Geometric series
A geometric series has the same ratio between the consecutive numbers. An example explains it much better. Let us say that the first number in the series is 20 and the common ratio is 2; then the series is:
20, 40, 80, 160, 320, 640, ….
Starting from 20 and ratio 0.5, the series will be:
20, 10, 5, 2.5, 1.25, ….
So, the series starting from the number a and common ratio r is:
a, ar, ar^2, ar^3, ar^4, …
The sum of the series can be calculated by a simple formula which can be remembered easily. For the sum of the first n-elements in the series we have:
a+ar+ar^2+...+ar^{n-1} = a(r^n-1)/(r-1)
An interesting thing happens when r<1. If we add up all the numbers in the series then it sums up to a number. For this the formula is easier:
a+ar+ar^2+...=a/(1-r)
Let us again take the above examples:
20+40+80+160+320=20(2^5-1)/(2-1)=620
and
20+10+5+2.5+1.25+....=20/(1-0.5)=40
Do not worry about all the hocus-pocus if you are not a fan of mathematics. You need to remember that when r<1 then the series when summed up forever gives a/(1-r) and if you want to sum n-elements in the series then the sum is a(r^n-1)/(r-1).
Calculating the terminal value
How is this useful, you ask? Let us suppose that you have $100 in your account and it is earning interest at the rate of 10%. How much money you will have in your account in five years ? This is easy to calculate. The following is the table with interest and the total at the end of every year until the end of five years.
So, at the end of five years you will have $161.051 in your account. But as you see, the calculation is quite cumbersome and difficult to do. Instead we can use the compounding formula
(initial sum)*(1+rate/100)^years
This gets us 100*(1.1)^5=161.051, which is the same as above. How do we get this formula?
In the first year, we will have 100+100*0.1=100*1.1
In the second year (100*1.1)+(100*1.1)*0.1=100*1.1^2
and so on.
Now, let us flip the question around. How much money do you need to put in the bank now so that you will have $200 at the end of five years? This is easy to do. If the money is $a then:
a*1.1^5=200
a=200/1.1^5
So, we need to put $200/1.1^5=$124.184265 in your account now.
Now, let us put this together. We will use what we learned so far to calculate the terminal value of a stock.
This way of valuing a business is called discounted cash flow. General Electric (GE, Financial) generated $26 billion in FCF in 2010. Let us suppose that if you put your money in the S&P, you can earn 10% compounded. Also, suppose that GE is expected to raise the FCF at the rate of 1% since the end of time. What is the amount of money you need to pay now for GE with 10% discounted rate and 1% FCF growth?
So, we need to find the following sum:
26+26*1.01/1.1+26*(1.01/1.1)^2+26*(1.01/1.1)^3+26*(1.01/1.1)^4+...
We now use our geometric series formula to calculate the terminal value. The first term in the series is 26 and the ratio is 1.01/1.1. This gets us 26/(1-1.01/1.1)=26*12.2222222=317.777778. So, you should pay around $317 billion for GE at the moment, assuming that GE will grow at 1% rate and you want a discount of 10%.
Calculating the exceptional growth period
To calculate the value of a stock with DCF model, we want to calculate the following:
To calculate the value of the company, we will first use our formula a(r^n-1)/(r-1) with n=5. Then we calculate the terminal value of the stock using the example of GE above. Let us take an example of Teva Pharmaceuticals (TEVA, Financial).
TEVA has FCF $3.4 billion, and let us say that it can grow it at the rate of 5% for the next four years, and then 2% (with the economy) until the end of time. With 10% discount we have
The total value is hence $(15.52+38.78) billion=$54.3 billion.
I hope there is no mistake in the calculations. But if you do find a mistake, please mention it in the comments. The idea is to understand the mathematics for some of the valuation models used to evaluate a company.
A geometric series has the same ratio between the consecutive numbers. An example explains it much better. Let us say that the first number in the series is 20 and the common ratio is 2; then the series is:
20, 40, 80, 160, 320, 640, ….
Starting from 20 and ratio 0.5, the series will be:
20, 10, 5, 2.5, 1.25, ….
So, the series starting from the number a and common ratio r is:
a, ar, ar^2, ar^3, ar^4, …
The sum of the series can be calculated by a simple formula which can be remembered easily. For the sum of the first n-elements in the series we have:
a+ar+ar^2+...+ar^{n-1} = a(r^n-1)/(r-1)
An interesting thing happens when r<1. If we add up all the numbers in the series then it sums up to a number. For this the formula is easier:
a+ar+ar^2+...=a/(1-r)
Let us again take the above examples:
20+40+80+160+320=20(2^5-1)/(2-1)=620
and
20+10+5+2.5+1.25+....=20/(1-0.5)=40
Do not worry about all the hocus-pocus if you are not a fan of mathematics. You need to remember that when r<1 then the series when summed up forever gives a/(1-r) and if you want to sum n-elements in the series then the sum is a(r^n-1)/(r-1).
Calculating the terminal value
How is this useful, you ask? Let us suppose that you have $100 in your account and it is earning interest at the rate of 10%. How much money you will have in your account in five years ? This is easy to calculate. The following is the table with interest and the total at the end of every year until the end of five years.
| Year | Balance at 01.01 | Interest earned | Total |
| 2011 | 100 | 10 | 110 |
| 2012 | 110 | 11 | 121 |
| 2013 | 121 | 12.1 | 133.1 |
| 2014 | 133.1 | 13.31 | 146.41 |
| 2015 | 146.41 | 14.641 | 161.051 |
So, at the end of five years you will have $161.051 in your account. But as you see, the calculation is quite cumbersome and difficult to do. Instead we can use the compounding formula
(initial sum)*(1+rate/100)^years
This gets us 100*(1.1)^5=161.051, which is the same as above. How do we get this formula?
In the first year, we will have 100+100*0.1=100*1.1
In the second year (100*1.1)+(100*1.1)*0.1=100*1.1^2
and so on.
Now, let us flip the question around. How much money do you need to put in the bank now so that you will have $200 at the end of five years? This is easy to do. If the money is $a then:
a*1.1^5=200
a=200/1.1^5
So, we need to put $200/1.1^5=$124.184265 in your account now.
Now, let us put this together. We will use what we learned so far to calculate the terminal value of a stock.
This way of valuing a business is called discounted cash flow. General Electric (GE, Financial) generated $26 billion in FCF in 2010. Let us suppose that if you put your money in the S&P, you can earn 10% compounded. Also, suppose that GE is expected to raise the FCF at the rate of 1% since the end of time. What is the amount of money you need to pay now for GE with 10% discounted rate and 1% FCF growth?
- In 2011 GE will generate $26*1.01 billion. For this we need to pay 26*1.01/1.1 billion now (the discount rate is 10%).
- In 2012 GE will generate $26*1.01^2 billion. For this we need to pay 26*1.01^2/1.1^2 and so on …
So, we need to find the following sum:
26+26*1.01/1.1+26*(1.01/1.1)^2+26*(1.01/1.1)^3+26*(1.01/1.1)^4+...
We now use our geometric series formula to calculate the terminal value. The first term in the series is 26 and the ratio is 1.01/1.1. This gets us 26/(1-1.01/1.1)=26*12.2222222=317.777778. So, you should pay around $317 billion for GE at the moment, assuming that GE will grow at 1% rate and you want a discount of 10%.
Calculating the exceptional growth period
To calculate the value of a stock with DCF model, we want to calculate the following:
- We have high visibility in the near term. So, we can expect that the company can grow at the rate of say 5% in the next 5 years.
- We have low visibility in the long term. So, we want to be safe and assume that the company will grow with the economy. This rate has been around 2%.
- To be safe, we want a discount rate of 10%.
To calculate the value of the company, we will first use our formula a(r^n-1)/(r-1) with n=5. Then we calculate the terminal value of the stock using the example of GE above. Let us take an example of Teva Pharmaceuticals (TEVA, Financial).
TEVA has FCF $3.4 billion, and let us say that it can grow it at the rate of 5% for the next four years, and then 2% (with the economy) until the end of time. With 10% discount we have
- Exceptional growth value 3.4+3.4*1.05/1.1+3.4*(1.05/1.1)^2+3.4*(1.05/1.1)^3+3.4*(1.05/1.1)^4=3.4*((1.05/1.1)^5-1)/(1.05/1.1-1)=15.52
- Terminal value: for this we need to find out the FCF at the end of four years. This is 3.4*(1.05)^4=4.13. The terminal value at the end of four years is 4.13/(1-1.02/1.1)=56.78 billion. But what should you pay now for $56.78 billion in four years? At a 10% discount rate this is $38.78n now.
The total value is hence $(15.52+38.78) billion=$54.3 billion.
I hope there is no mistake in the calculations. But if you do find a mistake, please mention it in the comments. The idea is to understand the mathematics for some of the valuation models used to evaluate a company.
