Lecture 5 How to Value Bonds and Stocks

Valuing Bonds How to value Bonds bond A bond is a certificate (contract) showing that a borrower owes a specified sum that will be repaid on a number of specified dates, along with a schedule of interest payments Pure discount bonds (zero coupon bonds) Level coupon bonds US government bonds Consoles

Pure discount bonds (zero coupon bonds) pure discount bond A pure discount bond paying F in T years, when the annual interest rate r in each 1,…,T year will have a value discount bond A discount bond of value PV paying F in T years has spot return (T -year spot rate) pure discount bond A pure discount bond makes one payment (the face value) at a specified date (the maturity date). The face value is also called principal or denomination

Level coupon bonds level coupon bond The value of a level coupon bond with face value F, coupon C and a maturity of T years will be, where r is the annual interest rate Most bonds issued by governments or corporations pay coupons C in addition to a face value F at maturity T

US government bonds US government bond A US government bond called “13 of November 1999” will have - a face value of $ an annual coupon of 13% of the face value $ coupons paid in May and in November $ 65 until November 1999 when the bond is redeemed for $1000 Suppose - it is November 1995, - the stated annual market rate is 10%, and hence the semi annual rate is 5%.

US government bonds (continued) The cash flows from the bond would be The value of this bond is

Consoles Consoles Consoles are bonds with no maturity date. console The value of a console with the coupon C at the interest rate r will be

Relationship between Bond values and Interest rates Determining Yields from Bond Prices: The yield to maturity is the interest rate that equates the PV of the payments on the bond to the current bond price. inversely Value of a bond depends inversely on interest rate r. Coupons reflect interest rates at issue time. Coupon rate is the market interest rate at the issue time. If r falls below the coupon rate, the bond sells at premium. If r rises above the coupon rate, the bond sells at discount.

Example The value of a 5% coupon 2 year bond with annual payments is The yield to maturity y on this bond solves Suppose that the current spot rate on a one year discount bond is 8% the current annual spot rate on a two year zero coupon bond is 10% I.e., market interest rate for year 1 is r 1 = 8%, for year 2 r 2 = 10%.

Example (continued) The value of a 12% coupon 2 year bond with annual payments is The yield to maturity y on this bond solves Therefore, higher coupon bonds have lower yield to maturity.

Term Structure of Interest Rates Recall our earlier example where the one year spot rate r 1 = 8% and the annual spot rate (or annual yield to maturity) on a two year zero coupon bond is r 2 = 10%. An individual investing $1 in a 2 year zero coupon bond will receive Notice that The term structure of interest rates relates the annual spot rates (yields to maturity) on zero-coupon government bonds to their terms to maturity.

Term Structure of Interest Rates (continued ) where f n is a forward rate over n-th year and r n is a n-year spot rate. An investor in the 2 year bond effectively invest in a 1 year bond at r 1 and “locks in” an investment for 1 year at f 2. Forward rates for later years can be calculated as : We can breakdown the 2 year spot rate r 2 into one year spot rate r 1 and forward rate f 2 for next year. More formally,

Estimating the Price of a Bond at a future date One year spot rate from year1 to year2 is unknown at date 0.

Estimating the Price of a Bond (continued) The price of Bond B at date 1 is unknown at date 0. Thus we consider expected value of Bond B at date 1, which is given by Now consider the following investment strategies at date 0. I : Buy a 1 year bond at date 0 II : Buy a 2 year bond at date 0 and sell it at date 1 Proceeds from the investment I at date 1 is

Estimating the Price of a Bond (continued) Proceeds from the investment II (expected) at date 1 If f 2 (=12.04%)= expected spot rate over year2, then I and II give the same proceed at date 1. So, the investors should be indifferent. If f 2 > expected spot rate over year 2, then the proceed from II is greater than I.

Estimating the Price of a Bond (continued) Expectation hypothesis Under Expectation hypothesis : (investors are assumed to be risk-neutral) f 2 = expected spot rate over year2 Liquidity Preference hypothesis Under Liquidity Preference hypothesis : (investors are assumed to be risk-averse : in order to induce risk averse investors to hold the riskier two year bonds, the market sets the forward rate f 2 over the second year to be above the spot rate expected over year2.) f 2 > expected spot rate over year2

How to value Stocks Consider a shareholder who intends to hold a stock for 1 year, earn a dividend D 1 and sell the stock for an expected price P 1. Fundamental equation of yield dividend + expected capital gain = opportunity cost If the required return on the stock is r, the price of the stock will be

How to value Stocks (continued) Note that P 1 is unknown now, and consequently we need to use its expected value, which can be computed if we know expected values of the dividend in 2 periods D 2 and the price of the stock in period 2, P 2. Substituting P 1 into the first Fundamental yield equation gives

How to value Stocks (continued) The current price of the stock P 0 can be obtained by repeating the above process. All future dividends D i affect the price P 0 even if the investor’s investment horizon is only one year.

Some Special Cases Zero Growth : Zero Growth : the share price of a stock that pays fixed dividend D in perpetuity should be For example, preferred stocks

Some Special Cases (continued) Constant Growth : Constant Growth : if the dividends are expected to grow at the constant rate g, then WW is expected to pay per-share dividend of $3 next year, growing at 8% forever. What is the price of the WW stock if the required return is 12% ?

Some Special Cases (continued) Differential Growth : A stock has just paid a dividend of $1, which is expected to grow at 20% for 5 years, 15% for 3 years, and then 8% for all future periods. Suppose the discount rate is 10%. Current stock price = = PV of the expected dividends for the first 8 years = PV of the expected dividends from 9 year on

Estimating the dividend growth rate g Consider a firm with a fixed retention ratio Such a firm would have and this in turn gives

Estimating the dividend growth rate g (continued) Earning next year = earning this year + increase in earning increase in earning = retained earning * expected gross return on retained earning at t Now, notice that Then we have

Estimating the dividend growth rate g (continued) growth rate of earnings (dividends) = retention ratio * return on retained earnings use the historical gross return on equity to approximate the expected gross return at t

Growth Opportunities Consider a company with a constant stream of earnings in perpetuity. Now suppose the dividend at date 1 is retained and invested in an investment project. The share value should increase by the NPV of the “growth opportunity”(NPVGO) induced by the investment project. If the firm pays all these earnings out as dividends to shareholders, then at all dates, earnings per share = EPS = d = dividends per share The share value at date 0, P 0 should be EPS/r.

Growth Opportunities (continued) Example : Example : Sam shipping with 100,000 shares outstanding expects to earn $1,000,000 per year in perpetuity, if it distributes all its earnings to shareholders. Suppose the appropriate discount rate r = 10%. Then The firm finds an investment opportunity that will cost $1 million at date 1, but will increase earnings in every subsequent period by $210,000. If the firm decides to retain the earning at date 1 and invest in the project, what is the share price? NPV of the investment opportunity at date 1 -1,000, ,000/0.1 = 1,100,000 NPV at date 0 1,100,000/1.1 = 1,000,000 or 10 per share

Growth Opportunities (continued) The share price with the investment project P 0 = EPS/r + NPVGO = = 110 The above share prices can be obtained from calculating PV’s of the future earnings with or without the investment opportunity.

Growth Opportunities (continued) Price-earning ratio(PER) Price-earning ratio (PER) P 0 / EPS = 1 / r + NPVGO / EPS PER depends positively on the growth opportunities. Hence, the stocks of firms retaining earnings to invest in growth opportunities do have higher PER. PER depends negatively on the discount rate r. Firms with risky earnings will therefore have lower PER. Reported accounting earnings are used. Conservative accounting rules leads to higher PER’s. For instances, Japanese firms PER’s