For the example in this exercise, we shall take the business of Home Depot. There are 3 main variables to consider in the calculation of the intrinsic value, namely, the book value, the historical return on shareholder's equity of the business, and the rate of return of the alternative choice of investment (in this case, it shall be the risk-free interest rate of government bonds).
In fiscal ended Jan 06, the book value for Home Depot was US$12.53. It was traded at about US$34 back then. The historical ROE is about 20%. And finally, the risk-free interest rate is 5.25%.
For an intrinsic value calculated for 10 years of earnings flow, the book value of Home Depot would have grown from $12.53 to approximately $77.6. Alternatively, if you have chosen to invest in risk-free bonds, the $34 which would otherwise be originally invested in Home Depot, would have grown to $56.72. Therefore, for the 10 years, the earnings from Home Depot generates about $65.07 ($77.6 - $12.53) while the earnings from the bond generates about $22.72. On the surface, the accumulated earnings from the business appears to be $65.07 for ten years before discounting it to present value. However, in order for the intrinsic value to be accurate, the accumulated earnings flow from bonds must be deducted from the earnings flow of the business, which gives a value of $42.35. Then this $42.35 must be discounted with an appropriate interest rate to get the present value of the intrinsic value today. In this case, the discount rate is 5.25%. So, after discounting for 10 years back to today, the intrinsic value of the earnings flow gives an intrinsic value of $25.39.
Then if you calculate the earnings flow for 20 years, the book value of the business would grow from $12.53 to $480.50 in 20 years. While the bond value would grow from $34 to $94.61 in 20 years. Thus, the earnings from the business is $467.97 while the earnings from the bond is $60.61. Again, the intrinsic value at the end of 20 years is $407.36, and then you need to discount it again at an appropriate rate to the present value, which gives $146.40.
Now to summarize the example for Home Depot. The intrinsic value based on 10 years earnings is $25.39, and 20 years is $146.40. The trading price is about $34 somewhere at the start of 2006. So if your time frame is 10 years, obviously, you are paying more than what the accumulated earnings in 10 years will be. But if you calculate based on 20 years, you get more than what you pay for. In fact, in you are paying $34 dollars for every $146.40 (in present value terms) of the next 20 years earnings flow.
And what is also clearly illustrated here in terms of book value is book value is meaningless as an indicator of intrinsic value if you should invest in a business. In fact here, you are paying $34 per $12.53 of book value which is a P/B ratio of 2.71. But it is capital well deployed since you will be getting $146.40 of future earnings in present intrinsic value today.
By the way, if you are wondering why is there a need to deduct the earnings which would otherwise be earned from bonds from the earnings of the business, here is the reason. In intrinsic value, it is in fact a comparison between two options. For example, between spending money in an education or to skip education and work straight away. So to compare which option is better in terms of future earnings, it is the difference in earnings between the earnings of having an education and the earnings of not having an education. To understand further on the basic concept of intrinsic value, do refer to one of my past subject on this.