# What is the reasoning for the Discount Factor formula?

I've previously come across dicount factors in my high school education but we had formula sheets so I never bothered actually learning it. In my university I have a business class in which we are now going over dicounted cash flows. Now that I'm a much more enthusiatic learner I would really like to know why the discount formula works and/or the reasoning behind it. The formula is as follows:

Discount factor = 1 / (1 + r)^t


where r is the discount rate and t is the amount of years.

This is more business related than math related. So if your wondering why I posted this here I figured that mathmaticians would be able to better explain the reason/origins of the formula, whereas all I've gotten from asking business realted experts is pretty much 'the fomula works so it's good enough for me' sort of answers.

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Also note that this is a simplified formula. The $t$ is a daycount fraction - which is different in different countries (e.g. there are 30E/360, Act/360, Act/Act). For example we system Act/Act which means $t=$(number of actual days)/(365 or 366 in case the period contains a leap day) -- fairly weird if you ask me. –  AD. Oct 18 '11 at 13:08

This is annually compounded interest ($r$ is the interest rate) run backwards in time (going back $t$ years).

If you start with $P_0$ dollars and are using interest compounded annually, then after 1 year $P_0$ has grown to $P_1=P_0+P_0r=P_0(1+r)$ dollars [$P_0$ earns $r$ percent interest so we get $P_0r$ dollars in interest]. To see what happens after 2 years, remember that with compounded interest you must "restart" the computation every year. So at the end of year 2, you'll have $P_2=P_1+P_1r=P_1(1+r)=P_0(1+r)(1+r)=P_0(1+r)^2$ dollars. Now we can see a pattern emerging and find that after $t$ years our original $P_0$ dollars has grown to $F=P_0(1+r)^t$ dollars.

Given a future value $F$, we can solve to find the present value $P_0$ and get $P_0=F/(1+r)^t$. Thus the discount factor appears!

http://en.wikipedia.org/wiki/Compound_interest

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The basic idea is that receiving a certain sum of money in the future, is equivalent to receiving a lesser sum now and investing it at a certain rate $r$. Suppose you get $100\$ $know and that you can safely invest it in something with return rate of$r=3\%$every year. Then after a year, you'll have$103\$= 100\$ \times (1+0.03) $. So getting$103\ in a year is the same as getting $100\$ $now. Or, the present value of that future payment is$100\$= 103\$ / (1+0.03)$. If you have more years, you just keep dividing by the same interest factor$(1+0.03)\$.

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The formula is used in a number of ways, two of which are distinguishable and particularly significant:

(i) The value of real money, as opposed to ideal money, is not stable over time, so the formula can be used to reflect this. The idea is to bring a series of cash flows occurring at different times into the same calculation, and for this they need to be measured in the same units (e.g. Pounds Sterling at Time 0) The simple factor you have cited assumes (in this context) a steady decline in value over time, which would be captured by such concepts as inflation - often thought of as the increasing price of goods and services, but equally well conceived as the declining purchasing power of money.

(ii) In evaluating investment opportunities, a business will want to know that they offer an adequate rate of return, higher than the cost of capital available to fund such projects, and also to compare different business opportunities against one another. Here the interest rate, or discount rate, can be treated as a hurdle rate - taking account of the outlays and revenues, will this investment generate cash/returns sufficient to justify the investment? Here you would discount at the required rate of return, and see whether the answer was positive or negative, before doing other things like a risk analysis. A closely related concept is the Internal Rate of Return, which is the rate which exactly matches the cash flow profile of the project and so gives a zero answer. Here the essential idea that having £100 now is more valuable to you than having £100 in five years time. If you have £100 now, you can keep it for five years, but you have other options too.

In each case the formula adjusts future cash flows to a comparable present value and therefore allows different cash flow profiles to be compared with one another using a consistent measure.

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