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

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

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

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

Example 1 n D 1 = $1.00 n D 2 = $1.25 n D 3 = $1.50 n P 3 = $50 n 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 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% P/YC/YNI/YPVPMTFV

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

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

Mathematical Formula for Finding PV PV = FV x (1+i) -n n PV = 1.00(1.10) (1.10) (1.10) (1.10) -3 n P 0 = $40.63 (same answer as we got using a financial calculator)

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

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

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

Constant Dividend (No Growth) P 0 = D p /K p n P 0 = Intrinsic value = Price today n D p = Preferred Dividend (fixed amount, doesn’t change) n K p = Required rate of return on P/S n Preferred stock is an example where the dividend is constant

Example 2 n 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? n P 0 = $1.25/.12 = $10.42

Dividends Growing at a Constant Growth Rate P 0 = D 1 /(K c - g) n P 0 = Intrinsic value = Price today n D 1 = Dividend expected 1 year from now n D 1 = Last dividend paid x (1 + g) n K c = Required rate of return n g = Constant annual dividend growth rate

Example 3 n 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? n P 0 = ($2 x 1.05)/( ) = $42

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

Example 4 n 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?

n P 4 = D 5 /K c -g n P 4 = $5/( ) = $125 n P 0 = P 4 (1 + K c ) -4 n P 0 = $125(1+.12) -4 = $79.44 maximum price you are willing to pay today

Valuing Non-public Corporations n Twitter article Twitter article

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

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

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