Bonds (101) Study Notes

Coupon Bond Valuation: Basics and Examples

  • Key concept: A coupon bond pays fixed periodic coupons plus the par value at maturity. The value (price) is the present value of all future cash flows discounted at the market yield.
  • General valuation formula: P = ext{PV of coupons} + ext{PV of par} = ext{PMT} \times \sum_{t=1}^{N} \frac{1}{(1+i)^t} + \frac{FV}{(1+i)^N} where:
    • $P$ = price of the bond
    • $i$ = yield per period (market interest rate)
    • $N$ = number of periods to maturity
    • $PMT$ = periodic coupon payment
    • $FV$ = face (par) value
  • Example 1 (Practice 1): 10-year bond, par $1000, annual coupon 10% ($PMT = 100$), yield $i = 10\%$.
    • Cash flows: $100$ each year for years 1–10, plus $1000$ at year 10.
    • Price calculation:
      VB=∑t=110100(1+0.10)t+1000(1+0.10)10VB = \sum_{t=1}^{10} \frac{100}{(1+0.10)^t} + \frac{1000}{(1+0.10)^{10}}
    • Result: $P = 1000$ (since coupon rate equals yield).
    • Intuition: When coupon rate equals yield, price equals par value.
  • Example 2 (Practice 2): 1-year bond, par $1000$, annual coupon 10% ($PMT = 100$), yield $i = 10\%$.
    • Price calculation:
      P=100+1000(1+0.10)1P = \frac{100 + 1000}{(1+0.10)^1}
    • Result: $P = 1000$.
  • Practice 2 takeaway: If the coupon rate equals the yield, the price tends toward par over the life of the bond.

Interest Rates, Maturity and Bond Values

  • Fundamental relation: Bond prices move inversely with market interest rates.
  • Long vs short maturity:
    • Longer maturity bonds exhibit larger price swings when yield changes (higher price sensitivity or interest-rate risk).
    • Shorter maturity bonds exhibit smaller price swings.
  • Example scenarios (illustrative values from the transcript):
    • When yield rises to $13\%$:
    • Long bond (10-year): price ≈ $837.21$.
    • Short bond (1-year): price ≈ $973.45$.
    • When yield falls to $7\%$:
    • Long bond (10-year): price ≈ $1210.71$.
    • Short bond (1-year): price ≈ $1028.04$.
  • Practical implication: Greater risk (volatility) for longer maturities due to higher interest-rate sensitivity.

Maturity and Bond Value (Par, Premium, Discount)

  • Given a 10% coupon and changing yields, bond values shift as follows:
    • Par value: $1000$.
    • Premium bond: price above par when yield < coupon.
    • Discount bond: price below par when yield > coupon.
  • Example price observations (from the transcript):
    • At yield 13% with various maturities: long ≈ $837.21$; short ≈ $973.45$.
    • At yield 7% with various maturities: long ≈ $1210.71$; short ≈ $1028.04$.
  • Conceptual takeaway: As yield drops, long-term bonds gain more price appreciation (premium increases more); as yield rises, long-term bonds lose more price value (price drops more).

Practice Question: Bond Price Behavior with Yield Changes

  • 15-year bond, annual coupon rate 8%, fixed until maturity, yield to maturity = 6%.
  • Which statement is correct?
    • a. The bond is currently selling at a price below its par value. (False, coupon > yield ⇒ price above par.)
    • b. If market rates remain unchanged, the price one year from now will be higher than today. (Unlikely when price is at a premium and time to maturity decreases; price generally drifts toward par.)
    • c. The bond should currently be selling at its par value. (False, coupon > yield ⇒ premium.)
    • d. If market rates remain unchanged, the price one year from now will be lower than today. (True, premium will amortize toward par as maturity approaches.)
    • e. If market rates decline, the price of the bond will also decline. (False: rates decline generally raise price.)
  • Correct answer: d. Explanation: With an 8% coupon and 6% yield, the bond trades at a premium. As time passes (and yield unchanged), the remaining coupons are the same but there is less time to receive them, pulling price toward par, so the price tends to fall over time.

Important Price–Yield Relationships

  • Key statements to memorize:
    • Bond values and interest rates are inversely related. When market rates rise, bond prices fall; when rates fall, prices rise.
    • Duration risk increases with longer maturity and lower coupon rate. Longer time to maturity and smaller coupon payments mean greater sensitivity to rate changes.
    • As a bond approaches maturity, its value converges toward its par value.
  • Quick diagram intuition (described in the transcript):
    • Market Rate and Coupon Rate influence Value and Par relations; higher market rates reduce value; higher coupon rates support higher value; par acts as a convergence anchor at maturity.

Yield to Maturity (YTM) Practice: Discount and Premium Bonds

  • Discount bond example:
    • A 9% coupon, 10-year, $1000 par bond sells for $887. Find YTM.
    • Cash flows: coupons of $90 each year for 10 years, plus $1000 at year 10, with price $-887$ today.
    • Calculator result (from transcript): YTM ≈ 10.91%10.91\%.
  • Premium bond example:
    • The same bond sells for $1,134.20. Find YTM.
    • Calculator result (from transcript): YTM ≈ 7.08%7.08\%.
  • Consistency: YTM for a discount bond should be greater than the coupon rate; for a premium bond, YTM should be less than the coupon rate. The results above align with price–yield relationships.

Current Yield, Capital Gains Yield, and Total Yield

  • Definitions:
    • Current Yield (CY):
      CY=INTP<em>0CY = \frac{INT}{P<em>0} where $INT$ is the annual coupon payment and $P0$ is the current price.
    • Capital Gains Yield (CGY):
      CGY=P<em>1−P</em>0P<em>0CGY = \frac{P<em>1 - P</em>0}{P<em>0} where $P1$ is the price one year later (assuming the same yield environment).
    • Total Yield (TY):
      TY=CY+CGY(=YTM, under the stated assumptions)TY = CY + CGY \quad (= \text{YTM, under the stated assumptions})
  • Practice results (from the transcript):
    • Discount bond: $P0 = 887$, $INT = 90$, next-year price $P1 = 893.87$.
    • CY ≈ 90887≈10.15%\frac{90}{887} \approx 10.15\%
    • CGY ≈ (893.87−887)/887≈0.76%(893.87 - 887)/887 \approx 0.76\%
    • TY ≈ 10.91%10.91\%
    • Premium bond: $P0 = 1134.20$, $INT = 90$, next-year price $P1 = 1124.52$.
    • CY ≈ 901134.20≈7.94%\frac{90}{1134.20} \approx 7.94\%
    • CGY ≈ (1124.52−1134.20)/1134.20≈−0.86%(1124.52 - 1134.20)/1134.20 \approx -0.86\%
    • TY ≈ 7.08%7.08\%
  • Note: TY equals the yield implied by the price path; CY and CGY provide components of that return.

Yield to Call (YTC) Practice (Callable Bonds)

  • Concept: For callable bonds, YTC is the yield assuming the issuer calls the bond at the next possible call date, at the call price.
  • Example (from transcript): A discount bond with 9% coupon, 10-year maturity, callable in 5 years at a call price of $1090, price $887.
    • Inputs for the callable scenario: N = 5, PV = -887, PMT = 90, FV = 1090.
    • Calculator result (from transcript): YTC ≈ 14%14\%.
  • Premium bond case (same bond, priced at 1134.20): YTC is not explicitly listed in the transcript for the premium callable case, but the method is the same: use the call date and call price to solve for the yield that equates the price to the PV of the cash flows up to the call.
  • Practical takeaway: YTC will generally be higher than YTM for bonds with high likelihood of being called when interest rates fall, because investors require compensation for the risk of early redemption at a higher price.

Semiannual Coupons: YTM (Practice)

  • Concept: When coupons are semiannual, the cash flows occur every half-year, and the yield is typically quoted as a nominal annual rate with semiannual compounding.
  • Example (from transcript): A 9% semiannual-coupon, 10-year, $1000 par bond sells for $887.
    • Setup for semiannual valuation:
    • N = 20 (semistructured periods)
    • PMT = 45 (since 9% of $1000 per year split into two payments: $90 per year → $45 every half-year)
    • FV = 1000
    • PV = -887
    • The calculator would solve for the yield per half-year, then double it to get the nominal annual YTM with semiannual compounding.
    • Transcript note: The output shown is listed as “18” (likely a display artifact for the annualized yield or a per-period value). A clear numerical solution (from standard practice) would yield a per-half-year rate around ≈ 5.5% and an annualized nominal YTM ≈ 11–12% (roughly, depending on rounding).
  • Practical takeaway: Semiannual coupon bonds typically have lower quoted YTM per half-year than the equivalent annual YTM of the same effective annual yield because of the different compounding convention; convert between per-half-year and annual rates accordingly when interpreting results.

Summary of Key Relationships and Practical Implications

  • Prices move inversely with yields: higher yields mean lower prices; lower yields mean higher prices.
  • Duration and interest-rate risk increase with longer maturity and lower coupon rates.
  • All else equal, bonds converge to par value as they near maturity.
  • Yield metrics tie together:
    • YTM reflects total expected return if held to maturity with reinvestment of coupons at the same rate.
    • Current yield captures cash return relative to price today.
    • Capital gains yield captures price change over the period as a percentage of price.
    • For many standard cases, TY ≈ YTM, but CY and CGY provide component insights.
  • Callable features can significantly alter YTM calculations; call risk can push YTC higher (as in the example with a 5-year call and price below par yielding ≈ 14%).
  • Semiannual compounding changes the interpretation of YTM; always convert per-period yields to the annual rate when comparing across instruments.

Practical exam takeaway: Be able to (a) compute price from coupon and yield, (b) interpret price changes with rate shifts across different maturities, (c) decompose yields into current and capital-gains components, (d) handle callable bonds via YTC, and (e) adjust for semiannual coupon structures when needed.