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Lecture 5Lecture 5
How to Value Bonds and Stocks
Valuing Bonds
How to value Bonds
A bondbond 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)
TrF
PV
1
A pure discount bondpure discount bond paying F in T years, when the annual interest rate r in each 1,…,T year will have a value
A discount bonddiscount bond of value PV paying F in T years has spot return
(T -year spot rate)
1
1
T
PV
Fr
A pure discount bondpure 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
TTr
TT
r
FAC
r
F
r
C
r
C
r
CPV
1
1111 2
The value of a level coupon bondlevel 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
A US government bondUS government bond called “13 of November 1999” will have
- a face value of $1000
- an annual coupon of 13% of the face value $ 130
- 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)
Date 96.5 96.11 97.5 97.11 98.5 98.11 99.5 99.11Payment 65 65 65 65 65 65 65 65+1000
95.1096
05.1
100065
05.1
1000
05.1
65
05.1
65
05.1
65
88
05.0
882
A
PV
The cash flows from the bond would be
The value of this bond is
Consoles
r
CPV
ConsolesConsoles are bonds with no maturity date.
The value of a consoleconsole 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.
Value of a bond depends inverselyinversely 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.
ExampleExample
The value of a 5% coupon 2 year bond with annual payments is
The yield to maturity y on this bond solves
06.914
1.1
100050
08.1
502
PV
%95.9 ;
1
100050
1
5006.914 2
yyy
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 r1 = 8% , for year 2 r2 = 10% .
Example (continued)Example (continued)
The value of a 12% coupon 2 year bond with annual payments is
73.1036
1.1
1000120
08.1
1202
PV
The yield to maturity y on this bond solves
%89.9 ;
1
1120
1
12073.1036 2
y
yy
Therefore, higher coupon bonds have lower yield to maturity.
Term Structure of Interest Rates
Recall our earlier example where the one year spot rate r1 = 8% and the annual spot rate (or annual yield to maturity) on a two year zero coupon bond is r2 = 10% .
An individual investing $1 in a 2 year zero coupon bond will receive
210.11$
Notice that 1204.108.11$10.11$ 2
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 fn is a forward rate over n-th year and rn is a n-year spot rate.
An investor in the 2 year bond effectively invest in a 1 year bond at r1 and “locks in” an investment for 1 year at f2. Forward rates for later years can be calculated as :
11
11
1
nn
nn
nr
rf
212
2 111 frr
We can breakdown the 2 year spot rate r2 into one year spot rate r1 and forward rate f2 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.
Initial Price Payment at maturity Interest Rate MaturityBond A $1,000 $1,080 8% 1 yearBond B $1,000 $1,210 10% 2 year
One year spot rate from year1 to year2 Price of Bond B at date1
6% 1210 / 1.06 = 1141.517% 1210 / 1.07 = 1130.84
14% 1210 / 1.14 = 1061.40
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
year2over ratespot expected1
$1210
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
108008.11000
Estimating the Price of a Bond (continued)
Proceeds from the investment II (expected) at date 1
year2over ratespot expected1
1.12041.08$1000
year2over ratespot expected1
1.10$1000 2
If f2 (=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 f2 > expected spot rate over year 2, then the proceed from II is greater than I.
Estimating the Price of a Bond (continued)
Under Expectation hypothesisExpectation hypothesis :
(investors are assumed to be risk-neutral)
f2 = expected spot rate over year2
Under Liquidity Preference hypothesisLiquidity 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 f2 over the second year to be above the spot rate expected over year2.)
f2 > expected spot rate over year2
How to value Stocks
Consider a shareholder who intends to hold a stock for 1 year, earn a dividend D1 and sell the stock for an expected price P1.
Fundamental equation of yieldFundamental equation of yield
dividend + expected capital gain = opportunity cost
rPPPDr
P
r
DP
0011
110
11
If the required return on the stock is r, the price of the stock will be
How to value Stocks (continued)
Note that P1 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 D2 and the price of the stock in period 2, P2.
rP
r
DP
1122
1
22
221
2210
111
111
1
1
r
P
r
D
r
D
r
P
r
D
rr
DP
Substituting P1 into the first Fundamental yield equation gives
How to value Stocks (continued)
The current price of the stock P0 can be obtained by repeating the above process.
ii
i
r
D
r
D
r
D
r
DP
1111 33
221
0
All future dividends Di affect the price P0 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
r
D
r
D
r
D
r
DP
320
111
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
gr
D
r
g
r
g
r
D
r
gD
r
gD
r
DP
2
2
3
2
20
1
1
1
11
1
1
1
1
1
1
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% ?
75$08.012.0
3$
gr
DP
Some Special Cases (continued)
Differential Growth : 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 = 11.61 + 95.33 = 106.94
PV of the expected dividends for the first 8 years = 11.61
33.951.1
36.204
08.010.0
08.4
1.1
188
PV of the expected dividends from 9 year on
Estimating the dividend growth rate g
Consider a firm with a fixed retention ratio
t
t
t
tt
E
D
E
DE
1
Such a firm would have
t
t
t
t
t
t
t
t
E
E
D
D
E
D
E
D 11
1
1
and this in turn gives
t
tt
t
tt
E
EE
D
DD
11
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
ettttt DEEE 11
Then we have
etet
t
tt
t
tetttt
t
tt
E
DE
E
EDEE
E
EEg
11
11
Estimating the dividend growth rate g (continued)
growth rate of earnings (dividends)
= retention ratio * return on retained earnings
1g
use the historical gross return on equity
to approximate the expected gross return at t
t1
et1
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.
NPVGOr
EPSP 0
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, P0 should be EPS/r.
r
EPSP 0
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?
1001.0
100
r
EPSP
NPV of the investment opportunity at date 1-1,000,000 + 210,000/0.1 = 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
P0 = EPS/r + NPVGO = 100 + 10 = 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 ratioPrice-earning ratio (PER)(PER)
P0 / 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