Time is Money: Maths Edition

Is $100,000 worth more today or in a year’s time? Most people would want to use that money to solve their current problems, like pay off their student debt, buy an apartment or a car, etc. Why would anyone want it later when they can have it now? Investors see this in a very similar light. Money is worth more now, because of its potential earning capacity. Simply put, the money you receive today can be invested and will earn interest which will mean that the sum of money after a year would most definitely be worth more than a $100,000 in a year’s time.

This is a great explanation by Investopedia:

 

So how is this concept useful and how do you apply it? Well, it shows you that time is money and through a few simple calculations, you will be able to see how to obtain the value of potential investments at different points in time. This also forms the basis of some variations of stock valuation.

 

Future Value

The first calculation is super simple. I’m sure you’ve all done this in your secondary school maths class. It is to find the future value of an investment that returns a fixed amount of money in multiple fixed intervals.

I hope I don’t bring back any bad memories doing this but here are a few basic question you can try to solve first and see if you get it right:

If you put $100,000 in a bank that returns 2% a year,

  1. how much would you have at the end of the first year?
  2. How much would you have after 15 years?

Obviously, the answer to the first question is 100,000(1+0.02) = $102,000.

The answer to the second question is 100,000(1+0.02)^15= $134,586.

Got it? Well done! You have learnt how to calculate the future value of an annuity (a series of continuous cash flows that lasts for a certain period of time. This will help you to estimate or even accurately predict what you will be receiving based on the decision that you make.

If you haven’t got it, or want a clearer picture, the equation is basically: Future Value = Present Value*(1+ Interest Rate per Time Period)^ No. of Time Periods. Now let’s move on to the next section, present value!

 

Present Value

If you can do the maths, to get present value, you can simply rearrange the equation to find it.

So the equation basically becomes: Present Value =  Future Value/ (1+ Interest Rate per Time Period)^ No. of Time Periods.

So if let say you received $100,000 in 5 years from now and you could’ve put that sum in a risk free bank for 2% per annum, how much would it be worth now?

100,000/(1+0.02)^5 = $90573.08

Tadah! Good job! You now know how to calculate the present value of future cash flows! Finding the present value of an investment is sort of like calculating how much you have to put in now to receive that sum in x number of years. 

 

Never Ending Cash Flows (Perpetuity)

You’re now ready for the next stage of this article/challenge (whatever you think it is). How do you calculate the present value of a never ending cash flow? So for example, what is the present value of an investment that pays $100 a year with an interest rate (also known as the discount rate when calculating present value) of 2%?

If you think about it, it is the addition of 100/(1+0.02) + 100/(1+0.02)^2 +100/(1+0.02)^3 … to infinity. This is what you call a perpetuity. It is a continuous constant cash flow that never ends. If you simplify the above equation, the formula is: Cash Flow/ Interest Rate (Discount Rate). So you will basically get 100/0.02 which is $5000. If the discount rate is increased, to let’s say 5%, the present value of the investment will fall. 100/0.05 = $2000. This is because the higher the interest rate, the lower the present value needs to be to make up the future value. (Much like the present value calculations just now)

If you’re a nerd and you you like maths, here’s how the formula is simplified. Don’t worry if you don’t get it.:

PV= Present Value

C= Cash Flow

r = Interest Rate

PV = C/(1+r) + C/(1+r)^2 + C/(1+r)^3 …

Multiply both sides by (1+r)

PV(1+r) =  C(1+r)/(1+r) + C(1+r)/(1+r)^2 + C(1+r)/(1+r)^3 …

PV(1+r) = C + C/(1+r) + C/(1+r)^2 + C/(1+r)^3 …

Then subtract PV from both sides:

PV(1+r) – PV = C + C/(1+r) + C/(1+r)^2 + C/(1+r)^3 … – C/(1+r) + C/(1+r)^2 + C/(1+r)^3 …

Then simplify:

PV*r = C

PV = C/r

Finally, we are on to the last stage, if you’ve made it here, give yourself a pat on the back. We are going to show you how to calculate a growing perpetuity. The same as the last example, except that it grows at a constant rate till eternity.  This is the basis of the discounted cash flow valuation where the value of the company is seen to be the sum of all future cash flows discounted back to present value. So how does one calculate this? If you think about it, the way to calculate a growing perpetuity is the same as a normal perpetuity except that you add the growth into every formula as shown below:

PV = C/(1+r) + C(1+g)/(1+r)^2 + C(1+g)^2/(1+r)^3 …

When you simplify it, you get PV = C/(r-g)

So for example, if the cash flow is $100 and it grows to by 5% forever and the interest rate is 7%. The present value is 100/ (0.07-0.05) = $5000.

Awesome! Now you’ve got the basics down, you’re now in a position to make slightly better decisions. Although finding a perfect perpetuity in real life is pretty much impossible, understanding it will eventually help you to value different investments in the future. I hope you enjoyed this article (it’s a little more technical and difficult than all the other articles so far) and pray that it wasn’t too dry for your tastes. If you have any questions we’ll be happy to answer them in the comments section.

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P.S. If you really are a maths nerd then the derivation for the growing perpetuity formula is shown below.

PV = C/(1+r) + C(1+g)/(1+r)^2 + C(1+g)^2/(1+r)^3 …

Simply it:

PV = C/(1+r) + C/(1+r)* (1+g / 1+r) + C/(1+r)* (1+g / 1+r)^2

You can see that (1+g / 1+r) is a term that repeats to infinity so it is an infinite geometric series (1/1-x). x= (1+g / 1+r)

So when you put it back into the equation it becomes:

PV = (C/(1+r))/ (1 – (1+g / 1+r))

Multiply by (1+r/1+r) to get

PV = C/ (1+r) – (1+g)

PV = C/ (r-g)

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