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elmama
An introduction to the time value of money

The time value of money is one of the most important concepts in the whole of financial theory. The time value of money can tell you what a set amount of money today will be worth in the future or what a set amount of money or series of payments to come in the future is worth today. This is crucial for many areas of finance such as is in the pricing of bonds and options.

Most people instinctively know that one dollar/pound/Euro etc ... today will not buy you the same amount of "stuff" in 5 years time.

The question is, how do we quantify all this? Well it turns out that we can do this quite easily in fact by formulating and using something called the Discount Factor. The discount factor is a multiplier we can apply to an amount of money now that will tell us what it's worth at some point in the future or vice versa.

I am not going into the maths of it here but will simply state that given an amount of money PV, its value FV after n periods of time is given by the formula

​FV = PV*(1+r)^t

Where 

t is the number of periods into the future i.e years, days etc...
r is the interest rate that you are able to achieve per period

The amount (1+r)^t is known as the Discount Factor.

So lets say annual interest rates are currently at 2.5% and someone asks you, would you rather I gave you 100 dollars now or 115 dollars in 5 years time we can calculate what your answer should be.

​t = 5
r = 2.5% = 2.5/100

PV = 100 dollars

​=> FV = 100*(1.025^5) = 113.14 dollars

​Clearly you would take up this offer as the best you can do after investing the money in a bank for 5 years is a little over 113 dollars.


Similarly, if you have a known future value amount FV and you want to calculate what its value is in today's terms (i.e its present value PV) you would divide the FV by the Discount Factor. So, for example, how much is 115 dollars given to you in 5 years time worth in today's terms given an annual interest rate of 5%.


We have


FV = 115 dollars

r = 5/100

t = 5


​PV = FV/(1.05)^5 = 90.10 dollars



NB The above discussion has concentrated on what is called discrete compounding. However the more general idea is called continuously compounded discouting. What's the difference? Simply that the discount factor changes from the expression (1+r)^t to exp(rt). This allows us to calculate the FV or PV of an amount at any time in the future (or past) not just at discrete times.