Session 4: Efficient Diversification
These notes are based on chapter 6 of
Bodie, Kane, and Marcus (2019),
Essentials of Investments, Eleventh Edition
pg 207
1
Systematic and unsystematic risk
• Market/Systematic(beyond our control ie: inflation rate/oil prices/ unemployment
rate)/Non-diversifiable Risk
• Risk factors common to whole economy
• Unique/Firm-Specific/Nonsystematic/ Diversifiable Risk
• Risk that can be eliminated by diversification
• Ie: within our control
• Issue in R and D team, internal problem
2
Figure 1 Risk as Function of Number of Stocks in Portfolio
3
Figure 2 Risk versus Diversification
4
Covariance, Correlation and portfolio risk
• Portfolio risk depends on covariance between returns of assets (
2 shares one from A and B) how returns of these two move along?
Uncertainties of return for these two assets move along
• Expected return on two-security portfolio
•
•
•
•
E (rp ) = W1r1 + W2 r2
W1 = Proportion of funds in security 1
W2 = Proportion of funds in security 2
r1 = Expected return on security 1
r 2 = Expected return on security 2
5
Asset Allocation with Two Risky Assets
• Covariance Calculations – direction return of assets (risk move)
S
Cov(rS , rB ) = ∑ p(i )[rS (i ) − E (rS )][rB (i ) − E (rB )]
i =1
• Correlation Coefficient – strength of that movement (-1 to +1)
negative will reduce overall risk
ρ SB
Cov(rS , rB )
=
σS × σB
Cov(rS , rB ) = ρ SB σ S σ B
6
Spreadsheet 1 Expected return
7
• E( r ) = p x r (column b x c)
• E( r ) s = 10%
• E( r ) b = 5%
• - 37 – 10 = - 47 (deviation from expected return)
8
Spreadsheet 2 Variance of Returns
9
Spreadsheet 3 Portfolio Performance (add 2
steps : use w1r1 method to calculate portfolio,
then use E ( r ) = pxr
10
• -20.2
• = E( r) = w1 x r1 + w2 x r2
• = 0.4 x (-37) + (0.6 x -9)
• = -20.2
11
Spreadsheet 4 Return Covariance (more
accurate than stan.deviation)
12
A Portfolio with Two Risky Assets
• RoR: Weighted average of returns on components, with
investment proportions as weights
• ERR: Weighted average of expected returns on
components, with portfolio proportions as weights
Variance of RoR: *** standard deviation for portfolio p =
correlation
13
Figure 3 Efficient Frontier******
14
The Optimal Risky Portfolio with a Risk-Free Asset
• Slope of CAL is Sharpe Ratio also call
reward to volatility ratio of Risky Portfolio
• Optimal Risky Portfolio
• Best combination of risky and safe assets to
form portfolio
15
The Optimal Risky Portfolio with a Risk-Free Asset
• Calculating Optimal Risky Portfolio
(weightage)
• Two risky assets
[ E (rB ) − rf ]σ S2 − [ E (rs ) − rf ]σ Bσ S ρ BS
wB =
[ E (rB ) − rf ]σ S2 + [ E (rs ) − rf ]σ B2 − [ E (rB ) − rf + E (rs ) − rf ]σ Bσ S ρ BS
wS = 1 − wB
16
Efficient Diversification with Many Risky Assets
• Efficient Frontier of Risky Assets
• Graph representing set of portfolios that
maximizes expected return at each level of
portfolio risk
• Three methods
• Maximize risk premium for any level standard deviation
• Minimize standard deviation for any level risk premium
• Maximize Sharpe ratio for any standard deviation or risk
premium
17
Figure 4 Efficient Frontier: Risky and Individual Assets
18
Figure 5 Scatter Diagram for Ford
19
• Steeper ford return is more responsive to
the market returns
20
A Single-Index Stock Market include sys and non-sys to estimate risk of a particular security
in a portfolio
• Index model
• Relates stock returns to returns on broad market index & firm-specific
factors. One common sys factor responsible to all covariance in all stock
return and all other variability due to firm specific factors
• Excess return
• RoR in excess of risk-free rate
• Beta (ford) systematic (the triangular slope)
• Sensitivity of security’s returns to market factor
• Firm-specific or residual risk
• Component of return variance independent of market factor
• Alpha (ie: market 4%, ford 4.3%, alpha 0.3%)
• Stock’s expected return beyond that induced by market index
21
• 100 securities – 100 expected return and
s.d
• Cov = 100 x 99/ 2 = 4950
22
A Single-Index Stock Market
• Excess Return
• 𝑹𝑹𝒊𝒊 = 𝜷𝜷𝒊𝒊 𝑹𝑹𝑴𝑴 + 𝜶𝜶𝒊𝒊 + 𝒆𝒆𝒊𝒊
Rm = market return
Alpha and ei : residual risk (non-systematic risk)
Where:
• β𝑖𝑖 𝑅𝑅𝑀𝑀 : component of return due to movements in overall market
• β𝑖𝑖 : security’s responsiveness to market
• α𝑖𝑖 : stock’s expected excess return if market factor is neutral, i.e. market-index
excess return is zero
• 𝑒𝑒𝑖𝑖 : Component attributable to unexpected events relevant only to this security
(firm-specific)
23
A Single-Index Stock Market
• Statistical and Graphical Representation of
Single-Index Model (want systematic)
• Ratio of systematic variance to total variance
•
24
A Single-Index Stock Market
• Using Security Analysis with Index Model
• Information ratio
• Ratio of alpha to standard deviation of residual
• Active portfolio
• Portfolio formed by optimally combining analyzed stocks
25
A Single-Index Stock Market NOT IMPORTANT
• Diversification in Single-Index Security Market
• In portfolio of n securities with weights
• In securities with nonsystematic risk
• Nonsystematic portion of portfolio return
•
• Portfolio nonsystematic variance
•
26
A Single-Index Stock Market (no need)
• Statistical and Graphical Representation of
Single-Index Model
• Security Characteristic Line (SCL)
• Plot of security’s predicted excess return from excess
return of market
• Algebraic representation of regression line
•
27
Session 7: Bond Prices and Yields
These notes are based on chapter 10 and
11 of Bodie, Kane, and Marcus (2019),
Essentials of Investments, Eleventh Edition
28
What is bond?
• A financial instrument issued by either a
company or the central or state
governments.
• The maturity period of a bond is more than
one year
• The par value of the bond is returned to the
investor at the end of the maturity period
• Income to a bond holder is interest which is
fixed till maturity
29
Bond Characteristics
• Face Value, Par Value
• Payment to bondholder at maturity of bond
• Coupon Rate
• Bond’s annual interest payment per dollar of
par value
• Zero-Coupon Bond
• Pays no coupons, sells at discount,
provides only payment of par value at
maturity
30
Type of Bonds
• Treasury Bonds and Notes
• Accrued interest and quoted bond prices
• Quoted prices do not include interest accruing
between payment dates
• Accrued interest =
𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑
𝟐𝟐
×
𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑
𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑
Example: Consider a bond with the following characteristics: Semi-annual
payments, coupon rate of 6%, $1,000 par value. If 45 days have passed since the
last coupon payment, what is the accrued interest?
45
A.I . =
× $30 = $7.42
182
31
Type of Bonds
• Corporate Bonds
• Call provisions on corporate bonds
• Callable bonds: May be repurchased by issuer
at specified call price during call period
• Convertible bonds
• Allow bondholder to exchange bond for
specified number of common stock shares
32
Type of Bonds
• Corporate Bonds
• Puttable bonds
• Holder may choose to exchange for par value
or to extend for given number of years
• Floating-rate bonds
• Coupon rates periodically reset according to
specified market date
33
Bond Characteristics
• Preferred Stock
• Commonly pays fixed dividend
• Floating-rate preferred stock becoming more
popular
• Dividends not normally tax-deductible
• Corporations that purchase other
corporations’ preferred stock are taxed on
only 30% of dividends received
34
Type of Bonds
• Other Domestic Issuers
• State, local governments (municipal bonds)
International Bonds
• Foreign bonds
Issued by borrower in different country than where bond sold
Denominated in currency of market country
• Eurobonds
Denominated in currency (usually that of issuing country) different
than that of market
35
Type of Bonds
• Premium Bonds
• Bonds selling above par value
• Discount Bonds
• Bonds selling below par value
36
Bond Pricing
• Bond value = Present value of coupons + Present
par value
• Bond value =
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶
𝑇𝑇
∑𝑡𝑡=1
(1+𝑟𝑟)𝑡𝑡
• T = Maturity date
+
𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
(1+𝑟𝑟)𝑇𝑇
• r = discount rate
• Bond price = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 ×
= 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶
1
𝑟𝑟
1−
1
1+𝑟𝑟 𝑇𝑇
+ 𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 ×
1
(1+𝑟𝑟)𝑇𝑇
× 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑟𝑟, 𝑇𝑇 + 𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 × 𝑃𝑃𝑃𝑃 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (𝑟𝑟, 𝑇𝑇)
37
Bond Pricing: Example
• What is the price of the following two bonds:
Bond A
Bond B
Maturity (T)
4 Years
30 Years
Coupon Rate (C)
5%
5%
Discount Rate (r)
8%
8%
Par Value (FV)
$1,000
$1,000
Bond A
Bond B
1 − (1 + r )
1 − (1 + r )
−T
−T
+ FV × (1 + r )
PV = C ×
+ FV × (1 + r )
PV = C ×
r
r
1 − (1.08) −30
1 − (1.08) −4
−4
$50 ×
$50 ×
=
+ $1, 000 × (1.08) −30
=
+ $1, 000 × (1.08)
.08
.08
= $165.61 + $735.03 = $900.64
= $562.89 + $99.38=$662.27
−T
Present Value of Coupons
−T
Present Par Value
38
Bond Pricing
• Prices fall as market interest rate rises
• Interest rate fluctuations are primary source
of bond market risk
• Bonds with longer maturities more sensitive
to fluctuations in interest rate
39
Figure 1 Inverse Relationship between Bond Prices and Yields
40
Table 1 Bond Prices at Different Interest Rates
41
Bond Pricing
• Bond Pricing between Coupon Dates
• Invoice price = Flat price + Accrued interest
• Bond Pricing in Excel
• =PRICE (settlement date, maturity date, annual
coupon rate, yield to maturity, redemption value
as percent of par value, number of coupon
payments per year)
42
Bond Yields
• Yield to Maturity
• Discount rate that makes present value of
bond’s payments equal to price.
• Current Yield
• Annual coupon divided by bond price
• Premium Bonds
• Bonds selling above par value
• Discount Bonds
• Bonds selling below par value
43
Spreadsheet 1 Finding Yield to Maturity
Semiannual
coupons
Settlement date
Maturity date
Annual coupon rate
Bond price (flat)
Redemption value (% of face value)
Coupon payments per year
Yield to maturity (decimal)
Annual
coupons
1/1/2000
1/1/2030
0.08
127.676
100
2
1/2/2000
1/2/2030
0.08
127.676
100
1
0.0600
0.0599
The formula entered here is =YIELD(B3,B4,B5,B6,B7,B8)
44
Bond Yields
• Yield to Call
• Calculated like yield to maturity
• Time until call replaces time until maturity; call
price replaces par value
• Premium bonds more likely to be called than
discount bonds
45
Bond Yields
• Realized Compound Returns versus Yield to
Maturity
• Realized compound return
• Compound rate of return on bond with all coupons
reinvested until maturity
• Reinvestment rate risk
• Uncertainty surrounding cumulative future value of
reinvested coupon payments
46
Bond Yields
• Yield to Maturity versus Holding Period
Return (HPR)
• Yield to maturity measures average RoR if
investment is held until bond matures
• HPR is RoR over particular investment period;
depends on market price at end of period
47
Realized compound yield
• A two-year bond with par value $1000
making annual coupon payments of $100 is
priced at $1000. What is the yield to
maturity of the bond? What will the realised
compound yield to maturity be if the oneyear interest rate next year turns out to be:
• 8%
• 10%
• 12%?
48
Realized compound yield
• As the bond’s price is the same as the par
value, the yield to maturity is 10%
($100/$1000).
• The realised compound yield for different
interest rates is calculated as follws:
49
Realized compound yield
• The bond is selling at par value. Its yield to
maturity equals the coupon rate, 10%. If the
first-year coupon is reinvested at an
interest rate of r per cent, then total
proceeds at the end of the second year will
be 100 × (1 + r) + 1100. Therefore, realised
compound yield to maturity will be a
function of r as given in the following table:
50
Realized compound yield
Realised yield to maturity =
r
Total proceeds
8%
$1208
1208 / 1000 − 1 = 0.0991 = 9.91%
10%
$1210
1210 / 1000 − 1 = 0.1000 = 10.00%
12%
$1212
1212 / 1000 − 1 = 0.1009 = 10.09%
proceeds / 1000 − 1
51
Duration of a bond
• Macaulay’s Duration (D)
• Measures effective bond maturity
• Used to understand the impact of interest rate changes on
the bond price
• Weighted average of the times until each payment, with
weights proportional to the present value of payment
• 𝑤𝑤𝑡𝑡 =
𝐶𝐶𝐶𝐶𝑡𝑡 /(1+𝑦𝑦)𝑡𝑡
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
• 𝐷𝐷 = ∑𝑇𝑇
𝑡𝑡=1 𝑡𝑡 × 𝑤𝑤𝑡𝑡
52
Spreadsheet 1 Calculation of Duration of Two Bonds
53
Interest Rate Risk
• Change in Bond (ΔP) Price and yield to
maturity (y)
Δ𝑃𝑃
•
𝑃𝑃
= 𝐷𝐷 ×
∗
𝐷𝐷
1+𝑦𝑦
Δ 1+𝑦𝑦
1+𝑦𝑦
• Modified Duration
• 𝐷𝐷 =
Δ𝑃𝑃
•
𝑃𝑃
Application
In the above example, If y of the
coupon bond increases by 1%,
the bond price will change by 2.78*0.01 = -2.78%.
= −𝐷𝐷 ∗ Δ𝑦𝑦
54
Convexity
• Convexity
• Curvature of price-yield relationship of bond
Δ𝑃𝑃
•
𝑃𝑃
= −𝐷𝐷 ∗ Δ𝑦𝑦 + 1⁄2 × 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 × (Δ𝑦𝑦)2
• 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 =
1
𝑝𝑝∗(1+𝑦𝑦)2
∗
∑𝑛𝑛𝑡𝑡=1
𝐶𝐶𝐶𝐶𝑡𝑡
(𝑡𝑡
1+𝑦𝑦 𝑡𝑡
+ 𝑡𝑡 2
• Why Do Investors Like Convexity?
• More convexity = greater price increases,
smaller price decreases when interest rates
fluctuate by larger amounts
55
Convexity
Example
y
4.50% Time
Cash flow
PV(CF) t+t^2
(t+t^2)*PV(CF)
coupon
10%
1
3
2.871
2
5.7416
par value
$100
2
3
2.747
6
16.4831
10
3
3
2.629
12
31.5467
4
3
2.516
20
50.3137
5
3
2.407
30
72.2206
6
3
2.304
42
96.7549
7
3
2.204
56
123.4512
8
3
2.110
72
151.8880
9
3
2.019
90
181.6842
10
103
66.325
110
7295.7006
n
88.131
convexity
8025.7846
83.39242
56
Debrief
• Types of bonds
• Yield to maturity
• Realized compound yield
• Duration and convexity
57
Session 8: Valuation of Equities
These notes are based on chapter 13 of
Bodie, Kane, and Marcus (2019), Essentials
of Investments, Eleventh Edition
58
Valuation by comparables
• Book Value
• Net worth of common equity according to a
firm’s balance sheet
• Limitations of Book Value
• Liquidation value: Net amount realized by selling
assets of firm and paying off debt
• Replacement cost: Cost to replace firm’s assets
• Tobin’s q: Ratio of firm’s market value to
replacement cost
59
Dividend discount models
• Intrinsic Value
• 𝑉𝑉0 =
𝐸𝐸 𝐷𝐷1 +𝐸𝐸(𝑃𝑃1 )
1+𝑘𝑘
• 𝑉𝑉0 =
𝐷𝐷1
1+𝑘𝑘
• For holding period H
+
𝐷𝐷2
(1+𝑘𝑘)2
+
𝐷𝐷𝐻𝐻 +𝑃𝑃𝐻𝐻
⋯+
(1+𝑘𝑘)𝐻𝐻
• Dividend Discount Model (DDM)
• Formula for intrinsic value of firm equal to
present value of all expected future dividends
60
Dividend Discount Models
• Constant-Growth DDM
• Form of DDM that assumes dividends will grow
at constant rate
• 𝑉𝑉0 =
𝐷𝐷1
𝑘𝑘−𝑔𝑔
• Implies stock’s value greater if:
• Larger dividend per share
• Lower market capitalization rate, k
• Higher expected growth rate of dividends
61
Dividend Discount Models
• For stock with market price = intrinsic value,
expected holding period return
• 𝐸𝐸 𝑟𝑟 = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
𝐷𝐷1
•
𝑃𝑃𝑜𝑜
+
𝑃𝑃1 −𝑃𝑃0
𝑃𝑃0
=
𝐷𝐷1
𝑃𝑃0
+ 𝑔𝑔
62
Dividend Discount Models: Two Stage Example
• Consider the following information:
• The firm’s dividends are expected to grow at g = 20% until t = 3 yrs.
• At the start of year four, growth slows to gs= 5%.
• The stock just paid a dividend Div0 = $1.00
• Assume a market capitalization rate of k = 12%
• What is the price, P0, of this stock?
P0
D0 × (1 + g )
D0 × (1 + g )t
D0 × (1 + g )t × (1 + g s )
+ ... +
+
t
(1 + k )
(1 + k )
(1 + k )t × ( k − g s )
$1× (1 + .2) $1× (1 + .2) 2 $1× (1 + .2)3 D0 × (1 + .2)3 × (1 + .05)
=
+
+
+
2
3
(1 + .12)
(1 + .12)
(1 + .12)
(1 + .12)3 × (.12 − .05)
= $1.07 + $1.15 + $1.23 + $18.45 = $21.90
63
Present value of growth opportunities
• Stock Prices and Investment Opportunities
• Dividend payout ratio
• Percentage of earnings paid as dividends
• Plowback ratio/earnings retention ratio
• Proportion of firm’s earnings reinvested in
business
• Present value of growth opportunities (PVGO)
• Price = No-growth value per share + PVGO
• 𝑃𝑃0 =
𝐸𝐸1
𝑘𝑘
+ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
64
Free Cash Flow Valuation Approaches
• Free Cash Flow for Firm (FCFF)
• 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 1 − 𝑡𝑡𝑐𝑐 + 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 −
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 − 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑖𝑖𝑖𝑖 𝑁𝑁𝑁𝑁𝑁𝑁
• EBIT = Earnings before interest and taxes
• 𝑡𝑡𝑐𝑐 = Corporate tax rate
• NWC = Net working capital
• Free Cash Flow to Equity Holders (FCFE)
• 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 − 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 × 1 − 𝑡𝑡𝑐𝑐 +
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
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13.5 Free Cash Flow Valuation Approaches
• Estimating Terminal Value using Constant
Growth Model
• 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 =
• 𝑃𝑃𝑇𝑇 =
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑇𝑇+1
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊−𝑔𝑔
1+𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑡𝑡
𝑇𝑇
∑𝑡𝑡=1
(1+𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊)𝑡𝑡
+
𝑃𝑃𝑇𝑇
(1+𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊)𝑇𝑇
• WACC = Weighted average cost of capital
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FCF Valuation Approaches: FCFF Example
• Suppose FCFF = $1 mil for years 1-4 and then is
expected to grow at a rate of 3%. Assume WACC = 15%
PT
FCFF
+
∑
t
(1 + WACC )T
t =1 (1 + WACC )
$1, 000, 000 × 1.03
4
$1, 000, 000
.15 − .03
= ∑
+
(1 + .15)t
(1 + .15) 4
t =1
= $ 7, 762, 527
=
FirmValue
T
• If 500,000 shares are outstanding, what is the predicted
price of this stock if the firm has $5,000,000 of debt?
P0
$7, 762, 527-$5,000,000
= $5.53
500, 000
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Free Cash Flow Valuation Approaches
• Market Value of Equity
• 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
• 𝑃𝑃𝑇𝑇 =
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑇𝑇+1
𝑘𝑘𝐸𝐸 −𝑔𝑔
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑡𝑡
𝑇𝑇
∑𝑡𝑡=1
(1+𝑘𝑘𝐸𝐸 )𝑡𝑡
+
𝑃𝑃𝑇𝑇
(1+𝑘𝑘𝐸𝐸 )𝑇𝑇
68
FCF Valuation Approaches: FCFE Example
• Suppose FCFE = $900,000 for years 1-4 and then is
expected to grow at a rate of 3%. Assume ke = 18%
PT
FCFE
+
Market Value=
of Equity ∑
t
t
+
+
(1
)
(1
)
k
k
t =1
e
e
T
$900, 000 × 1.03
4
$900, 000
.18 − .03
= ∑
+
t
(1 + .18) 4
t =1 (1 + .18)
= $ 2, 500,851
• If there are 500,000 shares outstanding, what is the
predicted price of this stock? Why can debt be ignored?
=
P0
$ 2, 500,851
= $5.00
500, 000
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Spreadsheet : FCF
70
Debrief
• Valuation by comparables
• Dividend discount model
• Present value of growth opportunities
• Free cash flow models
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