# 1. Present Value. 2. Bonds. 3. Stocks

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1 Stocks and Bonds

2 1. Present Value 2. Bonds 3. Stocks

3 1 Present Value = today s value of income at a future date Income at one future date value today of X dollars in one year V t = X t+1 (1 + i t ) where i t is the nominal interest rate on assets held during period t value today of X dollars in two years V t = X t+2 (1 + i t ) ( 1 + i e t+1 ) where i e t+1 is the interest rate expected to prevail in one year

4 Stream of income in the future, z t today, z t+1 in one year, z t+2 in two years and z t+3 in three years known stream z t+2 z t+3 V t = z t + z t i t (1 + i t ) ( 1 + i e )+ t+1 (1 + i t ) ( 1 + i e ) ( ) t i e t+2 unknown stream - replace known values of the future income with expectations z e t+2 z e t+3 V t = z t + ze t i t (1 + i t ) ( 1 + i e )+ t+1 (1 + i t ) ( 1 + i e ) ( ) t i e t+2

5 stream of income with interest rates constant V t = z t + ze t i t + ze t+2 (1 + i t ) 2 + ze t+3 (1 + i t ) 3 constant interest rates and constant payments of iz (interest multiplied by principle) forever beginning one year from now ( ( ) ( ) 1 3 V t = iz 1 + i ) = z 1 + i 1 + i note that the value of paying interest on debt forever (present-value sum) and the value of paying off the debt today (z) are equal!

6 Real stream of income - use real interest to take present-value Divide both sides of PV equation by current price to express real present value V t P t = z t P t + ze t i t 1 P t + z e t+3 z e t+2 (1 + i t ) ( 1 + i e t+1) 1 P t + (1 + i t ) ( 1 + i e ( ) t+1) i e t+2 P t Note that (1 + i t ) P t P e t+1 = 1 + r

7 Using this, real present value is stated as: V t P t = z t P t + ze t r t 1 + Pt+1 e zt+3 e z e t+2 (1 + r t ) (1 + r t+1 ) + (1 + r t ) (1 + r t+1 ) (1 + r t+2 ) 1 P e t+3 1 P e t+2 Definitions discount rate = i discount factor = 1 1+i

8 2 Bonds 2.1 Definitions Maturity - the length of time the bond promises payments to its holder Coupon bonds - bonds that promise multiple payments before maturity and one payment at maturity Discount bonds - bonds that promise a single payment at maturity Face value - the payment at maturity

9 2.2 Price of a discount bond Assume the bond will pay \$50 at the end of two years and that the interest rate is 4% and is expected to rise to 4.5% after one year. The price of the bond is the present value of that final payment. P 2t = \$50 (1 + i t ) ( 1 + i e t+1 ) = \$50 (1.04) (1.045) = \$46.01 Arbitrage - Will the investor prefer the two year bond above or two one year bonds? For every dollar you put in a two year bond, get 1 P two year bonds. 2t At the end of the first year, now you have P 1 one year bonds valued 2t at P e 1t+1 for a total value of P e 1t+1 P 2t.

10 For every dollar you put in a one year bond, get 1 + i t. Which do you prefer? or equivalently P e 1t+1 P 2t = 1 + i t P 2t = P e 1t i t The price of a two year bond must be the present-value of the expected price of the one-year bond. Note this ignores risk. Which option has risk in the first period?

11 Yield to maturity on an n-year bond or equivalently the n-year interest rate is defined as that constant annual interest rate that makes the bond price today equal to the present value of future payments on the bond. What is the yield to maturity on the two year bond above? solving for i 2t yields 4.25%. \$46.01 = \$50 (1 + i 2t ) 2 Note that \$50 (1 + i t ) ( 1 + i e t+1 ) = \$50 (1 + i 2t ) 2 Rearranging yields 1 + 2i 2t + i 2 2t = 1 + i t + i e t+1 + i ti e t+1

12 i 2t 1 2 ( it + i e t+1 ) Long-term rates are the average of expected future short-term rates (ignoring risk) Why are long-term interest rates currently so low? Risk - Since the two-year bond has a risky value in period 1 (why?) the alternatives of a two-year bond and two one year bond are not equally attractive if you don t like risk. Therefore, the two-year bond alternative must pay a slightly higher return, given by the risk premium (rp). P e 1t+1 P 2t = 1 + i t + (rp) t

13 performing the same steps as before, we find that with risk: i 2t 1 ( it + i e t rp ) t With risk, long-term rates are a little above the average of expected future short-term rates. Yield curve = interest rates on n-period bonds as n (maturity) increases. Due to risk, the yield curve generally slopes upward, implying that longer term bonds have higher interest rates. Under what circumstances might we have an inverted (slopes downward) yield curve?

14 3 Stock Prices The value of a stock (and hence its price) is determined by the present value of its dividends. The ex-dividend price (after the current dividend has been paid) is therefore: Q t = D e t i t + rp + D e t+2 (1 + i t + rp) ( 1 + i e t+1 + rp) D e t+3 + (1 + i t + rp) ( 1 + i e t+1 + ( rp) 1 + i e t rp) This expression is based on the fundamental value of the stock (the determinates of its true value).

15 Dividends tend to move with profits, implying that higher expected future profits raise a stock s price. Higher interest rates reduce the present-value of the stream of dividends, thereby reducing the stock s price. An alternative expression: Note that the price expected to prevail in one period is given by: Q e t+1 = D e t+2 ( 1 + i e t+1 + rp ) + D e t+3 ( 1 + i e t+1 + rp ) ( 1 + i e t+2 + rp) +...

16 Substituting this above yields: Q t = De t i t + rp + Q e t+1 (1 + i t + rp) Rational speculative bubbles - occur when asset prices are driven from fundamental values by expectations of every-rising asset prices

17 4 Summary The price of an asset is the present-value of the assets future payments Payment of interest on a bond forever equals the value of the bond Long-term interest rates are approximately equal to the average of expected future short-term rates plus a risk premium The yield curve typically slopes upward due to the risk premium on longterm assets

18 The price of a stock is the expected present value of future dividends Higher profits raise a stock s price Higher interest rates reduce a stock s price

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Bond Pricing - 1 Chapter 11 Several Assumptions: To simplify the analysis, we make the following assumptions. 1. The coupon payments are made every six months. 2. The next coupon payment for the bond is