Valuation of Stocks and Bonds
So far, we laid the foundations: Net Present Value (NPV) rule. How to find cash flows (numerator). How to discount them at the cost of capital (denominator).
Now we’re ready to jump into interesting applications!
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Valuation of Bonds and Stock
The Most Basic Formula in Finance:
Value of financial securities = PV of expected future cash flows
C1 C2 CN NPV C0 2 (1 r ) (1 r ) (1 r ) N N Ct NPV C0 (1 r ) t t 1
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Getting started…
To value securities or projects we Estimate future cash flows:
Size (how much) and Timing (when)
Discount future cash flows at the appropriate rate:
The rate should be appropriate to the risk presented by the security. It is the cost of capital.
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This is most easily done for bonds…
What is a bond?
A bond is a legally binding agreement between a borrower and a lender: Specifies the principal amount of the loan. Specifies the size and timing of the cash flows So we know all the necessary inputs for valuation.
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Example: A U.S. Treasury Bond
Consider a U.S. government bond listed as 6 3/8 of December 2009.
The Par Value of the bond is $1,000. Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). Since the coupon rate is 6 3/8 the payment is $31.875. As of January 1, 2002 the cash flows are:
$31.875 $31.875
$31.875
$1,031.875
12 / 31 / 09
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1 / 1 / 02 6 / 30 / 02
12 / 31 / 02
6 / 30 / 09
How to Value Bonds…
Identify the size and timing of cash flows. Discount at the correct discount rate.
If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate.
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Pure Discount (a.k.a Zero Coupon) Bonds
What’s needed for valuing pure discount bonds?
Time to maturity (T) = Maturity date - today’s date Face value (F) Discount rate (r)
$0
0
$0
$0
$F
2 T T 1 Present value of a pure discount bond at time 0:
1
F PV T (1 r )
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Example of a Pure Discount Bond:
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.
$0
0
$0
$0
$1,000
1
2
29
30
F $1,000 PV $174.11 T 30 (1 r ) (1.06)
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Level-Coupon Bonds
Information needed to value level-coupon bonds:
Coupon payment dates and time to maturity (T) Coupon payment (C) per period and Face value (F) Discount rate
$C
0
$C
$C
$C $F
2 T T 1 Value = PV of coupon payment annuity + PV of face value
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C 1 F PV 1 T T r (1 r ) (1 r )
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Example of a Level-Coupon Bond:
Find the present value (as of January 1, 2002), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent. On January 1, 2002 the size and timing of cash flows are:
$31.875 $31.875
$31.875
$1,031.875
12 / 31 / 09
1 / 1 / 02 6 / 30 / 02
12 / 31 / 02
6 / 30 / 09
$1,000 $31.875 1 PV 1 (1.025)16 (1.025)16 $1,089.78 .05 2 9
YTM and Bond Value
Bond Value
1300
When the YTM < coupon, the bond trades at a premium.
1200
1100
When the YTM = coupon, the bond trades at par.
1000
800 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 6 3/8 0.08 0.09 0.1
Discount Rate
When the YTM > coupon, the bond trades at a discount.10
Maturity and Bond Price Volatility
Bond Value
Consider two otherwise identical bonds.
The long-maturity bond will have much more volatility with respect to changes in the discount rate
Par
Short Maturity Bond
C
Discount Rate Long Maturity 11 Bond
Coupon Rate and Bond Price Volatility
Bond Value
Consider two otherwise identical bonds.
The low-coupon bond will have much more volatility with respect to changes in the discount rate
High Coupon Bond Discount Rate Low Coupon Bond
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The Behavior of Bond Prices – Overview:
1. 2.
3.
4.
Bond prices and market interest rates move in opposite directions. When coupon rate = YTM, price = par value. When coupon rate > YTM, price > par value (premium bond) When coupon rate < YTM, price < par value (discount bond) A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical.
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The Value of Common Stocks
The value of a stock equals the discounted stream of cash flows to shareholders Finance distinguishes Growth Stocks and Value Stocks (low growth, “cash cows”). We will look at different Types of Stocks Zero Growth Constant Growth Differential Growth In many cases, use “value by spreadsheet”.
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Case 1: Zero Growth
Assume that dividends will remain at the same level forever
Div 1 Div 2 Div 3
Since future cash flows are constant, the value of a
zero growth stock is the present value of a perpetuity:
Div 3 Div 1 Div 2 P0 1 2 3 (1 r ) (1 r ) (1 r ) Div P0 r
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Case 2: Constant Growth
Assume dividends will grow at a constant rate, g:
Div 1 Div 0 (1 g )
Div 2 Div 1 (1 g ) Div 0 (1 g ) 2 Div 3 Div 2 (1 .g ) Div 0 (1 g )3
. .
Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:
Div 1 P0 rg
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Case 3: Differential Growth
Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter. To value a Differential Growth Stock, we need to:
Estimate future dividends in the foreseeable future. Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2). Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate. 17
Case 3: Differential Growth
Assume that dividends will grow at rate g1
for N years and grow at rate g2 thereafter
Div 1 Div 0 (1 g1 )
Div 2 Div 1 (1 g1 ) Div 0 (1 g1 ) 2
Div N Div N 1 (1 g1 ) Div 0 (1 g1 ) N Div N 1 Div N (1 g 2 ) Div 0 (1 g1 ) N (1 g 2 )
. . .
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. . .
Case 3: Differential Growth
Dividends will grow at rate g1 for N years
and grow at rate g2 thereafter
Div 0 (1 g1 ) Div 0 (1 g1 ) 2
…
0 1 2
Div 0 (1 g1 ) N
Div N (1 g 2 ) Div 0 (1 g1 ) N (1 g 2 )
…
N
…
N+1
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Case 3: Differential Growth
We can value this as the sum of: an N-year annuity growing at rate g1
plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1
(1 g1 )T C PA 1 T r g1 (1 r )
Div N 1 rg 2 PB N (1 r )
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Case 3: Differential Growth
To value a Differential Growth Stock, we can use
Div N 1 C (1 g1 )T r g 2 P 1 T N r g1 (1 r ) (1 r )
Or we can cash flow it out, i.e. use a spreadsheet.
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What’s realistic?
Mature firms and mature industries are expected to grow with the overall economy at a rate of g = inflation + real economic growth where each of the two terms could be 1-3%. High growth firms grow at higher rates. However, this cannot be sustained forever. After T years of growth, the firm becomes mature – use differential growth formula.
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Differential Growth – Important Example
A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. What is the stock worth?
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With the Formula
Div N 1 rg T C (1 g1 ) 2 P 1 T r g1 (1 r ) (1 r ) N
$2(1.08)3 (1.04) 3 .12 .04 $2 (1.08) (1.08) P 1 3 .12 .08 (1.12) (1.12) 3
$32.75 P $54 1 .8966 3
(1.12)
P $5.58 $23.31
P $28.89
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A Differential Growth Example ctd.
$2(1.08)
0 1
$2(1.08)
2
2
$2(1.08)3 $2(1.08)3 (1.04)
…
3 4
The constant growth phase beginning in year 4 can be valued as a growing perpetuity at time 3.
$2.16
0 1
$2.33
2
$2.62 $2.52 .08
3
$2.16 $2.33 $2.52 $32.75 P0 $28.89 2 3 1.12 (1.12) (1.12)
$2.62 P3 $32.75 .08
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Estimates of Parameters in the Dividend-Discount Model
The value of a firm depends upon its growth rate, g, and its discount rate, r.
Where does g come from? Where does r come from?
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Formula for Firm’s Growth Rate “g”
g = Retention ratio × Return on retained
earnings
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Where does “r” come from?
The discount rate can be broken into two parts.
The dividend yield The growth rate (in dividends)
In practice, r is only estimated with error.
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Accounting for Growth Opportunities
Growth opportunities are opportunities to invest in positive NPV projects. The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends and the net present value of the growth opportunities. EPS1 P PVGO 29 r
The Value Driver Formula
EPS1 P PVGO r
is also know as value driver formula. • The first part represents the value of the firm deriving form operations already in place. • The second part represents the value of opportunities for future growth. • The bigger term is the driver of share value.
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The Dividend Growth Model and the Value Driver (PVGO) Model
We have two ways to value a stock:
The dividend discount model. The value driver formula: The price of a share of stock can be calculated as the sum of its price as a cash cow plus the per-share value of its growth opportunities.
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The Dividend Growth Model and the Value Driver (PVGO) Model
Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30%, a discount rate of 16-percent, and a return on retained earnings of 20-percent.
The dividend at year one will be $5 × .30 = $1.50 The retention ratio is .70 ( = 1 -.30), implying a growth rate in dividends of 14% = .70 × 20% From the dividend growth model, the price of a share is:
Div 1 $1.50 P0 $75 r g .16 .14
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The PVGO Model
First, we must calculate the value of the firm as a cash cow. Div $5
P0
1
r
.16
$31.25
3.50 .20 3.50 .16 $.875 $43.75 P0 rg .16 .14 Finally, P0 31.25 43.75 $75
Second, we must calculate the value of the growth opportunities.
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Price Earnings Ratio
Many analysts frequently relate earnings per share to price. The price earnings ratio is a.k.a the
multiple
Calculated as current stock price divided by annual EPS The Wall Street Journal uses last 4 quarter’s earnings
Price per share P/E ratio EPS
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More on P/E multiples
Firms whose shares are “in fashion” sell at high multiples. However, looking at the value driver formula, we expect growth stocks to have high P/E ratios. Firms whose shares are out of favor sell at low multiples. But the value driver formula tells us to expect low P/E multiples in value stocks. Bottom line: High P/E stocks are overvalued only if the market overestimates the value of growth opportunities.
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Other Price Ratio Analysis
Analysts frequently relate earnings per share to variables other than price, e.g.:
Price/Cash Flow Ratio
cash flow = net income + depreciation = cash flow from operations or operating cash flow current stock price divided by annual sales per share price divided by book value of equity, which is measured as assets - liabilities
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Price/Sales
Price/Book (a.k.a Market to Book Ratio)
More on these multiples
Price/Cash Flow Ratio
cash flows are highly unstable over time. Problem: this ratio varies a lot – not informative. Sales have been used during high tech bubble. Problem: They don’t tell us about profits!
Price/Sales
Price/Book (a.k.a Market to Book Ratio)
This one is quite popular in finance. It is a very useful alternative measure of growth vs. value.
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Summary and Conclusions
In this chapter, we used the time value of money formulae from previous chapters to value bonds and stocks.
The value of a zero-coupon bond is F PV T (1 r )
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Summary and Conclusions (continued)
The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity.
C 1 F PV 1 T r (1 r ) (1 r )T
The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price.
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Summary and Conclusions (continued)
A stock can be valued by discounting its dividends. There are three cases:
1.
2.
3.
Zero growth in dividends P0 Div 1 / r Constant dividend growth P0 Div 1 /( r g ) Differential dividend growth
Div N 1 C (1 g1 )T r g 2 P 1 T N r g1 (1 r ) (1 r )
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Summary and Conclusions (continued)
The value-driver formula (or PVGO formula) values a stock as the sum of its “cash cow” value plus the present value of growth opportunities. EPS1 P PVGO r
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