Bonds & Stocks and Their Valuation
• • • • Key features of bonds Bond valuation, yield & risk Features of common stock Determining common & preferred stock values
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Learning Objectives
• Understand typical features of bonds & stocks • Learn how information about bonds & stocks is reported. • Identify the main factors that affect the value of these securities. • Learn how to value these securities. • Understand how changes in the underlying factors affect the value of these securities.
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What is a bond?
• A long-term debt instrument in which a borrower agrees to make payments of principal and interest, on specific dates, to the holders of the bond.
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Bond markets
• Primarily traded in the over-thecounter (OTC) market. • Most bonds are owned by and traded among large financial institutions. • Full information on bond trades in the OTC market is not published, but a representative group of bonds is listed and traded on the bond division of the NYSE.
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Key Features of a Bond
• Par value – face amount of the bond, which is paid at maturity (assume $1,000). • Coupon interest rate – stated interest rate (generally fixed) paid by the issuer. Multiply by par to get dollar payment of interest. • Maturity date – years until the bond must be repaid. • Issue date – when the bond was issued. • Yield to maturity - rate of return earned on a bond held until maturity (also called the “promised yield”).
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Other types (features) of bonds
• Convertible bond – may be exchanged for common stock of the firm, at the holder’s option. • Warrant – long-term option to buy a stated number of shares of common stock at a specified price. • Putable bond – allows holder to sell the bond back to the company prior to maturity. • Income bond – pays interest only when income is earned by the firm. • Indexed bond – interest rate paid is based upon the rate of inflation.
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Bond Price Quotations
Bonds CaterpInc 93/8 01 Chryslr 10.95s 17 Citicp 6½ 04 ClevEl 8¾ 05 Coca-Cola 5¾ 05 CrayRs 61/8 11 Cur Vol Yld 8.3 9.8 6.5 8.7 7.3 cv 30 37 2 10 49 31 Close 112½ 111½ 99½ 1011/8 78¼ 79½ Net Chg - 1½ … + 1/8 - 5/8 +½ -¾
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Discounted Cashflow Valuation: Basis for Approach
t = n CF t Value = t t = 1 (1 + k)
where, • n = Life of the asset • CFt = Cashflow in period t • k = Discount rate reflecting the riskiness of the estimated cashflows
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Bond Valuation
• The bond’s fair value is the present value of the promised future coupon and principal payments. • At issue, the coupon rate is set such that the fair value of the bonds is very close to its par value. • Later, as market conditions change, the fair value may deviate from the par value.
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Example on Bond Valuation
Find the fair value of a bond with a $1,000 par value, a remaining life of 12 years, and a coupon rate of 9% per year paid semi-annually. Assume that at the present time, the required rate of return on the bond is 6% per year.
Semi - annual coupon payment coupon rate par value 2 0.09 $1,000 $45 2
Number of payments = 12 × 2 = 24 Semiannual required rate of return = 3%
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Example on Bond Valuation
Bond Value: B0 = PV(coupon payments) + PV(par value)
$45 $45 $45 $1,045 B0 1 2 3 (1.03) (1.03) (1.03) (1.03) 24
CPN (1 r / 2) 2 n 1 $1,000 B0 (r / 2)(1 r / 2) 2 n (1 r / 2) 2 n 2
B0 = $1,254.03
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Example on Bond Valuation
Find the fair value of a bond with a $1,000 par value, a remaining life of 12 years, and a coupon rate of 9% per year paid semi-annually. Assume that the required rate of return on the bond is 6% per year.
B (4.5% $1000) PVIFA3%, 24 $45 16.9355 $1,254 $1000 PVIF3%, 24 $1000 0.4919
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Bond Values and Required Rates of Return
Required Rate of Return 6.0 % 9.0 % 12.0 %
Bond Value $ 1,254.03 $ 1,000.00 $ 811.74
premium bond par bond discount bond
Coupon rate = 9% per year.
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Current Yield of a Bond
Annual Coupon Interest Current Yield Current Bond Price
• Assume the 9% coupon bond in the previous example is selling for $1076.23. • Its current yield is then $90/$1076.23 = 8.36% • The current yield ignores gain (or loss) resulting from the difference between the purchase price and the par value.
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Yield To Maturity (YTM)
• The Yield to Maturity is the APR (Annual Percentage Rate) that equates the bond’s market price to the present value of its promised future cash flows. • Assumes promised payments will be made in full and when promised.
FV P CPN n YTM FV P 2
CPN = annual coupon
FV = face (par) value P = current price
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Example on YTM of a Bond
Find the fair Yield to Maturity (YTM) of a bond with a $1,000 par value, a remaining life of 12 years, and a coupon rate of 9% per year paid semi-annually. The bond is currently selling for $1076.23.
FV P CPN n YTM FV P 2 $1000 $1076.23 $90 12 $1000 $1076.23 2 0.0806 8.06%
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A Comparison of Bond Returns
Bond Terms: Par Value = $1,000; Remaining life = 12 years Coupon rate = 9% / year paid semi-annually. Bond Price Current Yield 7.18 % 9.00 % 11.09 % Yield to Maturity 6.00 % 9.00 % 12.00 % Annual Percentage Yield 6.09 % 9.20 % 12.36 %
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$1,254.03 $1,000.00 $ 811.74
Bond values over time
• At maturity, the value of any bond must equal its par value. • If kd remains constant: • The value of a premium bond would decrease over time, until it reached $1,000. • The value of a discount bond would increase over time, until it reached $1,000. • A value of a par bond stays at $1,000.
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The price path of a bond
What would happen to the value of this bond if its required rate of return remained at 10%, or at 13%, or at 7% until maturity?
VB
1,372 1,211 1,000 837 775
30 25 20 15 10 kd = 10%.
kd = 7%.
kd = 13%. 5 0 Years to Maturity
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Bond Riskiness
• The YTM is the bond’s promised return. • But what if the bond issuer defaults? • Another source of risk lies with changing interest rates. • As the interest rate rises, the price of a fixed-coupon bond falls.
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Interest Rate Risk
• How does the value of a bond change as interest rates rise?
Bond values are inversely related to interest
rates.
• Changes in bond values as interest rates change is known as interest rate risk. • How much interest rate risk does a bond have?
It depends on the maturity of the bond.
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What is interest rate (or price) risk?
• Interest rate risk is the concern that rising kd will cause the value of a bond to fall.
% change 1 yr +4.8% $1,048 $1,000 -4.4% $956 kd 5% 10% 15% 10yr % change $1,386 +38.6% $1,000 $749 -25.1%
The 10-year bond is more sensitive to interest rate changes, and hence has more interest rate risk.
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Contrasting Short Term and Long Term Bonds
Price Changes as a function of Bond Maturities 20.00% 15.00%
% Change in Price
10.00% 5.00% 0.00% -5.00% -10.00% -15.00% 1 5 15 30 Bond Maturity
% Change if rate drops to 7% % Change if rate increases to 10%
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Bond Pricing Proposition 1
• The longer the maturity of a bond, the more sensitive it is to changes in interest rates.
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Contrasting Low-coupon and Highcoupon Bonds
Bond Price Changes as a function of Coupon Rates 25.00% 20.00% 15.00%
% Price Change
10.00% 5.00% 0.00% -5.00% -10.00% -15.00% -20.00% 0% 5% 10.75% 12% Coupon Rate
% Change if rate drops to 7% % Change if rate increases to 10%
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Bond Pricing Proposition 2
• The lower the coupon rate on the bond, the more sensitive it is to changes in interest rates.
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What is reinvestment rate risk?
• Reinvestment rate risk is the concern that kd will fall, and future CFs will have to be reinvested at lower rates, hence reducing income.
EXAMPLE: Suppose you just won $500,000 playing the lottery. You intend to invest the money and live off the interest.
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Reinvestment rate risk example
• You may invest in either a 10-year bond or a series of ten 1-year bonds. Both 10-year and 1-year bonds currently yield 10%. • If you choose the 1-year bond strategy: • After Year 1, you receive $50,000 in income and have $500,000 to reinvest. But, if 1-year rates fall to 3%, your annual income would fall to $15,000. • If you choose the 10-year bond strategy: • You can lock in a 10% interest rate, and $50,000 annual income.
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Conclusions about interest rate and reinvestment rate risk
Short-term AND/OR High coupon bonds Interest rate risk Reinvestment rate risk Low High
Long-term AND/OR Low coupon bonds High Low
• CONCLUSION: Nothing is riskless!
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Evaluating default risk: Bond ratings
Investment Grade Moody’s S&P Junk Bonds
Aaa
Aa A Baa
Ba B Caa C BB B CCC D
AAA AA A BBB
• Bond ratings are designed to reflect the probability of a bond issue going into default.
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Bond Values and Call Provisions
• Call Provision allows the issuer to pay off the bonds prior to maturity. • When bonds are called by the issuer,
they are purchased from the holder at the call
price.
the bonds are then retired. • The Yield-to-Call (YTC) is the bond’s expected return up to the call date.
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Yield to Call of a Bond
• Consider the 9% coupon, 12 year bond. • Assume that the bond is currently priced to yield 6% to maturity. • The bond price is $1,254.03.
CALCULATOR SOLUTION
Data Input
24 6 –1,254.03 45
Function Key
N I PV PMT
1,000
FV
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Yield to Call of a Bond
• If market rates do not change, five years after issue, the bond would sell for its fair value of $1,169.44.
CALCULATOR SOLUTION Data Input 14 6 –1,169.44 45 1,000 Function Key N I PV PMT FV
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Yield to Call of a Bond
• The rate of return from this investment is 6%.
CALCULATOR SOLUTION Data Input 10 Function Key N I PV PMT FV
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6 –1,254.03
45 1,169.44
Yield to Call of a Bond
• Now assume that the bond is callable five years after you purchase it. • The call price is $1,090.00. • For the call to be in-the-money, the call price must be less than the value that the bond would have if it were not callable (i.e., $1,169.44). • Thus, if the bond is called, your rate of return will be less than the yield to maturity.
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Yield to Call of a Bond
• The call price is $1,090.00. • The rate of return from this investment is 4.88%. • r = YTC = 4.88% • YTC < YTM
CV P nc YTC CV P 2 $1090 $1254.03 $90 5 $1090 $1254.03 2 0.0488 4.88% CPN
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Zero Coupon Bonds
• These do not pay any coupon interest. • The par value is returned to the bondholder at maturity. • These bonds are also known as: pure-discount bonds deep-discount bonds zeroes
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Valuing Zero Coupon Bonds
The required return on a 12 year zero coupon bond with a par value of $1,000 is 9%. What is the bond’s value today?
CALCULATOR SOLUTION
Data Input
12 9 –355.53
Function Key
N I PV PMT FV
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0
1,000
Features of Preferred Stock
• Claims of preferred stock holders are junior to claims of debt holders, but senior to those of common stock holders. • Limited voting rights compared to common stock. • Stock has a par value and a dividend rate. • Failure to pay the dividend does not force the issuing firm into bankruptcy.
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Preferred Stock Valuation
Value = PV of dividends
Consider a $100 par value preferred stock share with an 8% dividend rate (paid quarterly), and a 15 year life. Stockholders require a 12% rate of return. Find the fair value of each share today.
+ PV of par value
CALCULATOR SOLUTION Data Input 60 Function Key N I
12 –72.32
2
PV
PMT FV
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100
Preferred Stock Valuation
If the stock was perpetual, the value of the stock would be
$2 P0 $66.67 .12 / 4
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Features of Common Stock
• Represents residual ownership of the firm. • Common stockholders have important voting rights. • The issuer may pay dividends to common stockholders. However, it is not required to do so. Moreover, there is no pre-set dividend rate.
Future dividends are uncertain. We need a way to forecast future dividends.
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Obtaining Common Stock Information
• You can look in a newspaper such as The Wall Street Journal and find pages and pages of New York Stock Exchange (NYSE) quotes. • You can find out more about these stocks by looking in a stock and bond guide such as Standard & Poor’s. • There are hundreds of sites online, too.
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Stock Price Quotation
52-Weeks Hi Lo Stock Symbol Div Yld (%)
52 weeks Yld Net 85¼ 56 ¾ CocaCola Vol KO 1.00 1.2 Hi Lo Sym Div % PE 100s Hi Lo Close Chg 134 80 IBM .52 .5 21 143402 98 95 9549 -3 115 40 MSFT … 29 558918 55 52 5194 -475
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Fair Value of Common Stock Shares
• The fair value depends on only the expected future cash dividends on the stock. • The future selling price is not needed since this price will in turn depend on subsequent dividends.
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Dividend growth model
• Value of a stock is the present value of the future dividends expected to be generated by the stock.
D3 D1 D2 D P0 ... 1 2 3 (1 k s ) (1 k s ) (1 k s ) (1 k s )
^
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Constant growth stock
• A stock whose dividends are expected to grow forever at a constant rate, g.
D1 = D0 (1+g)1 D2 = D0 (1+g)2 Dt = D0 (1+g)t
• If g is constant, the dividend growth formula converges to:
D0 (1 g) D1 P0 ks - g ks - g
^
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Future dividends and their present values
$
D t D0 ( 1 g )
t
0.25
Dt PVDt ( 1 k )t
P0 PVDt
0 Years (t)
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What happens if g > ks?
• If g > ks, the constant growth formula leads to a negative stock price, which does not make sense. • The constant growth model can only be used if:
• ks > g • g is expected to be constant forever
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If D0 = $2, g is a constant 6%, and ks is 13%, find the expected dividend stream for the next 3 years, and their PVs.
0
g = 6%
1 2.12
ks = 13%
2 2.247
3 2.382
D0 = 2.00 1.8761 1.7599 1.6509
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What is the stock’s market value?
• Using the constant growth model:
D1 $2.12 P0 k s - g 0.13 - 0.06 $2.12 0.07 $30.29
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What is the expected market price of the stock, one year from now?
• D1 will have been paid out already. So, P1 is the present value (as of year 1) of D2, D3, D4, etc.
D2 $2.247 P1 k s - g 0.13 - 0.06 $32.10
^
• Could also find expected P1 as:
P1 P0 (1.06) $32.10
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^
What is the expected dividend yield, capital gains yield, and total return during the first year?
• Dividend yield • Capital gains yield • Total return (ks)
= D1 / P0 = $2.12 / $30.29 = 7.0%
= (P1 – P0) / P0 = ($32.10 - $30.29) / $30.29 = 6.0% = Dividend Yield + Capital Gains Yield = 7.0% + 6.0% = 13.0%
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What would the expected price today be, if g = 0?
• The dividend stream would be a perpetuity.
0 1 2 3
ks = 13%
...
2.00 2.00 2.00
PMT $2.00 P0 $15.38 k 0.13
^
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Supernormal growth: What if g = 30% for 3 years before achieving long-run growth of 6%?
• Can no longer use just the constant growth model to find stock value. • However, the growth does become constant after 3 years.
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Valuing common stock with nonconstant growth
0 k = 13% 1 s
g = 30% g = 30%
2
g = 30%
3
g = 6%
4
...
D0 = 2.00 2.301 2.647
2.600
3.380
4.394
4.658
3.045
46.114 54.107 = P0
^
$ P3
4.658
0.13 0.06
$66.54
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Find expected dividend and capital gains yields during the first and fourth years. • Dividend yield (first year)
= $2.60 / $54.11 = 4.81%
• Capital gains yield (first year)
= 13.00% - 4.81% = 8.19%
• During nonconstant growth, dividend yield and capital gains yield are not constant, and capital gains yield ≠ g. • After t = 3, the stock has constant growth and dividend yield = 7%, while capital gains yield = 6%.
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Nonconstant growth: What if g = 0% for 3 years before long-run growth of 6%?
0 k = 13% 1 s
g = 0% g = 0%
2
g = 0%
3
g = 6%
4
...
D0 = 2.00 1.77 1.57
2.00
2.00
2.00
2.12
1.39
20.99 25.72 = P0
^
$ P3
2.12
0.13 0.06
$30.29
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Find expected dividend and capital gains yields during the first and fourth years.
• Dividend yield (first year)
= $2.00 / $25.72 = 7.78%
• Capital gains yield (first year)
= 13.00% - 7.78% = 5.22%
• After t = 3, the stock has constant growth and dividend yield = 7%, while capital gains yield = 6%.
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If the stock was expected to have negative growth (g = -6%), would anyone buy the stock, and what is its value? • The firm still has earnings and pays dividends, even though they may be declining, they still have value.
D0 ( 1 g ) D1 P0 ks - g ks - g
^
$2.00 (0.94) $1.88 $9.89 0.13 - (-0.06) 0.19
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Find expected annual dividend and capital gains yields.
• Capital gains yield
= g = -6.00%
• Dividend yield
= 13.00% - (-6.00%) = 19.00%
• Since the stock is experiencing constant growth, dividend yield and capital gains yield are constant. Dividend yield is sufficiently large (19%) to offset a negative capital gains.
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Some Final Comments on Security Valuation
• Mathematical models of security valuation rely on estimates of various parameters. • The estimated value is only as good as the quality of the input parameters. • In an efficient market, the market price is a good estimate of the security’s value.
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