Week 3: Bond and Equity Valuation (Ch. 6, 7)

Companies finance themselves with a mix of debt and equity. Debt can be a regular bank loan or, for larger companies, bonds: loans that can be traded like shares. This week answers three questions. What is a fair price for a bond? What is a fair price for a share? And what determines the interest rates that are used to discount these cash flows? Both questions are applications of week 1 and 2: the value of a security is the present value of its expected cash flows.

The lecture starts with news on central bank rates: the US Federal Reserve and the ECB raised rates sharply in 2022 and 2023 to fight inflation and started cutting in 2024 and 2025. When rates were expected to keep rising (September 2022), investment-grade companies rushed to issue bonds before borrowing costs increased further. Interest rate decisions therefore affect financing costs and bond prices directly.

Definition and terminology of a bond

• Face value (par value, principal, F): the amount repaid at maturity, for example $10,000 on a US Treasury bond. If nothing is mentioned, assume a face value of 1,000.
• Coupon (C): the stated interest payment. The coupon rate is the annual coupon divided by the face value.
• Maturity (T): the date (or number of years) until the face value is repaid.
• Issuing price: when the bond is issued for the first time, the bondholder pays the issuing price; this is the amount actually loaned.
• Yield to maturity (YTM): the discount rate required in the market on the bond.

Example (Krueger Enterprises): 100,000 bonds of $10,000 each with a 5% annual coupon and 2-year maturity. $1 billion is borrowed; interest of $50 million is paid at the end of each year; at the end of year 2 both $50 million interest and the $1 billion principal are paid. The discount rate the market requires on the bond is its yield to maturity.

(Book) Debt differs from equity: it is not an ownership interest (no voting rights), interest is tax deductible while dividends are not, and unpaid debt is a liability that can lead to bankruptcy. The maximum reward on debt is fixed; the upside of equity is unlimited.

Bond features in the indenture (book)

The indenture is the written agreement between the corporation and its creditors (deed of trust); a trustee represents the bondholders. It contains the basic terms, the amount issued, the security, the repayment arrangements, the call provisions and the protective covenants.

• Registered versus bearer form: a registrar records owners and pays them directly; a bearer bond pays whoever holds it (hard to recover if lost, and the company cannot reach its bondholders).
• Security: collateral (pledged securities or assets) or mortgage securities (real property). An unsecured bond has no specific pledge; an unsecured debt with maturity under about 10 years is often called a note. (In the UK a debenture is secured, in the US unsecured.)
• Seniority: senior debt is paid before junior or subordinated debt in default; debt is never subordinated to equity.
• Repayment and sinking fund: a sinking fund is an account managed by the trustee to retire part of the bonds before maturity.
• Call provision: gives the firm the option to repurchase the bonds at a stated call price before maturity; the call premium is the call price minus par. A deferred call provision makes the bond call-protected for some years. A make-whole call pays bondholders about what the bonds are worth.
• Protective covenants: negative covenants ('thou shalt not': limit dividends, no extra debt, no pledging assets, no mergers) and positive covenants ('thou shalt': maintain minimum working capital, provide audited statements, maintain collateral).

Bond types

Three basic types:

(Book) Other types: government bonds (Treasury notes and bonds; normally no default risk because the government can print money, but Eurozone countries can default because the ECB controls the money supply); floating-rate bonds (coupons tied to an index, often with a floor and a cap, the 'collar'); inflation-linked bonds; catastrophe bonds (lose value after a stated catastrophe); bonds with warrants (right to buy shares at a fixed price, hence a low coupon); income bonds (coupons only if income suffices); convertible bonds (can be swapped for shares); put bonds (holder can force the issuer to buy back); and contingent convertibles (CoCo bonds). Most bonds trade over the counter, and the bond market (especially government bonds) has a far larger trading volume than the stock market.

Bond valuation

Example (Pixie plc): 10 years to maturity, annual coupon £80, face value £1,000, market rate for similar bonds 12%. PV of face value = 1,000/1.12^10 = £321.97. PV of coupons = 80 x [1/0.12 - 1/(0.12 x 1.12^10)] = £452.02. Bond value = £773.99.

Example (Treasury bond): 10 years, 4.25% annual coupon on $10,000 (coupon $425), market rate 1.5%. PV face value = 10,000/1.015^10 = $8,617; PV coupons = 425 x A(1.5%, 10) = $3,919; price = $12,536. The bond sells far above par because its coupon is much higher than the market rate.

Bond prices and market discount rates

Example (TNT): 10-year bond, face value €1,000, annual coupon 20% (€200). Price = 200 x A(r, 10) + 1,000/(1 + r)^10.

Bond values move in the opposite direction of market interest rates:

A price list of international bond issues (8 May 2008) shows bonds quoted below 100 (discount), above 100 (premium) and at 100 (par). Prices are quoted as a percentage of face value.

Bond prices and time to maturity

For the TNT bond (face value €1,000, coupon €200) the prices at different remaining maturities are:

As maturity approaches, the price of a premium bond falls towards par and the price of a discount bond rises towards par (the 'pull to par'); at maturity the bond is worth exactly its face value. A par bond stays at par if the market rate does not change. One year before maturity the price is (C + F)/(1 + r); t years before maturity it is C x A(r, t) + F/(1 + r)^t.

Quiz (slide): if the market interest rate increases, the value of a plain vanilla bond decreases because its fixed later payments are discounted at a higher rate. The coupon does not adjust.

Interest rate risk

Interest rate risk is the risk that a bond's value changes because market interest rates change. A graph of bond value against the interest rate for maturities of 1, 5, 10, 20 and 30 years (all with a 20% coupon) shows that all lines cross at the face value when r equals the coupon rate, but the longer the maturity, the steeper the line.

• All other things equal, the longer the time to maturity, the greater the interest rate risk, because the face value is discounted much more strongly over longer periods.
• All other things equal, the lower the coupon rate, the greater the interest rate risk, because a relatively larger share of the bond's cash flows (face value versus coupons) comes late in its life.

Both effects are summarised in the duration of a bond (the weighted average timing of its cash flows), treated in the course Investment Management. (Book) Interest rate risk increases with maturity, but at a decreasing rate. The 2012 Libor scandal (Barclays and other banks manipulating the benchmark rate) illustrates how even small changes in interest rates affect enormous amounts of contracts.

Yield to maturity

Example: price £108,042, face value £100,000, semi-annual coupon £3,000, maturity 5 years (10 periods). Solving 108,042 = 3,000 x A(r, 10) + 100,000/(1 + r)^10 gives r = 2.1% per six months. The YTM is typically stated in simple interest terms: 4.2% per year (2 x 2.1%).

Exercise: price $9,000, face value $10,000, annual coupon 10% ($1,000), 10 years. Solve 9,000 = 1,000 x A(r, 10) + 10,000/(1 + r)^10: YTM = 11.75%. Because the bond sells at a discount, the YTM is above the coupon rate. In Excel: put FV, coupon rate, T, C and r in cells, compute PV annuity =C*(1-(1/(1+r)^T))/r and PV FV =FV/(1+r)^T, sum them into the price cell, and let Solver set the price cell to 9,000 by changing r.

(Book) With semi-annual coupons, the coupon per period is half the annual coupon and the number of periods doubles. A zero coupon bond (Bocelli: €1,000 face value, 5 years, price €508.35) yields 7% per half year, 14% per year, and all interest (€491.65) is paid at maturity.

Inflation and interest rates

Hyperinflation shows why the distinction matters: in Germany in 1923 banknotes of 100 billion marks were issued and children played with piles of worthless notes; in Zimbabwe in 2008 ten zeros were removed from the currency, after which the exchange rate still went from about 1,000 to 300 trillion Zimbabwe dollars per US dollar in a few months.

(Book) The Fisher effect says investors look at real returns: if inflation rises, nominal rates rise so that the real rate stays the same.

Discounting: nominal or real. Never mix rates and cash flows. Nominal cash flows are discounted at the nominal rate; real cash flows at the real rate. Both approaches give the same answer.

Example: nominal discount rate 14%, expected inflation 5%, nominal cash flows -1,000 now, 600 in year 1 and 650 in year 2.

• Nominal approach: NPV = -1,000 + 600/1.14 + 650/1.14^2 = 26.47.
• Real approach: (1) make cash flows real: 600/1.05 = 571 and 650/1.05^2 = 590; (2) real discount rate = 1.14/1.05 - 1 = 8.57%; (3) NPV = -1,000 + 571/1.0857 + 590/1.0857^2 = 26.47.

Term structure of interest rates

An upward-sloping term structure (long rates above short rates) arises when inflation is expected to rise and the interest rate risk premium adds to it. A downward-sloping (inverted) term structure arises when expected inflation falls enough to outweigh the interest rate risk premium. A humped shape is also possible. The euro area AAA yield curve on 12 September 2025 rose from about 1.9% at short maturities to about 3.3% at 20 to 25 years. The Treasury yield curve plots yields on government coupon bonds against maturity; it is almost the same as the term structure (which uses pure discount bonds). Central bank rates were cut sharply after the financial crisis (Lehman, 2008), the euro crisis and COVID, and raised in 2022 to 2023 to fight inflation.

(Book) Beyond the term structure, a corporate bond's yield contains a default risk premium (compensation for the possibility of default; lower ratings mean higher yields), a taxability premium (compensation for unfavourable tax treatment) and a liquidity premium (compensation for lack of liquidity). So a bond yield reflects six things: the real rate plus premiums for expected inflation, interest rate risk, default risk, taxability and illiquidity. Note that the yield on a risky bond is a promised yield: if the issuer defaults, the realised return is lower.

Bond ratings and default risk

A price list shows two Swiss franc bonds where the one with the higher coupon (BNG, 4.875%) sells for less (99.992) than one with a lower coupon (IBK, 2.25%, 100.72). Normally a higher coupon means a higher price; the explanation is differences in maturity and especially in default risk: investors demand a higher yield from a riskier borrower.

Investment-grade bonds are rated at least BBB (S&P, Fitch) or Baa (Moody's); below that bonds are speculative, 'junk' or 'high-yield'. D means default. Ratings only concern the possibility of default, not interest rate risk: a AAA bond can still be very volatile. Bonds that fall from investment grade to junk are 'fallen angels'. Sovereign ratings depend on political risk, economic strength, government debt and monetary and fiscal flexibility; corporate ratings mainly on financial risk and the ability to meet debt payments. Examples: Fitch downgraded France in September 2025 amid political turmoil, raising its borrowing costs. The lecture also points to the role of rating agencies in the 2007 to 2008 financial crisis (securitisation and CDOs that received too-high ratings).

Equity: returns versus prices

Price is the absolute price quoted in a market (for example Unilever at $39.20). Return is the change in an investment's value (39.20/39.99 - 1 = -1.98%).

Share price as the PV of dividends

If you hold a share for one year: P0 = D1/(1 + R) + P1/(1 + R). But P1 = D2/(1 + R) + P2/(1 + R), and so on. Substituting repeatedly gives

(Book) Shares are harder to value than bonds because even the promised cash flows are unknown, equity has no maturity, and the required return cannot easily be observed. A firm that never pays anything to shareholders would be worthless; 'non-dividend-paying' firms are valued on the dividends they are expected to pay later. Three special cases of dividend patterns: zero growth, constant growth and differential (non-constant) growth.

Zero growth

Constant growth: the dividend growth model

• The last dividend of €8 was just paid (not received by a new buyer) and dividends grow 10% per year, R = 20%: P0 = 8 x 1.1/(0.20 - 0.10) = €88.
• The next dividend is €8 and then grows 10% per year: P0 = 8/(0.20 - 0.10) = €80.

Reading carefully whether the given dividend is D0 (just paid) or D1 (next) is a classic exam trap. The derivation uses the geometric series with q = (1 + g)/(1 + R).

Three-minute exercise (British Sky Broadcasting): last dividend £0.176, growth 4% forever, discount rate 10%. Dividend after 8 years: 0.176 x 1.04^8 = £0.2409. Share price today: 0.176 x 1.04/(0.10 - 0.04) = £3.05.

If g were equal to or larger than R, the model would give nonsense (the value would be infinite), so the model only works if R > g.

Differential (non-constant) growth

A firm may grow very fast for a few years (g1, possibly even above R) and then settle at a lower sustainable rate g2 < R. The value is the PV of a growing annuity for the high-growth phase plus the PV of a delayed growing perpetuity for the low-growth phase.

Example (Irresistible Snoep NV): next year's dividend €2; from year 2 dividends grow 15% for 4 years; after that 10% per year; R = 20%.

With the growing annuity formula: 2/(0.20 - 0.15) x [1 - (1.15/1.20)^5] = 7.67. Low-growth phase: D6 = 2 x 1.15^4 x 1.1 = 3.8478; P5 = 3.8478/(0.20 - 0.10) = 38.48; PV = 38.48/1.2^5 = 15.46. Value of the share = 7.67 + 15.46 = €23.13. (The slides show 7.76 and 23.22; the table on the same slide gives 7.67, which is the correct figure.) The most common mistake is identifying the start of constant growth wrongly: the growing perpetuity formula applied to D6 gives the value at t = 5, which is discounted 5 years, not 6.

(Book) Other cases: a firm that pays its first dividend of €0.50 in year 5, growing 10% afterwards, R = 20%: P4 = 0.50/0.10 = €5 and P0 = 5/1.2^4 = €2.41. The two-stage model with g1 for t years and g2 afterwards is the same as the differential growth model.

Components of the required return (book)

Preference shares (book): have priority over ordinary shares for dividends and in liquidation, usually pay a fixed dividend as a percentage of a stated value, often without voting rights. Dividends are usually cumulative (unpaid dividends carry forward as arrears and must be paid before ordinary dividends) but are not debts of the firm. Preference shares resemble debt (fixed payments, ratings, sometimes sinking funds) but are equity for tax purposes. A preference share with a fixed dividend forever is valued as a perpetuity: P = D/R. Dealers trade from their own inventory (bid price below ask price; the spread is their profit); brokers only match buyers and sellers. The P/E ratio is linked to growth: firms with high growth opportunities have high P/E ratios.

Caveats and equity valuation in practice

Difficulties in practice: future cash flows are uncertain; the return the market expects is uncertain; the life of the investment is uncertain (equity can theoretically last forever); we need good forecasts. Therefore the fundamental value of equity is uncertain. Retail investors in practice look at financial health (earnings, revenue growth, profitability), valuation ratios (P/E, price-to-book, price-to-sales), growth projections (analyst forecasts, management guidance), market sentiment (fear versus greed, the VIX), economic indicators (interest rates, inflation, GDP growth), news and events, and technical analysis (chart patterns, moving averages; scientifically there is no evidence that this works). Bubbles and crashes, such as the Nasdaq peak in March 2000 followed by a collapse of about 78%, show how far prices can deviate from fundamentals for a while.

Wrap-up of chapters 4 to 7: every valuation is a present value. Single payment: PV = Ct/(1 + r)^t and FV = C0(1 + r)^t. Multiple different cash flows: PV = sum of Ci/(1 + r)^i. Level cash flows: fixed perpetuity C/r, growing perpetuity C/(r - g), fixed annuity C/r - C/r x 1/(1 + r)^T, growing annuity C/(r - g) - C/(r - g) x ((1 + g)/(1 + r))^T. Bonds combine an annuity and a single payment; shares are perpetuities, growing perpetuities or combinations of a growing annuity and a delayed growing perpetuity.