the three cases, we need two discount rates, and consequently we’ll have two Net Present Values. The discount rates are so chosen so that one NPV is below Zero, and the other above Zero. (For the bottom breakers who ask why – Plot a graph of NPVs on the graph, with Discount Rate on Y Axis, and NPV on X axis, you’ll observe that the line isn’t straight, but curved. As a result, the slope between two points on the line keeps on changing.)

How to choose the two discount rates? There are no rules here except that one discount rate should result in positive NPV, and the other negative NPV. My recommendation would be first use either 10% or 12%, and increase / decrease either 4% or 5% for the other rates. 99% of the cases in the examination will have an IRR between 10% and 15%. Or use a simpler approach. Use 10%. Increase it by 10% or decrease by 5%.

For all the three models, I use the following streams of cash flows.

Initial Investment at T₀ = 1,00,000; Inflows at T₁, T₂ and T₃ are 50,000, 40,000 and 30,000 respectively. Two discount rates chosen are 10% and 12% at which NPVs are Rs.1050 (Positive) and Rs.2113 (Negative) respectively. (Btw, what do they, "NPV", communicate, apart from “accept” or “reject”?)

The actual IRR works out to 10.65%.

Method One – Formula Route

(Widely followed by many people. Google around you’ll get this formula.)

Primarily meant for students who are at difficulty understanding the principles of intrapolation. 

R₁ = Lower Discount Rate and R₂ = Higher Discount Rate

V₁ = Net Present Value at R₁ and V₂ = Net Present Value at R₂

For given set of numbers, by applying the above formula —

Remark: Often I have found students struggling to follow the denominator computation. “Minus of Minus is Plus.” Funny, but true. Many, if not most, don’t remember this. Some don't know this!

Method Two – BSP Graph

I call it that way since BSP sir was the first person I had seen using the method. I haven’t seen any body else using the same or any book which had this idea before he designed it for students who had difficulty remembering the formula. This is essentially the graphical representation of the above formula.

Left Tail Route:

X         = 10 + 1050/3163 × 2

            = 10 + 0.664 = 10.664

Right Tail Route:

X         = 12 – 2113/3163 × 2

            = 12 – 1.336 = 10.664

Method Three – Greatly inspired from the Binomial Model and Risk Neutral Model for determining the probability for valuing “Option Contracts”. It is essentially the modified BSP Graph method. In developing this approach, I have tried to keep the computation as low as possible, and reducing the possibility of making errors, and targeting the IRR straightway. This follows the weighted average route, where the variables are the two discount rates chosen, and the weights 'difference' on the opposite side.

My personal favourite is the third one for obvious reasons.

So that’s about it. The three approaches for determining the IRR. Share your thoughts on refining the above models or creating newer models. Please don't share your grievances on how IRR is making your life difficult. For that, go pray.