What is IRR?

The basics

The Internal Rate of Return, or IRR for short, is a measure of your investment performance, and is expressed as percent return per year.  It is essentially equal to the (annualized) interest rate a bank would have to pay you to duplicate the performance of your portfolio.

IRR takes into account the amount of time that has elapsed since making an investment.  For example, if you purchased $1000 worth of stock today, and one year from today its market value is $1200, your gain would be 20%.  Your IRR would also be 20%, because exactly one year would have elapsed since you made the investment.

On the other hand, if you had sold the stock after six months for $1200, although your gain would still be 20%, your IRR would be 44%.   To see why, you must understand that the calculation of IRR "annualizes" your investment return.  In this example, if you were to continue to invest your money with the same degree of success as before, you would take your six-month balance of $1200 and earn another 20% in another six months, giving you $1200 x 1.2 = $1440:  a 44% gain in one year.

The mathematics

When analyzing a complex series of purchases and sales, it is useful to look at your investment activity as a cash flow diagram.  A cash flow diagram is nothing more than a timeline on which arrows are drawn to represent cash transactions.  When you pay cash out (for example, purchasing a stock or making a bank deposit), the cash flow is negative and the arrow is drawn pointing downwards; when you receive cash (for example by selling a stock or making a withdrawal) the cash flow is positive and the arrow points upwards.  The simple example above is represented by the diagram at right.

Mathematically, the IRR is defined as the interest rate r (or "discount rate") that would make the "present value" of the series of cash flows equal to zero.  The "present value" of a particular cash flow is defined as the amount of cash (either positive or negative) divided by (1+r)^t, where t is the time the cash flow occurred relative to the present, expressed in years.  The present value of a series of cash flows is simply the sum of the present values of each individual cash flow.  Applying this to the above example, and using r = 0.44, we would obtain:

	Transaction	cash	t, years	(1+r)t	PV
	initial purchase	-1000	0	1	-1000
	sale after 6 months	+1200	0.5	1.2	+1000
	Total present value:				0

When there are multiple purchases, sales, dividends, etc. the cash flow diagram can become arbitrarily complex.  The general equation can be expressed as follows:

where C_i are the cash flows and t_i the times.  The value of r that solves this equation is the IRR for the series of transactions.

In some circumstances there can be multiple values of r that solve the equation, so that a unique IRR cannot be determined.  The solver utilized in Dan's Portfolio Tracker does not attempt to detect the presence of multiple solutions, but simply returns the first solution found.

Cash-carrying vs. cashless portfolios

In Dan's Portfolio Tracker, cash-carrying portfolios are treated differently from cashless portfolios when calculating IRR.  In a cashless portfolio, every transaction (except stock splits) is treated as a cash flow:  stock purchases are negative, while stock sales and dividends are positive.  In a cash-carrying portfolio, on the other hand, the only cash flows considered are deposits and withdrawals from the cash account.

To see why, think about what happens when a stock is purchased.  When a stock is purchased in a cash-carrying portfolio, the cash required is removed from the cash account, and the portfolio value is unaffected (except for the commission fee).  This is in contrast to a cashless portfolio, in which purchase of a stock increases the portfolio value by the cost of the stock.  In the latter case the portfolio holder would have to put out some cash; in the former, the cash comes from within the portfolio itself.

These implementation details are transparent to the user, however.  All you need to do is decide whether or not to carry a cash balance, and the program will take care of the rest.  If you do decide to maintain a cash balance, however, please remember to make at least one deposit.  The program will allow you to go into debt to make stock purchases, but if there are no deposits at all then there are no cash flows from which to calculate the IRR.