**Theory of Stock Valuation** • Same theory as bond valuation • Find PV of future cash flows • Use investor’s required rate of return as the discount rate in finding PV

**Cash Flows from Owning Stock** • Dividends • Capital gain (loss) from selling at a higher (lower) price than you paid for the stock

**Difficulties in Valuing Stock** • 1) Future cash flows not known • 2) Stock has no maturity - infinite life of corporation • 3) No way to easily observe the rate of return that the market requires

**Stock Valuation Symbols** • D = dividend • Subscript tells when dividend is expected to be paid/received • P = price • Subscript tells when price is expected to be paid/received • Kc = investor’s required rate of return

**Example 1** • D1 = $1.00 • D2 = $1.25 • D3 = $1.50 • P3 = $50 • If you require a 10% rate of return, what is the most you will pay for this stock?

**Using Financial Calculator** Sum PVs to get -40.63 $40.63 is max price you are willing to pay for this stock if you require a 10% rate of return. Pay more than $40.63 → Return < 10% Pay less than $40.63 → Return > 10%

**BUT…future stock cash flows are not known with certainty** • Future dividends aren’t known with certainty • Dividends may be estimated, but it will only be an estimate • Future selling price isn’t known with certainty • How to overcome these problems?

**Future Selling Price** • Can prove mathematically that it doesn’t matter that we don’t know what we can sell a stock for in the future • Need to use mathematical formula for finding PV to prove this point

**Mathematical Formula for Finding PV** PV = FV x (1+i)-n • PV = 1.00(1.10)-1 + 1.25(1.10)-2 + 1.50(1.10)-3 + 50(1.10)-3 • P0 = $40.63 (same answer as we got using a financial calculator)

**Theoretical Determination of Future Selling Price** • The future selling price (Pn) is based on what the next investor will pay for the stock. • The next investor is valuing the stock based on the present value of his/her expected future dividends and future selling price. • The next investor follows the same process, etc., etc., etc.

**Since stock never matures, the actual determination of the** next selling price can be put off indefinitely. • If the actual determination of the future selling price is pushed far enough out into the future, its present value will eventually approach zero.

**With PV of future selling price dropping off to zero, value** of stock becomes the PV of its dividend stream. • The question now becomes, how can you find the PV of an unending stream of dividends? • Can do it if you make assumptions about how dividends grow from year to year.

**Constant Dividend (No Growth)** P0 = Dp/Kp • P0 = Intrinsic value = Price today • Dp = Preferred Dividend (fixed amount, doesn’t change) • Kp = Required rate of return on P/S • Preferred stock is an example where the dividend is constant

**Example 2** • If you require a 12% rate of return, what is the maximum price you will pay for a share of preferred stock that pays a $1.25 annual dividend? • P0 = $1.25/.12 = $10.42

**Dividends Growing at a Constant Growth Rate** P0 = D1/(Kc - g) • P0 = Intrinsic value = Price today • D1 = Dividend expected 1 year from now • D1 = Last dividend paid x (1 + g) • Kc = Required rate of return • g = Constant annual dividend growth rate

**Example 3** • How much would you pay for a share of common stock if the last dividend paid was $2.00 per share, dividends are expected to grow at a constant annual rate of 5%, and you require a 10% rate of return? • P0 = ($2 x 1.05)/(.10 - .05) = $42

**What if a company isn’t paying dividends?** • Just because a company is not currently paying dividends doesn’t mean that they never plan to. • Estimate when first dividend will be paid and at what rate dividends will grow. • Find price for year prior to first dividend. • Discount future price back to present.

**Example 4** • You estimate that a company that is not currently paying dividends will pay a $5 dividend per share at the end of 5 years and that dividends will grow at a constant annual rate of 8% thereafter. If you require a 12% rate of return, what is the maximum price you will pay for the stock today?

**P4 = D5/Kc-g** • P4 = $5/(.12-.08) = $125 • P0 = P4(1 + Kc)-4 • P0 = $125(1+.12)-4 = $79.44 maximum price you are willing to pay today

**Valuing Non-public Corporations** • Twitter article

**Estimate total revenue** • # users = 250 M by 2013 • Revenue per user = $2 by 2013 • 250 M * $2 = $500 M total rev by 2013

**Borrow ratios from comparable firm** • Google’s profit margin = .27 and Google’s PE = 20 • .27 * $500 M = $135 M profit • $135 * 20 = $2.7 B total value (as measured by price * # shares)

**Discount future value back to present** • Use 20% as appropriate rate for small, risky, high growth company • N = 4; I/Y = 20; PMT = 0; FV = $2.7B • PV = $1.3 Billion estimated value for Twitter