**Reply:** 1

**From:** Tony Dixon

Hi Ray,

No one seems to have answered your question so I thought I'd have a crack.

The hand-waving answer

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The key to the IRR is recognising that money has a time value - a dollar today is worth more than a dollar tomorrow, and a lot more than a dollar in 20 years time, due to inflation.

Assume 1 dollar today = $20 in 20 years time.

Then if you entered into an investment where you paid $1 today and received a certain amount after 20 years, then it should be obvious that a payout of $15 is a loss (a negative return) in real terms, whereas a payout of $25 is a profit (a positive return) in real terms.

A payout of $20 mean that your real return was zero! You made no money from the deal.

The actual (real) rate of return depends greatly on inflation.

The mathematical answer

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Assume annual interest rates are i%.

So (ignoring risk), an investment of an amount C will be returned after one year with an additional payment of i*C.

After one year you now have total capital of

C + i*C = C*(1 + i)

Reinvesting this amount for n years, you will have compounding of your interest (i.e. your interest earns extra interest).

So you will end up with an amount (ignoring risk)

FV = C*(1 + i)^n

This is the "future value" of your initial capital C.

Turning this argument around, we can work the "present value" of future cashflows received in the n-th year (expenses, rents, capital gains) by multiplying through by the "discounting factor"

1/(1 + i)^n.

i.e.

PV = FV/(1 + i)^n

So in the simple example above we have the FV of $1 given by (for an inflation rate e = 16.1%).

$1 = $20/(1 + e)^20

A general payout X has an IRR i satisfying

$1 = $X/[ (1+e)^20 * (1 + i)^20 ]

So how do you calculate the IRR for a property?

First write down your purchase costs.

Then guess what all your future cashflows will be for each year in the future.

Assume an initial value for rent. Assume that it will grow with the estimated inflation rate - this takes into account the inflation effect so the 'e' factor is absorbed into the cashflows.

Similarly with expenses.

Then you need to figure out what you will sell the property for. To do that you need to estimate capital growth.

Add all these cashflows together and set the sum equal to zero.

e.g.

pay lawyer $1 in year 0 => cashflow = -1

receive an inflation-adjusted(!) rent of $2 after 3 years,

=> cashflow = +$3 in year 3.

So the equation of value in today's dollars is

-$1 + $3/(1+IRR)^3 = 0

In general we need to solve for the IRR numerically, which is why we need a computer.

Phew.

I think I prefer the hand-waving answer.

Hope this gives you some idea of what is going on.

cheers, Tony