Security Valuation

1. Security Valuation DCF Approach to Stock and Bond Valuation

2. DCF Valuation • In general, the value of a security is the sum of the present values of the (expected) future cash flows that accrue to the owner (or purchaser) of that security. • Expected cash flows since we ultimately will have to be concerned about risk. • The presence of risk implies we must use a discount rate consistent with market conditions and appropriate for the risk involved. • Different discount rates may be appropriate for different cash flows generated by the same asset. • Discount all future cash flows to the present and the sum of the present values equals the securities current value. • Correct application of this approach may be used to value all the securities you will come across. For some securities (i.e. options and other derivatives) there are easier ways.

3. Bond Valuation - Terminology • Face or par value (F), is the promised payment at maturity. • Coupon interest: The bond is quoted as a coupon rate of C per year and usually makes actual payments of C/2 every 6 months. C/F is defined as the coupon interest rate, a rate that is constant over the life of the bond. • Call provision, call protection, call premium. • Default risk. • Discount rate, r, the market determined appropriate rate of return. • Yield to maturity (yield) is the single discount rate that equates the present value of the bond’s promised payments and its market price, V. • Current yield is defined as C/V.

4. Pure Discount Bonds • Pure discount bonds are the simplest bonds possible. They pay no interest and promise only to return the face value at maturity. • Wrinkle: in the bond market we always presume semi-annual compounding. • T-bills (up to a year in maturity) and Strips. • The value of a three year $1,000 face value treasury strip, when the market rate is 6%, is

5. Level Coupon Bonds • Level coupon bonds are the most common type of bond. They pay a semiannual (fixed) coupon, quoted as C per year, face of F, T years to maturity, and an annual discount rate of r. • We can think of there being two components of this valuation: • The coupon payments comprise an annuity. • The lump sum payment of the face value at maturity is like a pure discount bond.

6. T-1 Yr 1 2 T 0 C/2 C/2 C/2 C/2 C/2 C/2 C/2 C/2 +F Level Coupon Bonds cont… • The two pieces: • Annuity of C/2 for 2T periods (a period is 6 months). • Lump sum of F at the end of 2T periods. • Technical note: r is the stated annual discount rate (YTM) and we are using semiannual compounding. This is the current value assuming you receive the first coupon in 6 months.

7. Bond Pricing Example • Dupont issued 30-year bonds with a coupon rate of 7.95%. These bonds currently have 28 years remaining to maturity (the next coupon is to be paid in 6 months) and they are rated AA. Newly issued AA bonds with similar maturities currently have a yield to maturity of 7.73%. The bonds have a face value of $1000. What is the value of this Dupont bond today?

8. Bond Pricing Example cont… • Annual coupon payment = 0.0795*$1000=$79.50 • Semiannual coupon payment = $39.75 = $79.50/2 • Semiannual discount rate = 0.0773/2 = 0.03865 • Number of semiannual periods = 28*2 = 56 • Why is this more than $1,000?

9. Bond Prices and Interest (Discount) Rates • When The Discount Rate Is Equal To The Coupon Rate The Bond Will Sell At Par • When The Discount Rate Is Above The Coupon Rate The Bond Will Sell At A Discount To Par • When The Discount Rate Is Below The Coupon Rate The Bond Will Sell At A Premium To Par • At The Instant Before Maturity All Bonds Sell At Par • Why Do These Relations Hold? • What Feature Of A Bond Is The Primary Determinant Of Its Price Sensitivity To Interest Rate Changes?

10. Bond Prices and Time to Maturity Discount Rates Or Yields to Maturity What is the coupon rate?

11. Term Structure of Interest Rates • We have been talking (and we will commonly continue to talk) as if the interest rate is constant across all future periods. • One look at the WSJ bond pages and you know I’m lying to you. • Its not only because I enjoy doing so. • We shouldn’t leave this discussion without introducing the term structure of interest rates. • The term structure of interest rates is the structure of yields on debt instruments which differ only in their times to maturity.

12. The Yield Curve http://finance.yahoo.com/bonds

13. The Yield Curve

14. Measuring the Term Structure • We have been dealing with “spot rates,” and thinking of the same “spot rate” for each maturity (the same r for all time periods). What do we do knowing they can differ? • The formulas are the same, we just need to include a subscript to denote different maturities. A T year zero: • Knowing the face value and price you can calculate the spot rates or knowing face and the spot rate you can calculate the price of the zero’s with different maturities.

15. Examples • Suppose that a two year zero has a face of $1,000 and a current price of $800. What is the two-year spot rate? • Once we have computed all the spot rates (term structure) we can find the current value of any stream of cash flows. • A government bond that matures in 2 years promises to pay a coupon of 6%. The spot rate for a ½ year cash flow is 6% for a 1 year cash flow is 6.2%, for an 18 month cash flow is 6.4% and for a 2 year cash flow is 6.5%. Find the current value (price) of the bond.

16. Yield to Maturity (or Call) • The yield to maturity is the discount rate that equates the bond’s current price with its stream of promised future cash flows. It is useful in that it is a single descriptive interest rate entirely intrinsic to the bond. • This is the yield that you would receive for the maturity of the bond if you held the bond to maturity (and were able to reinvest the coupon payments at this rate). • Introduces reinvestment risk. • The yield to call is the discount rate that equates the bond’s current price with its stream of promised cash flows until the expected call date (a discussion for advanced classes). • Given two bonds, equivalent in all respects except that one is callable, which bond will have a higher price?

17. YTM – Example • On 9/1/95, PG&E bonds with a maturity date of 3/01/25 and a coupon rate of 7.25% were selling for 92.847% of par, or $928.47 per $1,000 of face value. What is their YTM? • Semiannual coupon payment = 0.0725*1000/2 = $36.25. • Number of semiannual periods to maturity = 30*2 – 1 = 59.

18. YTM – Example cont… • r/2 can only be found by trial and error. However, calculators and spread sheets have algorithms to speed up the search. • Searching reveals that r/2 = 3.939% or a stated annual rate (YTM) of r = 7.878%. • This is an effective annual rate of:

19. Reinvestment and YTM • The yield to maturity as an approximation to the return of bond. It’s calculation assumes you reinvest the coupons at the YTM rather than the prevailing future spot rates. • Take our 2 year 6% coupon bond with a current price of $990.95 as an example. • The spot rates were between 6% and 6.5%. • The yield to maturity is 6.48979% or 3.2449% for 6 months. • Now assume we “invest” the $30 coupons at this rate till the maturity of the bond. The final value at maturity is $30(1.032449)3 + $30(1.032449)2 + $30(1.032449) + $1030 = $1125.968 which is a return on $990.95 equal to the YTM for 2 years compounded semiannually: $990.95(1.032449)4.

20. Common Stock Valuation - Terminology • Dt =dividend per share of stock at time t. • P0=market price of the stock at time 0 (now). • Pt=market price of the stock at time t. (Prior to date t, this would be the expected price.) • g=expected growth rate in dividend payments. • rs=required rate of return (the S is for stock). • [P1 - P0]/P0= capital gain rate during period 1.

21. Common Stock Valuation - Terminology • Dividend yield is the (past) annual dividend over the current price. • PE (price earnings ratio) is the current price divided by the trailing year’s earnings (sum of the last 4 quarters announced earnings). • Get to know how to read the information contained in the WSJ stock quotes, it can be surprisingly informative.

22. Common Stock Valuation • What would you pay for a share of stock today? • To answer this question, ask: why would you buy it? • Suppose you have a one year holding period horizon. • D1 and P1 represent expectations. • We can rearrange this to see that required return is expected dividend yield plus the expected capital gain yield:

23. Common Stock Valuation cont… • What determines P1? • An investor purchasing the stock at time 1 and holding it until time 2 would be willing to pay: • Substitute this into the equation for P0 from the last slide and find:

24. Common Stock Valuation cont… • Repeat this process N times and find: • If we continue to apply the same logic (let N get really big) we find that: • The current market value (price) of a share of stock is the present value of all its expected future dividends.

25. Stock Valuation if Dividends Display Constant Growth (Forever) • If the dividend payments on a stock are expected to grow at a constant rate, g, and the discount rate is rs, then the value of the stock at time 0 is • g must be less than rs for this to be valid. • If g = 0 this collapses to the perpetuity formula. • If g is negative this also works for shrinking dividends. • Labeled the Gordon growth model. • Why would prices change?

26. Example • Geneva Steel just paid a dividend of $2.10. Geneva’s (once per year) dividend payments are expected to grow at a constant rate of 6%. The appropriate discount rate is 12%. What is the current price (or value) of Geneva Stock? • D0 = $2.10  D1 = $2.10(1.06) = $2.226

27. Non-constant Growth in Dividends • Firms often go through lifecycles. • Fast growth. • Growth that matches the economy. • Slower growth or decline. • A super normal growth stock is one that is experiencing rapid growth. But, supernormal growth is, by definition, only temporary.

28. Valuation of Non-constant Growth Stocks • Could just derive all expected dividend payments individually and discount them. Tedious. • Find the present value of the dividends during the period of rapid growth. • Project the stock price at the end of the rapid growth period. This will be the discounted value of the subsequent dividends. Discount this price back to the present. • Add these two present values to find the intrinsic value (price) of the stock.

29. Example • Batesco Inc. just paid a dividend of $1. The dividends of Batesco are expected to grow by 50% next year (time 1) and 25% the year after that (year 2). Subsequently, Batesco’s dividends are expected to grow at 6% in perpetuity. • The proper discount rate for Batesco is 13%. • What is the fair price for a share of Batesco stock?

30. 0 1 2 3 4 g1=50% g2=25% g3=6% g4=6% ...... 1.50 1.875 1.9875 2.107 Example cont… • First, determine the dividends. Draw the timeline! • D0 = $1 g1 = 50% • D1 = $1(1.50) = $1.50 g2 = 25% • D2 = $1.50(1.25) = $1.875 g3 = 6% • D3 = $1.875(1.06) = $1.9875

31. Example cont… • Supernormal growth period: • Constant growth period. Value at time 2: • Discount Pc to time 0 and add to Ps: • What if supernormal growth lasted 5 yrs at 50%?

32. Stocks That Pay No Dividends • If investors value dividends, how much is a stock that pays no dividends worth? • A stock that will literally never pay dividends in any form has a value of zero. • In actuality, a company that has not paid dividends to date can be worth a lot, if the company had good investment opportunities or if it is accumulating assets that can be liquidated and the proceeds eventually distributed. • McDonald’s started in the 1950’s but paid its first dividend in 1975. The market value of McDonald’s stock was in excess of $1 billion prior to 1975. • Anyone familiar with Intel’s history? Or Microsoft’s?

33. Valuing Operations Instead of Dividends • Stocks can be (and often are) valued based on earnings and/or operating cash flows instead of dividends. • Let OCF denote operating cash flow (after taxes and after all accrual and working capital corrections). • Let F denote the net cash flow to the firm from future financings (new debt and equity issues less any debt repaid or equity repurchased). • Let I denote net new capital investment to be undertaken by the firm. • Then using the cash flow identity, future dividends can be written as Dt = OCFt + Ft – It. • So if we can value the firm by discounting future dividends we can also value it by discounting future operating cash flows, financing flows, and requisite capital investments instead of dividends.

34. Valuing Operations cont… • We have been using the present value of the expected future dividends as the current price of the stock. • Given the cash flow identity we can also look at the present value of the right hand side of the equality. • The present value of the first term is the present value of the cash flow that is expected to be generated by the firms existing assets. Since we are concerned with equity, we must remove cash flows assigned to existing debt. • Let PVA denote the present value of all the future cash flows expected to be generated by the firms existing assets. • Let PVL denote the present value of the portion of these expected future cash flows required to service the firm’s existing debt.

35. Valuing Operations cont… • Let NPVGO represent the net present value of the firm’s future investments, I. This is the present value of the operating cash flows the new investments will create less the present value of the cash outflows that will be required to develop them. • Let NPVF represent the net present value of the firm’s future financing transactions, F. This is the present value of the proceeds from future financings less the present value of the resulting obligations --- interest and principal for debt, added dividends for equity (a good starting point is NPVF=0: why?). • If we put it all together:

36. Valuing Operations cont… • The cash flow identity tells us that the discounted dividend model must be equivalent to: P0 = PVA - PVL + NPVGO + NPVF • Equivalent, even though it does not directly involve dividend projections at all! • Observations regarding RWJ’s Chapter 5 discussion: • They assume no future financings. (More generally, NPVF = 0 is usually a very good approximation.) • They assume no existing debt, so PVL = 0. • They assume that existing assets generate cash flow as a perpetuity in the amount of EPS per period. So, PVA = EPS/rs. • So, with their special restrictions, we have: P0 = EPS/rs + NPVGO.

37. Time 0 1 2 3 4 ...... $1 million $1 million $1 million $1 million Xcorp Example • Suppose that Xcorp’s current assets will produce operating cash flows of $1 million per year in perpetuity. The discount rate for Xcorp is 15%. • What is the market value of Xcorp’s equity?

38. 0 1 2 3 4 ...... 0 million 0 million 0 million 1.75 million XCORP Example cont… • Now suppose that Xcorp has an R&D project that will require cash infusions of $1 million in each of the next three years. Subsequently, the project will generate additional cash flow of $0.75 million per year in perpetuity. Xcorp’s net cash flow including the project is: • What is the market value of Xcorp’s equity with the project?

39. 0 1 2 3 4 ...... 1 million 1 million 1 million 1 million 0 1 2 3 4 ...... -1 million -1 million 0.75 million -1 million XCORP Example cont… • Xcorp’s cash flow can be divided up into two pieces: • The cash flow from current assets (we know this value): • Plus the cash flows from the new project:

40. XCORP Example concluded • The NPV of the project at time 0 is: • Xcorp’s value with the project is: • The value of the firm rises by the NPV of the project.

41. The Discounted Dividend and NPV Approaches Are Equivalent • Fresno Corporation has 1 asset and 1 growth opportunity. • The existing asset is a factory which generates operating cash flow (OCF) of $100,000 per year. This will continue for 10 years only, with no salvage value. • The growth opportunity would require an investment of $2 million at t=2, and will return $350,000 to Fresno in each year from t=3 to t=10. • The growth opportunity will be funded by selling $2 million in zero coupon bonds at t=2. The bonds will be repaid at t=10. • Fresno currently has 100,000 equity shares outstanding and pays a dividend of $1 per share annually. • For simplicity, assume that interest/discount rates are zero.

42. Valuing Fresno’s Operations • P0 = PVA - PVL + NPVGO + NPVF • PVL = 0 (No existing debt). • NPVF = 0 (Fair terms on the future financing) • PVA = $100,000x10 = $1,000,000 (no discounting) • NPVGO = $350,000x8 - $2,000,000 = $800,000 • P0 = $1,000,000 + $800,000 = $1,800,000 in total, or $18 per share.

43. Valuing Fresno’s Dividends • Fresno pays $100,000 per year in regular dividends. At t = 10 Fresno will pay off its debt and pay out all remaining cash as a liquidating dividend. • We need to determine the size of the liquidating dividend, and obtain the present value (no discounting) of the dividend stream.

44. Valuing Fresno’s Dividends cont… • Fresno will pay 9 dividends of $100,000 each, plus a liquidating dividend of $900,000. • With a zero discount rate the present value of the dividend stream is $1,800,000, or $18 per share. • The equivalence in the two approaches does not depend on the zero discount rate assumption only on the cash flow identity. • So, use whichever method is easier to implement!