Bonds and Their Valuation – Comprehensive Study Notes

What Is a Bond?

• Long-term debt instrument.
  • Borrower promises to pay principal (par value) + interest (coupon) on predetermined dates.
  • Think of it as an IOU between issuer (borrower) and investor (lender).
• Predominantly traded over-the-counter (OTC).
  • Ownership concentrated among large financial institutions.
  • The Wall Street Journal (print & online) lists Treasury, corporate, municipal, investment-grade, high-yield, and convertible issues.

Bond Market Structure

• OTC nature → prices negotiated dealer-to-dealer rather than on centralized exchange.
• Liquidity supplied by institutional investors (mutual funds, pension funds, insurance companies, banks).
• Market transparency improved by electronic quotation services but still less than equities.
• Corporate bonds categorised into:
  • Investment-grade (higher credit quality, lower yield).
  • High-yield or “junk” (lower credit quality, higher yield).

Core Features Embedded in Every Bond

• Par (Face) Value
  • Amount repaid at maturity; conventionally $\$1{,}000$ for U.S. corporates.
• Coupon Interest Rate
  • Stated nominal rate; usually fixed.
  • Dollar coupon = Coupon Rate × Par.
• Issue Date
  • Calendar date the bond is first sold.
• Maturity Date (N)
  • Years until final principal repayment.
• Yield to Maturity (YTM, $r_d$)
  • Internal rate of return if held to maturity and coupons reinvested at that rate.
  • Also called “promised yield.”

Special Contractual Provisions

• Call Provision
  • Gives issuer right to redeem early at a call price (par ± premium).
  • Valuable to issuer when market yields fall (refinancing motive), harmful to investors.
  • Usually includes:
  • Deferred call (protection period before first call).
  • Declining call premium (premium shrinks as maturity approaches).
• Sinking Fund
  • Contractually forces issuer to retire portion of principal periodically.
  • Two execution methods:
  1. Call a fixed percentage at par (favoured when r_d <\text{coupon} → bond trades at premium).
  2. Open-market purchase (favoured when r_d >\text{coupon} → bond trades at discount).
  • Reduces credit risk & average maturity but hurts investors if yields decline after issuance.
• Other Embedded Options / Structures
  • Convertible: exchangeable for issuer’s stock at holder’s discretion.
  • Warrants: detachable long-term call options on stock.
  • Putable: holder may force issuer to repurchase before maturity.
  • Income Bond: coupons paid only if issuer earns enough income.
  • Indexed (TIPS-style): coupon or principal adjusts with inflation.

Opportunity Cost of Debt Capital

• Discount rate used in valuation = rate available on alternative investments of equal risk: $r_i = r^* + IP + MRP + DRP + LP$ where:
  • $r^*$ = real risk-free rate.
  • $IP$ = inflation premium.
  • $MRP$ = maturity risk premium.
  • $DRP$ = default risk premium.
  • $LP$ = liquidity premium.

Time-Value Logic: Valuing Any Financial Asset

• General PV model:
  $V = \sum<em>{t=1}^{N} \frac{CF</em>t}{(1+r)^t}$
• For a standard coupon bond: $V<em>B = \sum</em>{t=1}^{N} \frac{C}{(1+r<em>d)^t} + \frac{M}{(1+r</em>d)^N}$
  • $C$ = annual coupon payment.
  • $M$ = maturity (par) value.

Worked Examples (Annual Coupons, $M=\$1{,}000$, $N=10$)

1. 10% coupon, $r_d = 10\%$
   • PV = $\$1{,}000$ (par).
   • Excel: =PV(0.10,10,100,1000).
2. 13% coupon, same $r_d$
   • PV = $\$1{,}184.34$ (premium because coupon > YTM).
3. 7% coupon, same $r_d$
   • PV = $\$815.66$ (discount because coupon < YTM).

How Prices Evolve Over Time (if $r_d$ constant)

• Premium bond price ↓ toward par as maturity nears.
• Discount bond price ↑ toward par.
• Par bond price remains at par.

Measuring Yields

• Yield to Maturity (YTM): Solve for $r_d$ in PV equation.
  • Example: 9% coupon, 10-yr bond priced at $\$887$ ⇒ $YTM = 10.91\%$ (=RATE(10,90,-887,1000)).
  • If price $=\$1{,}134.20$ ⇒ $YTM = 7.08\%$.
• Current Yield (CY): $CY = \dfrac{C}{P_0}$.
• Capital Gains Yield (CGY): price appreciation rate.
  • Relationship: $YTM = CY + CGY$.
• Effective Annual Rate (EAR) for non-annual coupons:
  $EAR = (1 + r_{nom}/m)^m - 1$
  where $m$ = # compounding periods.

Risk Dimensions in Bond Investing

Price (Interest-Rate) Risk

• Value sensitivity to changes in $r_d$.
• Longer maturity ⇒ higher duration ⇒ more price risk.
  • Example table (1-yr vs 10-yr, par = $\$1{,}000$):
  • At $r_d=5\%$: 10-yr price $=\$1,386$ (+38.6%), 1-yr $=\$1,048$ (+4.8%).
  • At $r_d=15\%$: 10-yr $=\$749$ (-25.1%), 1-yr $=\$956$ (-4.4%).

Reinvestment Risk

• Uncertainty about rate available to reinvest interim CFs.
• Higher for short-maturity or high-coupon bonds.
• Lottery metaphor: invest $\$500,000$ to live off interest.
  • Strategy A: roll one-year bonds; if rates drop to 3%, income falls from $\$50,000$ to $\$15,000$.
  • Strategy B: lock in 10-yr bond at 10% and secure $\$50,000$ annually.
• Conclusion matrix:
  • Short-term/high-coupon: low price risk, high reinvestment risk.
  • Long-term/low-coupon: high price risk, low reinvestment risk.

Semiannual Coupon Bonds

• Conversion rules:
  • $N_{semi} = 2N$
  • $r<em>{periodic} = r</em>{nom}/2$
  • $C_{periodic} = C/2$
• Example valuation: 10-yr, 10% semiannual coupon, $r_d = 13\%$
  • $N=20, I/YR=6.5, PMT=50$ ⇒ PV $= -\$834.72$ (=PV(.065,20,50,1000)).
• EAR comparison:
  • Semiannual EAR at 10% nominal: $EFFECT(0.10,2)=10.25\%$.
  • Prefer semiannual over annual if quoted nominal equal because of higher EAR.
• Cross-price equivalence: if semiannual priced at $\$1{,}000$ (EAR = 10.25%), annual bond offering same EAR should price at $\$984.80$.

Callable Bonds & Yield to Call (YTC)

• YTC mirrors YTM calculation but replace:
  • $N$ with years until first call.
  • $FV$ with call price.
• Example: 10-yr, 10% semiannual, price = $\$1,135.90$, callable in 4 yrs at $\$1,050$.
  • Compute semiannual periodic rate: 3.568%.
  • $YTC_{nom}=3.568\%\times2=7.137\%$, $EAR = 7.26\%$.
• Likelihood of call:
  • Premium bonds (coupon > market yield) more likely to be called → investors should expect to earn YTC.
  • Par or discount bonds → expect YTM.

Credit (Default) Risk

• Promised return > expected return because of possibility of default.
• Influenced by:
  • Financial ratios: Debt/Capital, TIE, Current Ratio.
  • Contract provisions: secured vs unsecured, senior vs subordinated, sinking fund, maturity.
  • Qualitative factors: earnings stability, regulation, litigation risk, pension & labour obligations.
• Security Types (priority descending):
  • Mortgage bonds (secured by real assets).
  • Debentures (unsecured).
  • Subordinated debentures (junior claim).
• Ratings Agencies (probability of default proxy):
  • Moody’s: $Aaa, Aa, A, Baa$ (investment grade); $Ba, B, Caa, C$ (junk).
  • S&P: $AAA$ → $D$.

Bankruptcy Framework

• Governed by U.S. Federal Bankruptcy Act.
• Chapter 11 (Reorganization)
  • Company petitions to halt creditor actions; 120 days to file reorg plan.
  • Trustee supervises; management often remains.
  • Must prove “worth more alive than dead.” Otherwise court orders liquidation.
• Chapter 7 (Liquidation)
  • Assets sold; proceeds distributed by legal priority:
  1. Secured creditors (from collateral proceeds).
  2. Trustee/admin costs.
  3. Wages (subject to cap).
  4. Taxes.
  5. Unfunded pension liabilities.
  6. Unsecured creditors.
  7. Preferred shareholders.
  8. Common shareholders.
• Unsecured creditors usually recover little in liquidation, motivating them to vote for reorganization plans that cut their claims but allow firm survival.

Summary Connections & Implications

• Bond pricing integrates time value, credit quality, liquidity, optionality.
• Understanding embedded options (call, put, convert) critical for true yield comparison.
• Risk management balances price risk vs reinvestment risk; duration matching & immunization strategies stem from these concepts.
• Bankruptcy procedures influence recovery rates and hence default risk premiums.
• Real-world investors constantly arbitrate among bonds of different coupons/maturities/compounding frequencies to equalize effective yields.