Make sure to lock the cell with $ notation, for cell ID pointing to
first row’s date value
Use drag and auto-fill for other values of the discounted cashflow, though your spreadsheet might actually prompt the same with smart fill.
Finally, invoke SUM() to compute summation of all discounted cashflow values in last column.
This video should help with above steps:
Final result should resemble this:
Verifying XIRR value from summing up discounted cashflows - Dark Mode
Verifying XIRR value from summing up discounted cashflows - Light Mode
You would be tempted to manually use 3.52% or 0.0352 instead of relying on output of xirr() formula. As you’d see, it would add up to a finite positive or negative value.
Due to formatting, we are only seeing two places after decimal point. But actual computed value is something slightly different from exact 3.52%.
How to Solve the XIRR Equation
Given a value of XIRR, we can verify if it’s indeed satisfying the equation of NPV being zero.
But given an equation, as above, can we derive the value of
Turns out, even if it’s a single-variable equation; it’s not as simple as solving an algebraic equation.
In essence, we need to use advanced approaches such as Newton-Raphson approximation to solve such equations, iteratively.
It starts with a guess for the right value of XIRR. Then it plugs that value back into the NPV equation, to extract a new approximate value.
After repeating this and extracting newer approximate values, it only gets more and more precise, through hit and trial.
Eventually, the value stabilizes reaching correct answer for the problem at hand.
A full treatment of Newton-Raphson method is beyond scope for this wiki, but even a cursory search in popular search engines would reveal enough materials that cover it in some depth.
The third argument in xirr() function is rate_guess, which if you provide, can speed up XIRR computation by reducing number of iteration it takes to reach correct value of XIRR.
But it’s best avoided; because if you plug in a value as guess for rate, which is far removed, it would only serve as a means to increase number of iterations it takes to find the value of XIRR.
Why is XIRR annualized growth rate?
value is set to 1 year or 365 days.
We could, however, change that to 1 month or 1 decade; and get rates per month, or rates per decade.
Now that we’ve a formal mathematical understanding of what XIRR is; it’s instructive to understand XIRR visually.
This would help us build a mental model of XIRR, intuitively, because it might not always be possible to have enough information to invoke xirr() function and obtain exact value of it.
Two Transactions - CAGR and XIRR
Let’s begin with a simple use-case, where a set of cashflows is basically only two transactions.
One purchase, and one sell or redeem-able account value.
It could be a lumpsum purchase transaction for a bond or a stock or even a mutual fund / ETF, and after the first purchase, there have been no transactions on that account.
Let’s consider an investor had purchased HDFC (NSE: HDFC) stock at a price of
, on 1st April 2016, for quantity of 1 share.
Then total invested amount is same as share price at the time of purchase.
Also assume this investor has continued to hold on to this share of HDFC. As on today, it’d have a different value from its valuation at the time of purchase.
We can model it as one purchase transaction (negative cashflow, because money is going away from end user), and a potential redemption transaction of today’s value.
At the time of writing this, this value is
per share, as on 1st April 2021.
It effectively means, if the investor were to sell the share when price is ₹2515.20, they’d receive ₹2515.20 in liquid cash in their account. We’re ignoring STT (Security Transaction Tax), brokerage, and other fees for now.
Then XIRR of these transactions can be computed easily, by inserting these numbers from the table into a spreadsheet, invoking the xirr() formula.
We’d get XIRR as
XIRR of only two transactions - Dark Mode
XIRR of only two transactions - Light Mode
At the same time, we could obtain CAGR of underlying asset, HDFC shares, over 5 years, which also comes out to be
CAGR computed with RRI function - Dark Mode
CAGR computed with RRI function - Light Mode
They are the same!
You might have noticed that output of RRI() function is slightly different from output of XIRR function. This is because those two dates are not exactly 5 years apart, you’ve to also count the leap days in between.
CAGR of an asset between two dates, can be computed using XIRR function, by creating a purchase transaction at the start date, and an imaginary sell transaction on the end date.
This is why most spreadsheet or excel applications don’t include a dedicated CAGR function ****You only have to invert the sign of first row entry of value!
Discounted cashflow for only two transactions - Dark Mode
Discounted cashflow for only two transactions - Dark Mode
We can mathematically prove it as well.
Recall that formula of CAGR is:
is number of years, where we’re considering 1 year as unit of time.
Then equivalent transactions would be of values
Plugging these in the NPV equation for XIRR, we get
Rearranging, and keeping in mind that power raised to
, we get
Therefore, CAGR and XIRR would have same values.
We've simply found a case, where value of CAGR for an asset between two dates; can be same as XIRR of an imaginary portfolio of two transactions, on those two dates.
This would be true for all investible assets, for any two dates; as long as it can be modeled as two transactions (one purchase, one sell).
But CAGR of a portfolio, is a meaningless metric to look for. Asset has CAGR, portfolios have XIRR.
This makes sense intuitively as well.
XIRR of a portfolio where investor has invested only once and redeemed once (or can redeem full amount if they choose to), is same as rate of growth of underlying asset over same period - which is CAGR.
More than Two Transactions
What if an investor has more transactions, mix of buy / sell etc.?
This is usually how normal portfolios look like.
Even if one were to make a bank fixed deposit, there can be year-end TDS, resulting in cash outflow from the deposit account; which would end up having multiple transactions in that portfolio.
Discounting cashflows for XIRR in case of more that two transactions - Dark Mode
Discounting cashflows for XIRR in case of more that two transactions - Light Mode
Dotted line shows actual numeric value, of each transaction’s cashflow.
While, solid lines indicate how these values would be after discounting using the decay formulation.
You can see the decay as a result of discounting.
To understand why the decay over time moves like that, we can plot similar graphs from our spreadsheets
Select the three columns of data on dates, cashflow, and discounted cashflow that we’d worked on earlier, to validate XIRR.
Insert chart from menubar or toolbar
Switch to bar chart
Here’s a short video to help you out, though it’s quite straight-forward, if you’ve followed along thus far:
Final plot should look like this:
Cashflow and Discounted Cashflow vs Years - Dark Mode
Cashflow and Discounted Cashflow vs Years - Light Mode
While the blue lines in the above plot are of same heights for positive cashflow values and negative values; the red lines are where we see values being discounted more and more over time.
Recap and Wrapping Up
We learned various aspects of XIRR in the last chapter, using an example and output of in-built xirr() function.
Now we can cross-validate these, how these follow from definition of XIRR naturally:
XIRR doesn’t depend on exact dates, rather relative differences between dates of transactions
This is because in NPV equation, the power terms are normalized differences in time, and not absolute values of time points (
XIRR doesn’t depend on exact value of cashflow amounts, only relative scaling within those numbers.
In other words, if every cashflow amounts were multiplied by some arbitrary number, it won’t change the value of XIRR.
This is readily verifiable; as in the NPV equation, the right hand side is
. Hence multiplying both sides by a constant co-efficient has no effect.
Moving up payout schedule improves XIRR
Moving up schedule of payout (positive cashflows) means the discounted value of those transactions become higher; as
To keep the equation at zero, now the base of the fraction would have to go up, which is
. In other words, XIRR would have to go up.
Similarly, we can refer to the NPV equation of XIRR formulation, and validate our erstwhile understanding of XIRR behavior.
You might have noticed that we didn’t do any semi-log plots with cashflow values.
This is because NPV equation is an equation that lists formulas as sum of decayed entities.
There’s no mathematical formula to simplify
, that relates to
in any way, in general.
Next chapter we would learn about applying XIRR in financial decision-making, in the wild. We’d consider real-world scenarios, and approach decision-making on those, by computing their XIRR; and comparing with XIRR of alternative investment avenues.