Corporate Finance – Bonds & Shares

Course Timeline, Topics & Required Reading

• Week 1 – Concepts in Finance

  • Prior knowledge: Berk & DeMarzo (BDM) Ch. 1

  • Problem-set references: Finance Text- & Workbook Ch. 2:1-9,11-15,17-19

• Week 2 – Capital Budgeting

  • BDM Ch. 2–5

  • Workbook Ch. 3:1-6,9; Ch. 4:1-2; Ch. 5:1-6

• Week 3 – Bonds & Shares (current lecture)

  • BDM Ch. 7-8; Workbook Ch. 6:1-2,6-8; Ch. 7:1,3-6; Ch. 8:1-7; Ch. 9:1-4; Ch. 13:1-3,11; Ch. 12:1-6,8-9; Ch. 14:1-4

• Week 4 – Mean-Variance Analysis & Portfolio Theory (BDM Ch. 6 + 9 + 10 + 11)

• Week 5 – CAPM & Cost of Capital (BDM Ch. 11 + 12)

• Week 6 – Efficient-Market Hypothesis & M&A (BDM §9.5 + Ch. 28)

• Week 7 – Discount Rates for International Projects + Review (BDM Ch. 1-12 + 28; Workbook Ch. 1-9,12-15)

• Week 8 – Reflection week

• Week 9 – Assessment (company selection, partial exam, assignment, team evaluation, final exam)

Bonds vs. Shares – Conceptual Contrast

• Bonds

  • Liability of the corporation; bondholders are creditors.

  • Promised, tax-deductible interest payments; principal repaid at maturity.

• Shares (Common Stock)

  • Represent equity/ownership; shareholders are residual claimants.

  • Dividends are discretionary and not tax-deductible; no legal repayment of contributed capital.

• Implication: capital-structure choice affects taxation, cash-flow flexibility & default exposure.

  • (Source: Pollard et al., 2007, p. 630)

Cash-Flow Patterns of Bonds

• Zero-coupon bond: single payment of face value (FV) at maturity.

• Coupon bond: periodic coupon $C=\text{coupon rate}\times FV$ plus FV at maturity.

• Graph illustrates identical FV paid at $T=5$ but different interim cash-flows for 5 % coupon vs. zero-coupon.

Present-Value (PV) Framework

• General pricing formula $P=\sum<em>{t=1}^{T} \frac{CF</em>t}{(1+r)^t}$

  • $r$ = discount rate / spot rate matching maturity.

• Example 1 – Flat term structure $r=10\%$

  • Bond A: 1 yr, 12 % coupon, FV = 1 000

  • $P=\frac{120+1000}{1.10}=1\,018.18$

  • Bond B: 2 yr, 5 % coupon

  • $P=\frac{50}{1.10}+\frac{50+1000}{1.10^2}=913.22$

  • Bond C: 3 yr, zero coupon

  • $P=\frac{1000}{1.10^3}=751.31$

Term Structure of Interest Rates

• Definition: relationship between maturity $T$ and its corresponding spot rate $r_T$ (yield curve).

• Shapes: flat, upward-sloping (normal), downward-sloping (inverted), humped.

• Numerical illustration (increasing):

  • $r<em>1=10\%\;,\;r</em>2=12\%\;,\;r_3=14\%$

  • $100\,(1+r_T)^T$ yields:

  • $T=1\Rightarrow110$, $T=2\Rightarrow125.44$, $T=3\Rightarrow148.15$

Extracting Spot Rates (Bootstrapping)

• Method 1 – Using observable zero-coupon prices

  • Example 3: prices 909.09, 797.19, 674.97 for 1-,2-,3-year zeros (FV = 1 000)

  • $r_1=\frac{1000}{909.09}-1=10\%$

  • $r_2=(\frac{1000}{797.19})^{1/2}-1=12\%$

  • $r_3=(\frac{1000}{674.97})^{1/3}-1=14\%$

• Method 2 – Using coupon bonds sequentially

  • Solve for $r<em>1$ from 1-year bond, then $r</em>2$ from 2-year bond, etc. (Example 4 provides numerical walk-through.)

Price Relationships: Par, Premium, Discount

• Flat curve assumed.

  • If $\text{coupon rate}=r$ ⇒ trades at par ($P=FV$).

  • If \text{coupon rate}>r ⇒ premium (P>FV).

  • If \text{coupon rate}<r ⇒ discount (P<FV).

• Two illustrative tables (3- & 10-year, r = 5 % or 10 %).

Yield-to-Maturity (YTM)

• Solves $P=\sum<em>{t=1}^{T}\frac{CF</em>t}{(1+y)^t}$ for $y$.

• Function of observed price & cash-flows, not vice-versa.

• Flat term-structure ⇒ $y=r$.

• For zero-coupon, $y=r_T$ (spot rate of same maturity).

• Example: $r<em>1=4\%,\;r</em>2=8\%$; 2-yr 5 % bond price $94.83$ → $y=7.9\%$.

• Pricing grid highlights negative (mis-)interpretation of causality.

Forward Rates

• Definition: implied future short rate contracted today.

• Law of one price equates strategies:
  $(1+1f<em>2)=\frac{(1+r</em>2)^2}{(1+r_1)}$

• General formula: $(1+n!f<em>{n+1})=\frac{(1+r</em>{n+1})^{n+1}}{(1+r_n)^n}$

• Numerical example:

  • $r<em>1=5.3\%\;,\;r</em>2=11.8\%$ ⇒ $1f_2=18.7\%$.

Corporate Bonds & Credit Risk

• Default possibility reduces price; investors demand risk premium.

• Pricing with expected cash-flow and required return $r=\text{risk-free}+\text{risk premium}$.

  • Avant zero-coupon: default 50 %, recovery 90 %.

  • Expected CF = $0.5\times1000+0.5\times900=950$

  • Required return $5.1\%$ ⇒ $P=\frac{950}{1.051}=903.90$

  • Promised YTM $y=10.63\%$; default spread $y-r_f=6.63\%$.

• Expected return equals required return: mixture of default & non-default pay-offs.

Credit Ratings

• Investment-grade: Aaa/AAA, Aa/AA, A/A, Baa/BBB.

• Speculative (junk): Ba/BB down to C/D.

• Historical average vs. recession default rates (Moody’s study):

  • AAA ≈ 0 %; BB ≈ 2.2 % (avg) & 8 % (recession); CCC ≈ 48 % (recession).

• Spreads widen with deteriorating rating and with state of economy.

Equity Valuation – Infinite-Horizon DDM

• General identity
  $P<em>0=\sum</em>{t=1}^{\infty}\frac{\text{div}_t}{(1+r)^t}$

• Gordon-Growth (constant $g$)
  P0=\frac{\text{div}1}{r-g},\quad g<r

• Zero-growth special case: $P<em>0=\frac{\text{div}</em>1}{r}$.

• Example (Fledging Electronics): $r=15\%,\;\text{div}<em>1=5,\;P</em>1=110 \Rightarrow P_0=100$.

Philips Example – Sensitivity to g

• Given $r=6.5\%$, $\text{div}_1=0.85€$.

  • If $g=3\%\Rightarrow P_0=24.29€$.

  • If $g=4.55\%\Rightarrow P_0=43.59€$ ~ matches 30 Dec 2020 market price (43.71€).

• Demonstrates high valuation leverage to growth assumptions.

Growth, Retention & ROIC (BDM 9; Schauten & Steenbeek 2015)

• Sustainable growth: $g=\text{retention ratio}\times ROIC$.

• Share-price impact depends on differential between ROIC and required return $r$.

  • Value-creating growth if ROIC>r.

  • Value-neutral if $ROIC=r$.

  • Value-destroying if ROIC<r.

• Crane Sporting Goods illustration:

  • Base: $EPS<em>1=6,\;payout=100\%,\;r=10\%,\;P</em>0=60$.

  • Cut payout to 75 % (retain 25 %).

  • If $ROIC=12\%$ ⇒ $g=3\%$ ⇒ $P_0=64.29$ (value ↑).

  • If $ROIC=8\%$ < $r$ ⇒ $g=2\%$ ⇒ $P_0=56.25$ (value ↓).

Relative Valuation – Multiples

• Price/Earnings: $\frac{P<em>0}{EPS</em>1}=\frac{k}{r-g}$ (where $k=\text{payout ratio}$).

  • Example: competitor PE = 26; if $EPS<em>1=2.50€$ ⇒ implied $P</em>0≈65€$.

• Shiller-PE web reference for market context.

Key Formula Compendium

• Bond price: $P=\sum<em>{t=1}^{T}\frac{CF</em>t}{(1+r_t)^t}$

• YTM: solve $P=\sum<em>{t=1}^{T}\frac{CF</em>t}{(1+y)^t}$.

• Forward rate: $(1+n!f<em>{n+1})=\frac{(1+r</em>{n+1})^{n+1}}{(1+r_n)^n}$.

• Gordon model: $P<em>0=\frac{div</em>1}{r-g}$; growth link $g=\text{retention}\times ROIC$.

• PE linkage: $\frac{P<em>0}{EPS</em>1}=\frac{k}{r-g}$.

Practical & Ethical Notes

• Misinterpreting YTM as causal driver of price can mislead investors; correct view: price (market) ⇒ YTM (implied).

• Growth strategies should be assessed on value creation, not EPS optics; managerial incentive systems should align with ROIC-based value metrics.

• Credit-rating methodologies offer transparent risk differentiation but can lag rapid credit events; investors must complement with market-implied spreads.

Course Timeline, Topics & Required Reading

• Week 1 – Concepts in Finance - Prior knowledge: Berk & DeMarzo (BDM) Ch. 1

  • Problem-set references: Finance Text- & Workbook Ch. 2:1-9,11-15,17-19

• Week 2 – Capital Budgeting - BDM Ch. 2–5

  • Workbook Ch. 3:1-6,9; Ch. 4:1-2; Ch. 5:1-6

• Week 3 – Bonds & Shares (current lecture) - BDM Ch. 7-8; Workbook Ch. 6:1-2,6-8; Ch. 7:1,3-6; Ch. 8:1-7; Ch. 9:1-4; Ch. 13:1-3,11; Ch. 12:1-6,8-9; Ch. 14:1-4

• Week 4 – Mean-Variance Analysis & Portfolio Theory (BDM Ch. 6 + 9 + 10 + 11)

• Week 5 – CAPM & Cost of Capital (BDM Ch. 11 + 12)

• Week 6 – Efficient-Market Hypothesis & M&A (BDM §9.5 + Ch. 28)

• Week 7 – Discount Rates for International Projects + Review (BDM Ch. 1-12 + 28; Workbook Ch. 1-9,12-15)

• Week 8 – Reflection week

• Week 9 – Assessment (company selection, partial exam, assignment, team evaluation, final exam)

Bonds vs. Shares – Conceptual Contrast

• Bonds - Represent a liability of the corporation, meaning bondholders act as creditors who lend money to the company.

  • They are characterized by promised, tax-deductible interest payments (coupons) and a principal amount that is repaid to the bondholder at maturity.

• Shares (Common Stock) - Represent equity or ownership in the corporation; shareholders are thus residual claimants, meaning they have a claim on the company’s assets and earnings only after creditors (like bondholders) have been paid.

  • Dividends, which are payments to shareholders, are discretionary (declared by the board of directors and not guaranteed) and are not tax-deductible for the company; there is no legal requirement for the company to repay the initial capital contributed by shareholders.

• Implication: The choice between debt (bonds) and equity (shares) in a company's capital structure significantly affects its taxation, cash-flow flexibility, and exposure to default risk.

Cash-Flow Patterns of Bonds

• Zero-coupon bond: This type of bond does not pay periodic interest. Instead, it makes a single payment of its face value (FV) to the bondholder at its maturity date. These bonds are typically sold at a discount to their face value.

• Coupon bond: This bond pays periodic interest payments, known as coupons, throughout its life. The coupon amount $C$ is calculated as $\text{coupon rate}\times FV$ (face value). In addition to these periodic payments, the face value is also repaid at maturity.

• Graph illustration (implied): A visual representation would show that while both 5% coupon bonds and zero-coupon bonds might pay an identical face value at $T=5$, the coupon bond provides interim cash flows, unlike the zero-coupon bond.

Present-Value (PV) Framework

• Definition: The present value framework is used to determine the current worth of a future stream of cash flows or a single future payment, discounted at a specific rate.

• General pricing formula: The price $P$ of an asset (like a bond) is the sum of the present values of all its future cash flows ($CFt$). It is calculated as $P=\sum</em>{t=1}^{T} \frac{CF_t}{(1+r)^t}$, where $r$ is the discount rate (or spot rate) which matches the maturity of the cash flow ($t$).

• Example 1 – Flat term structure $r=10\%$

  • Bond A: A 1-year bond with a 12% coupon and a face value of 1,000.

    • The present value $P$ is calculated by discounting the sum of the final coupon payment and face value ($120+1000$) back one year at a 10% discount rate: $P=\frac{120+1000}{1.10}=1\,018.18$

  • Bond B: A 2-year bond with a 5% coupon.

    • The present value $P$ is calculated by discounting the first coupon pago ($50$) back one year and the second coupon plus face value ($50+1000$) back two years, both at a 10% discount rate: $P=\frac{50}{1.10}+\frac{50+1000}{1.10^2}=913.22$

  • Bond C: A 3-year zero-coupon bond.

    • The present value $P$ is calculated by discounting the face value ($1000$) back three years at a 10% discount rate: $P=\frac{1000}{1.10^3}=751.31$

Term Structure of Interest Rates

• Definition: This describes the relationship between the maturity ($T$) of a debt instrument and its corresponding yield or spot rate ($r_T$). It is often visualized as a yield curve.

• Shapes: The yield curve can have various shapes:

  • Flat: Short-term and long-term rates are similar.

  • Upward-sloping (normal): Long-term rates are higher than short-term rates, typically seen during economic expansion, indicating market expectations of future economic growth and inflation.

  • Downward-sloping (inverted): Short-term rates are higher than long-term rates, often signaling impending economic recession as investors expect lower future interest rates.

  • Humped: Medium-term rates are higher than both short-term and long-term rates.

• Numerical illustration (increasing): If spot rates increase with maturity ($r1=10\%, r2=12\%, r_3=14\%$), then the future value of 100 invested for increasing periods will reflect these higher rates: $\begin{aligned} &amp;T=1\Rightarrow100(1+0.10)^1=110 \ &amp;T=2\Rightarrow100(1+0.12)^2=125.44 \ &amp;T=3\Rightarrow100(1+0.14)^3=148.15 \end{aligned}$

Extracting Spot Rates (Bootstrapping)

• Definition: Bootstrapping is a method used to derive the zero-coupon yield curve (spot rates) from the prices of coupon-paying bonds or observable zero-coupon bond prices. It effectively extracts the pure discount rates for different maturities.

• Method 1 – Using observable zero-coupon prices: If the prices of zero-coupon bonds for different maturities are known, the spot rates can be directly calculated.

  • Example 3: For 1-, 2-, and 3-year zero-coupon bonds with a face value of 1,000 and given prices (909.09, 797.19, 674.97):

    • $r1$ (1-year spot rate) is derived from the 1-year zero-coupon bond: $r</em>1=\frac{1000}{909.09}-1=10\%$

    • $r2$ (2-year spot rate) is derived from the 2-year zero-coupon bond: $r</em>2=(\frac{1000}{797.19})^{1/2}-1=12\%$

    • $r3$ (3-year spot rate) is derived from the 3-year zero-coupon bond: $r</em>3=(\frac{1000}{674.97})^{1/3}-1=14\%$

• Method 2 – Using coupon bonds sequentially: This method involves starting with the shortest-maturity coupon bond to find its spot rate, then using that rate to calculate the spot rate for the next maturity, and so on. (Example 4 provides a numerical walk-through of this sequential process.)

Price Relationships: Par, Premium, Discount

• Assumes a flat yield curve for simplicity (meaning $r$ is constant across all maturities).

• At Par: If a bond's coupon rate is equal to the market discount rate ($r$), the bond will trade at par, meaning its price ($P$) will be equal to its face value ($FV$). This is because the bond's cash flows provide the exact return investors require.

• Premium: If a bond's coupon rate is greater than the market discount rate ($r$), the bond will trade at a premium, meaning its price ($P$) will be greater than its face value ($FV$). This occurs because the bond offers more attractive interest payments than what is currently available in the market for similar risk, making it more valuable.

• Discount: If a bond's coupon rate is less than the market discount rate ($r$), the bond will trade at a discount, meaning its price ($P$) will be less than its face value ($FV$). This is because the bond offers less attractive interest payments compared to current market rates, making it less desirable and thus trading at a lower price.

• Two illustrative tables (3- & 10-year, r = 5% or 10%) would further demonstrate these price relationships across different maturities and market rates.

Yield-to-Maturity (YTM)

• Definition: Yield-to-Maturity ($y$) is the total return an investor can expect to receive if they hold a bond until it matures. It is the discount rate that equates the present value of the bond's promised future cash flows (coupons and face value) to its current market price ($P$).

• Calculation: YTM is solved for $y$ in the equation: $P=\sum<em>{t=1}^{T}\frac{CF</em>t}{(1+y)^t}$.

• Fundamental Relationship: YTM is a function of the bond's observed market price and its contractual cash flows (coupon payments and face value), not the other way around. The price determines the YTM, not vice-versa.

• Flat term-structure: If the term structure of interest rates is flat, then the YTM ($y$) will be equal to the spot rate ($r$) for all maturities.

• Zero-coupon bond: For a zero-coupon bond, its YTM ($y$) is simply equal to the spot rate ($r_T$) for the same maturity.

• Example: Given spot rates $r1=4\%$ and $r2=8\%$, a 2-year 5% coupon bond with a price of 94.83. Solving for $y$ in the PV formula would yield a YTM of $y=7.9\%$.

• Pricing grid highlights negative (mis-)interpretation of causality: It's crucial to understand that the market price is determined by the collective preferences and expectations of investors, and the YTM is then derived from that observable price. Misinterpreting this can lead to flawed investment decisions.

Forward Rates

• Definition: A forward rate is an interest rate for a future period that is agreed upon today. It is an implied future short-term rate, derived from the current term structure of interest rates.

• Law of one price equates strategies: The concept of forward rates is based on the idea that in an efficient market, two investment strategies with identical risk and identical outcomes should have the same cost. Specifically, investing for $n$ years at today's $n$-year spot rate should yield the same result as investing for $n-1$ years at the $(n-1)$-year spot rate and then investing for the subsequent year at the implied 1-year forward rate (${n-1}fn$).

  • For instance, investing for 2 years at the 2-year spot rate ($r2$) should yield the same as investing for 1 year at the 1-year spot rate ($r1$) and then for the second year at the implied 1-year forward rate starting in year 1 ($1f2$). This is expressed as: $(1+<em>1f</em>2)=\frac{(1+r<em>2)^2}{(1+r</em>1)}$

• General formula: The future 1-year forward rate starting at time $n$ ($nf{n+1}$) can be calculated from current spot rates as: $(1+<em>nf</em>{n+1})=\frac{(1+r<em>{n+1})^{n+1}}{(1+r</em>n)^n}$

• Numerical example: If $r1=5.3\%$ and $r2=11.8\%$, the implied 1-year forward rate starting in year 1 ($1f2$) is: $(1+<em>1f</em>2)=\frac{(1+0.118)^2}{(1+0.053)}\Rightarrow<em>1f</em>2=18.7\%$

Corporate Bonds & Credit Risk

• Default possibility: Unlike risk-free government bonds, corporate bonds carry credit risk, meaning there's a possibility that the issuer may fail to make promised interest or principal payments (default).

• Impact on price: The possibility of default reduces the bond's price compared to a risk-free bond with identical cash flows, as investors require compensation for taking on this additional risk.

• Risk premium: Investors demand a risk premium, which is an additional return beyond the risk-free rate, to compensate them for bearing credit risk. Thus, the required return ($r$) for a corporate bond is: $r=\text{risk-free}+\text{risk premium}$.

• Pricing with expected cash-flow: Pricing of corporate bonds accounts for the probability of default and the expected recovery rate in case of default. The price is based on the expected cash flows, not just the promised ones.

  • Avant zero-coupon example: If an Avant zero-coupon bond has a 50% chance of default and a 90% recovery rate (meaning 90% of the face value is recovered if default occurs), the expected cash flow at maturity (assuming FV=1000) is: $0.5\times1000+0.5\times(0.90\times1000)=500+450=950$.

  • If the required return is 5.1%, the price $P$ would be: $P=\frac{950}{1.051}=903.90$

  • Promised YTM and default spread: The promised YTM ($y$) (calculated based on promised 1000 payment, not expected) would be higher, say $10.63\%$. The difference between this promised YTM and the risk-free rate ($rf$), here $y-rf=6.63\%$, represents the default spread, which is the extra return investors demand for bearing the bond's credit risk.

• Expected return equals required return: For an investor, the expected return on a corporate bond must equal their required return, taking into account the probability-weighted outcomes of both default and non-default scenarios.

Credit Ratings

• Definition: Credit ratings are assessments of the creditworthiness of a bond issuer, typically provided by rating agencies (e.g., Moody's, S&P, Fitch). They indicate the likelihood of an issuer defaulting on its debt obligations.

• Investment-grade: These are bonds considered to have a relatively low risk of default. Ratings typically include Aaa/AAA, Aa/AA, A/A, and Baa/BBB (from highest to lowest quality within this category).

• Speculative (junk): These bonds carry a higher risk of default and are considered speculative investments. Ratings range from Ba/BB down to C/D.

• Historical average vs. recession default rates (Moody’s study): Default rates vary significantly by credit rating and economic conditions. For instance:

  • AAA-rated bonds historically have a default rate of approximately 0%.

  • BB-rated bonds have an average default rate of around 2.2%, which can surge to about 8% during a recession.

  • CCC-rated bonds show very high default rates, reaching approximately 48% during a recession.

• Spreads widen: The credit spreads (the difference in yield between a corporate bond and a risk-free bond of similar maturity) typically widen as the credit rating deteriorates and also tend to increase during economic downturns, reflecting higher perceived risk.

Equity Valuation – Infinite-Horizon DDM

• Dividend Discount Model (DDM): This model values a company's stock based on the present value of its expected future dividends. It assumes that the intrinsic value of a stock is determined by the sum of all future dividend payments, discounted back to the present.

• General identity: The current stock price ($P0$) is the sum of the present values of all future dividends ($divt$) expected to be paid, discounted at the required rate of return ($r$): $P<em>0=\sum</em>{t=1}^{\infty}\frac{\text{div}_t}{(1+r)^t}$

• Gordon-Growth model (constant $g$): A widely used version of the DDM that assumes dividends grow at a constant rate ($g$) indefinitely. It is applicable when the dividend growth rate is stable and less than the required rate of return ($g<r$): $P<em>0=\frac{\text{div}</em>1}{r-g}$ where $div_1$ is the expected dividend in the next period.

• Zero-growth special case: If there is no expected growth in dividends ($g=0$), the formula simplifies to a perpetuity: $P<em>0=\frac{\text{div}</em>1}{r}$.

• Example (Fledging Electronics): Given a required return $r=15\%$, an expected dividend next year $div1=5$, and an expected price at the end of year 1 $P1=110$. The current price $P0$ can be found by discounting the expected dividend and future price: $P0=(div1+P1)/(1+r) = (5+110)/(1+0.15) = 100$.

Philips Example – Sensitivity to g

• This example demonstrates how sensitive a stock's valuation (price $P_0$) is to small changes in the assumed constant dividend growth rate ($g$) in the Gordon Growth Model.

• Given a required return $r=6.5\%$ and next year's dividend $div_1=0.85€$.

  • If $g=3\%$, the calculated price $P_0=24.29€$.

  • If $g=4.55\%$, the calculated price $P_0=43.59€$, which closely matches the actual market price of 43.71€ on 30 Dec 2020.

• This illustrates that even small differences in the expected long-term growth rate can lead to significant variations in the calculated intrinsic value of a stock, making growth assumptions critical in equity valuation.

Growth, Retention & ROIC (BDM 9; Schauten & Steenbeek 2015)

• Sustainable growth ($g$): The rate at which a company can grow its earnings and dividends internally without issuing new equity or changing its financial leverage. It is determined by the company's reinvestment policy and its profitability. The formula is: $g=\text{retention ratio}\times ROIC$ (Return on Invested Capital).

• Share-price impact: The effect of a company's growth strategy on its share price depends crucially on the relationship between its Return on Invested Capital (ROIC) and its required rate of return ($r$).

  • Value-creating growth: Occurs if $ROIC>r$. When a company can reinvest its earnings at a rate higher than its cost of capital, each dollar retained and reinvested creates more than a dollar of shareholder value.

  • Value-neutral growth: Occurs if $ROIC=r$. Reinvestment at a rate equal to the cost of capital neither adds nor destroys value.

  • Value-destroying growth: Occurs if $ROIC<r$. Reinvesting earnings at a rate lower than the cost of capital actually reduces shareholder value.

• Crane Sporting Goods illustration: This example demonstrates the impact of reinvestment on stock price based on ROIC.

  • Base case: Current EPS $EPS1=6$, 100% payout ratio (no retention), and a required return $r=10\%$. Price $P0=60$ (since all EPS is paid as dividend, $P0=EPS1/r$).

  • Scenario: The company cuts its payout ratio to 75%, meaning it retains 25% of its earnings for reinvestment.

    • If $ROIC=12\%$ (which is $>r$): $g=0.25\times0.12=3\%$. The stock price increases to $P_0=64.29$ (value is created).

    • If $ROIC=8\%$ (which is $<r$): $g=0.25\times0.08=2\%$. The stock price decreases to $P_0=56.25$ (value is destroyed).

Relative Valuation – Multiples

• Relative valuation: This method estimates a company's value by comparing it to similar companies (or its own historical performance) using various financial ratios or multiples. The idea is that similar assets should sell for similar prices.

• Price/Earnings (P/E) ratio: One of the most common valuation multiples, it expresses the relationship between a company's share price and its earnings per share. It can be linked to the Gordon Growth Model:
  $\frac{P<em>0}{EPS</em>1}=\frac{k}{r-g}$ where $k$ is the payout ratio (dividends per share / earnings per share).

  • Example: If a competitor's P/E ratio is 26, and a company has expected earnings per share ($EPS1$) of $2.50€$, its implied stock price ($P0$) would be approximately $65€$ ($26 \times 2.50€$).

• Shiller-PE web reference for market context: This refers to resources like the Cyclically Adjusted Price-to-Earnings (CAPE) ratio, which is a broader market valuation multiple that averages earnings over 10 years to smooth out business cycle effects, providing context for overall market valuations.

Key Formula Compendium

• Bond price: The value of a bond is the present value of its future cash flows (coupons and face value) discounted by the appropriate spot rates for each maturity: $P=\sum<em>{t=1}^{T}\frac{CF</em>t}{(1+r_t)^t}$

• YTM: Yield-to-Maturity is the single discount rate ($y$) that equates a bond's observed market price to the present value of its promised cash flows: solve $P=\sum<em>{t=1}^{T}\frac{CF</em>t}{(1+y)^t}$.

• Forward rate: The implied future spot rate for a period starting at time $n$ and lasting for one period, derived from existing spot rates: $(1+<em>nf</em>{n+1})=\frac{(1+r<em>{n+1})^{n+1}}{(1+r</em>n)^n}$.

• Gordon model: For a stock with dividends growing at a constant rate $g$: $P<em>0=\frac{div</em>1}{r-g}$. The growth rate $g$ is linked to a company's reinvestment policy and profitability: $g=\text{retention}\times ROIC$.

• PE linkage: The Price/Earnings ratio can be expressed in terms of payout ratio ($k$), required return ($r$), and growth rate ($g$): $\frac{P<em>0}{EPS</em>1}=\frac{k}{r-g}$.

Practical & Ethical Notes

• Misinterpreting YTM: It is a common mistake to view YTM as a causal factor driving a bond's price. The correct understanding is that the market's collective valuation of the bond, reflected in its observable price, is what determines the implied YTM. Investors should derive YTM from market price, not predict price based on an assumed YTM.

• Growth strategies assessment: Managerial decisions regarding growth should be evaluated based on their capacity to create shareholder value, which occurs when the Return on Invested Capital (ROIC) exceeds the required return ($r$). Managers should avoid pursuing growth solely for the sake of improving short-term Earnings Per Share (EPS) figures if that growth is value-destroying (i.e., $ROIC<r$). Incentive systems should be aligned with ROIC-based metrics to encourage value-creating growth.

• Credit-rating methodologies: While credit ratings offer a standardized and transparent way to differentiate credit risk, they can sometimes lag behind rapid changes in an issuer's financial health or market sentiment. Therefore, investors should supplement credit ratings with real-time market-implied credit spreads (the extra yield demanded by the market for a risky bond compared to a risk-free one) to get a more current assessment of default risk.