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In the previous section, we covered more qualitative means of stock valuation and, in this section, we wish to turn to more numerical ways to determine valuation. The types of quantitative valuation types that we will discuss include the following:

• **Discounted Cash Flow.** This is the most theoretically complete valuation method and, fundamentally, underlies most other valuation methods. Other valuation methods can be seen as a short-hand or highly simplified approximation to discounted cash flow.

• **Ratio Analysis.** Valuation using ratio analysis, or the ratio of the share price to some other statistic, is one of the most common valuation methods. The advantage of ratio analysis is the ease with which it can be calculated and, thus, its simplicity.

• **Yield Analysis.** Yield analysis is a less commonly used valuation method. It is helpful from the standpoint that it aids in valuation comparison between different kinds of investments. It allows valuation to be reduced to a universal statistic.

• **Expected Value.** Expected value is actually a valuation method one step beyond the valuation methods mentioned above in that it utilizes the other valuation methods to quantify a scenario analysis.

We will now discuss each of these valuation types in more detail below.

**10.1. Discounted Cash Flow** – the base of everything

Theoretically, the value of an investment is equal to the total amount of money which an investor can expect to receive or own from that investment in all future years. In most cases, this would refer to money “owned” by way of his stock ownership since few companies pay out all their cash flows in dividends. Because money received in the future is generally worth less to people than money received now, money expected to be received in the future is discounted to a value which reflects that lower worth, hence the name discounted cash flow. Mathematically, the discounted cash flow calculation can be represented as follows:

Equity Value =

“FCFE” stands for Free Cash Flow to Equity. “r” is the required rate of return to equity investors in the firm, also called the cost of equity. In other words, the total value of a company’s shares is equal to the present value (or discounted value) of the total cash flows expected to be generated by the company. We will discuss the two most important concepts related to this valuation technique, cash flow and discounted value, below.

CASH FLOW

In concept, it is quite reasonable to expect that the value of a company should mathematically be equal to the sum total of cash which one expects the company to generate over its lifetime (hopefully, into infinity). However, the issue sometimes becomes more complicated by the definition used for cash flow. In the equation we show above, we have used the most direct way of calculating equity value from cash flow – that is using cash flow that is “owned” by equity. This means, literally, the cash generated each year after paying out all cash expenses, notably including interest and principal payments as well as taxes. This cash flow is sometimes termed cash flow to equity.

Another variation is cash flow to the firm, or cash that is “owned” by both equity and debt. In this case, the cash used to pay back interest and principal on debt is added back to the cash flow in each year. The value calculated would be the total value of the debt and equity in the firm. The cash flow would need to be discounted by the weighted average cost of capital (weighted by the proportion of debt and equity and using the appropriate equity and debt discount rates). Finally, to get equity value using this method, total debt would have to be subtracted from the total firm value calculated by this method.

Finally, before we turn to a discussion of discounted values, we do want to point out that in the equation, we have summed the total cash flow generated into infinity. If there is a reason to think the company would stop operation in a certain year, then the subsequent years would generate no cash flow and the calculation would be simplified. However, in most cases, the assumption would be that the company will be a going concern into infinity. In such a case, at some point, perhaps after five or ten years, the subsequent cash flows would have to be calculated using a short-hand since forecasting individual year cash flows into infinity would not be possible. Say, if we have decided to explicitly forecast ten years of cash flows, then in the eleventh year, we would use a formula to generate a “terminal value”. The formula we would use is as follows:

Terminal value cash flow =

“r” is again the cost of equity or return to equity investors. “ ” is the growth rate in FCFE forever.

Reworking the equity value equation, would then yield the following formula:

Equity value = +

The growth rate “ ” is now the growth rate that would apply forever after the terminal year. During the periods from 1 to “n”, you could explicitly forecast FCFE in each year. Usually this period would consist of an extraordinary growth period. At the end of this period, or after “n”, the terminal value formula could be used to calculate total FCFE for all the subsequent years. It is assumed that growth would stabilize to a constant value “ ”.

DISCOUNTED VALUE (OR PRESENT VALUE)

Since this sort of valuation is called a discounted cash flow valuation or DCF, we will use the term discounted value here. However, this term is really interchangeable with present value to some extent. It is generally accepted that money received in the future is less valuable than money received today. As a result, cash flows generated in later years are discounted to a lower value than cash flows generated nearer to the present.

This concept is sometimes not so obvious to grasp, although we see its impact in some basic items. For instance, when you deposit money in a bank, you receive interest for each year the money is deposited. Similarly, when you borrow money, you must pay interest for each year the money is borrowed. These interest payments are related to the concept of discounted cash flow. If you hadn’t deposited the money in the bank, you could use the money for other purposes. Since it is tied up, you get some compensation. In addition, since there is also some risk that you might not get all of your money back, you are also compensated to some degree for this risk. Debtors must also compensate their creditors for the risk that the debtor may default. The higher the risk the higher the interest rate.

Discounted value or present value, is then some future value which is discounted by an appropriate discount rate to its present value, or the value today. The discount rate we use in the formula for our DCF calculation is called the cost of equity. This is the annual return which an equity holder would expect from holding onto the equity of the company in question. The more risky the company, the higher the cost of equity would be expected to be.

Generally, the cost of equity is calculated using the capital asset pricing model. The model is based on many assumptions, but it basically measures risk by measuring the covariance of a company’s equity returns against a market index return. This covariance is called Beta. The formula to calculate cost of equity is as follows:

Cost of equity = Risk free rate + Beta × (Equity market risk premium)

The risk free rate is the rate that can be earned on a riskless asset. Generally, this is taken to be a long-term government bond rate, such as a US 10-year government bond. Beta is more difficult to calculate, and usually is provided by a statistical information service. The equity market risk premium is the expected return of the market index less the risk free rate, or the excess return expected to be generated by putting money into the market index, rather than the riskless asset (i.e. 10-year government bond). The market index used is a suitable benchmark index. It is usually a very broad-based equity index such as the S&P 500 index.

Discounted cash flow valuation is quite an involved process and there are many technical details involved in its calculation. If you truly have an interest in exploring this sort of valuation further, we would suggest further study. A very good book which covers this topic is “Damodaran on Valuation” written by the NYU professor Aswath Damodaran (see Suggested Reading List).

**10.2. Ratio Analysis – Short-Hand for DCF**

The most common ratio used for valuation is the P/E (Price/Earnings) ratio. Ratio analysis, in general, applies to the variety of methods of judging valuation by comparing the price of a share with a given company statistic. The company statistic could be from the financial statements of the company – such as net income (earnings) or revenues from the income statement or book value from the balance sheet. The ratio could also use a company statistic not directly related to something in the financial statements but some other numerical value related to the company such as number of customers. We discuss a variety of the different valuation ratios below:

• **Price/Earnings.** The price/earnings (P/E) ratio is the price of one share of stock divided by the earnings per share of the company. The higher the P/E ratio, the more expensive a company’s stock is. However, a higher P/E ratio does not necessarily mean a stock is not good value. The P/E ratio can be used to compare similar companies in the same industry. If the ratios are different, there should be good reasons for the difference. If there appear to be no good reasons for the difference, a company with a low P/E ratio might be a good value and one with a high P/E relative to its peers may be expensive. Across industries, the P/E ratio is often used in conjunction with earnings growth rates. Higher P/E ratios are generally seen as justified by higher earnings growth while low P/E ratios are seen as justified by lower earnings growth. Some investors may even convert the P/E ratio into another ratio, the P/E to growth (or PEG) ratio.

• **Price/Revenues.** The price/revenues ratio is another way to look at valuation. Again, good reasons should exist for why similar companies in the same industry have different price/revenues ratios. Oftentimes, the price/revenues ratio is used in conjunction with a profitability measure such as operating profit margin or net profit margin. The higher the price/revenue ratio, the higher the profit margin should be. If company earnings are being distorted by unusual items, the price/revenues ratio may be useful as a way to compare different shares without having to make adjustments to each of the company’s earnings to account for special items. The price/revenues ratio is also useful if some of the companies in the comparison group are making losses. In this case, a price/earnings ratio would not be useful as it would give a negative number.

• **Price/Book.** The price/book (P/B) ratio takes the share price and divides it by the per share book value (usually the total shareholder’s equity divided by total shares outstanding). Usually, the P/B ratio is used in conjunction with the return on equity (ROE) figure for a company. The ROE is the company’s earnings divided by total shareholder’s equity and is a measure of how efficient a company uses its equity capital. The P/B ratio is often used where capital efficiency is a key driver of company profitability. For instance, banks may often be compared using the P/B ratio, since a bank’s product is money and its profitability can be measured by how efficiently it can make money from the money which comprises its equity base. In industries which are very asset intensive and where the book value is an accurate representation of the value of the company’s assets (rather than an accounting value which has drifted from the true value of the company’s assets), the P/B ratio may be especially useful. For instance, if the book value represents the true value of the invested capital in a company and the P/B ratio is below 1, then it means the shareholder is buying assets at a price below what they could be sold for. This does not necessarily mean the shares are a good investment, especially if the company is losing money. However, a low P/B ratio in such a case may be an indicator that an opportunity may exist for a good investment if the company’s prospects are good.

• **Price/EBITDA.** EBITDA (earnings before interest, taxes, depreciation and amortization) is a proxy for operating cash flow. This ratio can be used for companies where analysis of operating cash flow is important. It is also useful for comparing companies with different capital structures (i.e. – different levels of debt), different tax structures (due to tax credits or geographic location) and companies at different points in their facilities upgrade cycles (in other words, a company with very old equipment may have a much lower depreciation expense than a company with new equipment). The price/EBITDA ratio can help to minimize the impact of the differences cited above. It is often used by private equity firms as a private equity firm is likely to load up an acquisition target with more debt. If the price/EBITDA ratio is low and a company currently does not have a lot of debt, this may make it a candidate for a buy-out as it could likely be leveraged up without too much impact to its near-term (although maybe not long-term) value.

• **Price/Output.** For companies which produce somewhat commoditized products, such as electricity or steel, it may be useful to compare a company’s share price to its total production of the given product per share. Since the company’s products can be expected to fetch a reasonably predictable price per unit, the price/output ratio can be a useful comparison method. If there are large differences in the price/output ratio, you should seek to understand the reason for these differences. For instance, different steel companies may have different efficiencies in their production processes, which would result in varying levels of profitability. However, if the profitability in a commodity industry could be expected to converge over time due to common production methods, and there are big differences in the price/output method, then this sort of ratio analysis may help locate investment opportunities.

• **Price/Customers.** In industries with a large number of customers and where the customers exhibit relatively predictable behavior, the price/customers ratio can be used to compare companies. This ratio can be useful in the early stages in a company’s development although it can also be misused in the early stages of a company’s development. A new mobile phone company, in its early stages, may have marketing costs which exceeds its earnings from existing customers, which would cause the company to exhibit losses, making a P/E ratio unreliable. In addition, the average monthly revenue from customers may not have stabilized as some of the customers may be benefiting from special teaser rates. In such a case, the price/revenues ratio may not be an accurate reflection of the company’s worth. As a result, the price/customers ratio becomes a practical way of measuring company value. This is only a useful measure if the company’s customers can eventually be expected to produce profits in-line with other companies you might be comparing the start-up to (i.e. – similar mobile phone companies in more developed, stable markets, etc.). During the internet bubble, many internet companies were valued on price/customers ratios since many internet companies were not making profits, or revenues for that matter. In many of these cases, the price/customers ratio was abused because there was no foreseeable way of turning those customers into profitable customers.

In each of the ratios above, the price used should be the latest market price for the shares of the company in question. For the denominator in the ratio, you can use either a historical (actual) figure or you can use an estimate of its value in the future. It is important, however, that all ratios being compared use a denominator from the same time frame. If you are relatively confident in your estimates for the figures in question, then perhaps ratio analysis based on these estimates may be the most useful as the share price should track the company’s future performance more than its past. However, if estimates are unreliable or difficult to construct, then ratios based on historical figures can be used instead. One good thing about using historical figures is that it takes out any bias that can creep into estimates since the historical figures cannot be manipulated.

10.3. **Yield Analysis – A Way To Compare Different Investments**

Yield analysis is a less commonly used method of valuation but one that is useful for making rough comparisons between different types of investments. In particular, yield analysis can be used to compare stocks to bonds or stocks to cash. Since a bond has a fixed rate of rate of return whether the bond is a corporate bond or a government bond, yields are the most common way of judging whether a bond is expensive or cheap. Similarly, the value of holding cash can also be measured by the interest yield to be expected by holding cash. Comparing this interest yield to a stock’s P/E or P/B ratio, doesn’t really make an easy to interpret comparison.

• **Earnings Yield.** Earnings yield is a company’s earnings per share divided by its share price per share. We would use expected earnings for the full year divided by the company’s current share price. In fact, earnings yield is directly related to the P/E ratio as it is the direct inverse of the P/E ratio. However, earnings yield is a number which can be more readily compared to a bond’s interest yield.

• **Dividend Yield.** Dividend yield is a company’s dividend per share divided by its share price. We would use the expected amount of the next dividend to be distributed divided by the company’s current share price.

• **Bond Yield.** For bond yield, we would use the yield to maturity (YTM) as opposed to the current yield (annual interest divided by bond price). The YTM includes an annual amount to account for the difference between the bond’s current price and the value of the bond at maturity. Bond prices fluctuate up and down to adjust to market interest rates. Since a bond’s interest payments are fixed, the adjustment must come in the bond price.

As mentioned earlier, yield analysis can allow different types of investments (i.e. bonds and stocks) to be compared to each other. If yields are different you must analyze the reasons behind the yield gap and determine whether the gap is justified or not. When comparing bond yields to earnings yield or dividend yield, you must also decide which bond duration to use for comparison. Given the fact that stocks usually represent ownership in a company that intends to be a going concern for as long as possible, it makes sense in many cases to use a long-term bond yield, such as a 10-year bond. Let’s take a 10-year government bond yield and compare it to the dividend yield of company A. If the two yields are equal, we would then try to judge whether the stock had some other compensating factors to account for the offsetting higher risk as compared to the very low risk of the government bond. If company A had a reasonably high growth rate and its share price and dividend yield could be expected to grow at a double-digit rate in the next few years, then you might conclude that company A’s share price was a very good value relative to the bond.

**10.4. Expected Value**

It might be helpful to review section 9.3 “Scenario Analysis” before continuing. Here, we will show you how to take the qualitative valuation technique of scenario analysis one step further into a quantified valuation. We will call this quantified scenario analysis an expected value valuation.

To perform an expected value valuation, you will need to review the assumptions behind your worst-case, base case and best-case scenarios. Then, based on these underlying assumptions for the company in question, use one of the quantitative valuation methods already described to assign a share price value to each of your scenarios. For instance, you may find that for company A, your worst-case scenario yields a share price target of US$12 per share, while your base case share price target yields a value of US$19 per share, and your best-case share price target yields a value of US$25 per share.

Having constructed share price targets for each of your scenarios, you then need to assign to each scenario a probability of occurrence. The sum of the three probability estimates for each scenario should be 100%. For instance, we might assign a probability of 30% to the worst-case scenario, a probability of 50% to the base case scenario and a probability of 20% to the best-case scenario. There is not well defined method for determining these probabilities and, at the end of the day, these numbers will be a reflection of your judgment more than anything else.

The next step would then be to multiply the various probabilities by the per share values derived for each scenario. We show the math calculation in Exhibit 2 below using the numbers from our examples above. For the worst-case scenario, we multiply 0.30 (30% probability) by the per share value estimate of US$12, to get US$3.60. We do the same for the base case and best-case scenarios, as shown, and get values of US$9.50 and US$5.00, respectively. The final step is to add up all three probability weighted share values ($3.60+$9.50+$5.00) to get one sum: US$18.10.

Exhibit 2: Sample Expected Value Calculation

The US$18.10 per share value would then be our valuation estimate for company A’s shares using the expected value method. This share does not necessarily represent a value at which we think the shares would settle at given that it is an amalgam of scenarios. However, it allows us to judge whether investing in a given share is likely to turn out favorable or unfavorable – whether the odds are in our favor, so to speak. If the current share price for company A is far enough below US$18 such that our expected return is satisfactory, we would be willing to invest in company A.

The expected value method of valuation is particularly suited to those situations where the potential outcomes for a given company’s development could be quite divergent. However, we believe the expected value technique is suitable for almost any investment and that it would be helpful to think in these terms for any investment, as this better reflects the reality that a range of potential outcomes is possible.

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