Chapter 7 — Bond Pricing (MBA Foundations)

Learning Goals

  • Understand what a bond is: coupon, maturity, price, yield
  • Explore the relationship between bond prices and interest rates
  • Practice computing bond prices, current yield, yield to maturity (YTM)
  • Use Excel and online calculators for real examples

Bond Basics

A bond is a loan you make to a company or government. You receive periodic interest payments (“coupons”) and the face value (par, usually $1,000) at maturity.

TermMeaning
CouponAnnual interest payment, e.g., 5% of $1,000 = $50
PriceWhat you pay for the bond today (may be below or above $1,000)
Yield to MaturityThe annualized return if you hold the bond until maturity

4) Price–Yield Curve (hold everything else constant)

Tip: This keeps par, coupon, maturity, and frequency fixed while sweeping the YTM. Expect a downward-sloping curve.

Excel (How-to + Recipes)

Quick how-to

  • Price (present value of coupons + par): =ABS(PV(rate, nper, pmt, fv))
    Use rate = YTM/frequency, nper = years*frequency, pmt = coupon*par/frequency, fv = par.
  • Yield to Maturity (solve for rate): =RATE(nper, pmt, -Price, fv)*frequency
    Return is annualized by multiplying the per-period rate by frequency.
  • Current Yield (quick ratio): =(coupon*par)/Price
  • Duration (Excel built-in): =DURATION(Settle, Maturity, coupon, ytm, frequency, 1)   and   =MDURATION(...)

Copy/paste recipes

  • Price (semi-annual): =ABS(PV( y/2 , n*2 , coupon*par/2 , par ))
  • YTM (semi-annual): =RATE( n*2 , coupon*par/2 , -Price , par )*2
  • Current Yield: =(coupon*par)/Price
  • Duration (semi-annual): =DURATION(Settle, Maturity, coupon, ytm, 2, 1)   /   =MDURATION(...)
  • Zero price from YTM (semi-annual): =ABS(PV( y/2 , n*2 , 0 , par ))

Examples

  • Price a 5-yr, 4% annual coupon at 5% yield: =ABS(PV(5%, 5, 40, 1000)) → $957.88
  • YTM for 10-yr, 5% semi-annual coupon, price $950: =RATE(10*2, 0.05*1000/2, -950, 1000)*2 → ≈ 5.63%

Worked Examples (sanity checks) — with math + Excel

  1. YTM from price (semi-annual): 10-year, 5% coupon, par $1,000, price $950.
    🔍 Show solution (math + Excel)

    Given: N = 10 years, frequency = 2 ⇒ periods Np = 20; coupon/period = 0.05×1000/2 = $25.

    Math setup: Let i be the per-period yield. Solve 950 = Σ_{t=1}^{20} 25/(1+i)^t + 1000/(1+i)^{20}. Numerical root gives i ≈ 0.02815 ⇒ YTM = 2×i ≈ 5.63%.

    Excel: =RATE(10*2, 0.05*1000/2, -950, 1000)*2 → 5.63%

  2. Zero-coupon yield (semi-annual): 10-year zero priced $456.39.
    🔍 Show solution (math + Excel)

    Math: Price = PV of par only: 456.39 = 1000/(1 + r/2)^{20} ⇒ 1 + r/2 = (1000/456.39)^{1/20} ⇒ r = 2*((1000/456.39)^{1/20} − 1) ≈ 8.00%.

    Excel: =RATE(10*2, 0, -456.39, 1000)*2 → 8.00%

  3. Current Yield vs YTM: 5% coupon priced at $850.
    🔍 Show solution (math + Excel)

    Current yield (approx): CY = annual coupon / price = 0.05×1000 / 850 = 5.88%.

    Why YTM > CY here? Discount price < par implies a capital gain component at maturity, so YTM exceeds current yield.

    Illustrative YTM (assume 10y, semi-annual): =RATE(10*2, 0.05*1000/2, -850, 1000)*2 → ≈ 7.12% (changes with maturity).

Tip: sanity-check sensitivity with Excel =DURATION()/=MDURATION() for the same inputs.

Interactive Demo — Bond Pricing Game (in-page)

Answer within ±$1.00 for prices or ±0.05% for yields to be marked correct. Click “Show solution” to see math and the Excel formula for the exact setup.

Problem

Scoreboard

Correct: 0   |   Attempts: 0   |   Accuracy: —

Real-World Yield Curve (live link + quick Q&A)

Visit: U.S. Treasury Yield Curve or the interactive UST Yield Curve explorer.

⚠️ Teaching note (updated Sept 12, 2025): The curve is currently not inverted. The 2-year is ~3.56% and the 10-year is ~4.06% (spread ≈ +0.50%). The 2s/10s curve had been inverted for roughly ~2 years (mid-2022 → late-2024), the longest stretch in modern records.
What does a normal (upward-sloping) curve usually mean? (click to reveal)
  • Story: Investors demand higher yields to lock money up longer → inflation/term risk priced into long bonds.
  • Economy signal: Expansion baseline; policy not tight.
  • Student takeaway: Long-term rates > short-term. If you need cash soon, avoid long duration (bigger price swings). If your horizon is long, you can earn more but accept interest-rate risk.
What does an inverted curve mean? (click to reveal)
  • Story: Short rates (anchored by Fed policy) are high; markets expect future cuts → long yields below short yields.
  • Economy signal: Often a slowdown/recession risk signal; credit conditions tight.
  • Student takeaway: T-bills/CDs can yield more than 10-year Treasuries. Defensive posture and shorter duration make sense for near-term goals.
What does a humped curve mean? (click to reveal)
  • Story: Mixed signals—front end tight, mid maturities softer, long end higher (inflation/term premium).
  • Economy signal: Transition/uncertainty (policy shifts, inflation doubts).
  • Student takeaway: Consider a barbell (some short, some long) instead of concentrating in the middle.

Practice

  1. Compute price of a 3-year zero-coupon bond at 5% yield.
  2. Find current yield for a 10-year 6% coupon bond selling at $950.
  3. Use Excel to solve YTM for IBM 5-year 4% coupon bond priced at $920.
  4. Graph price vs yield for a 10-year bond (Excel data table).

Homework of Chapter 7 (due by 7/24/2025)

  1. Firm AAA’s bonds price = $850. Coupon rate is 5% and par is $1,000. The bond has six years to maturity. Calculate current yield? (5.88%)
    🧩 Excel hint (click) CY ≈ annual coupon / price → =(5%*1000)/850 (format as %).
  2. Zero coupon bond: 10 years, price $250 → YTM? (14.35%)
    🧩 Excel hint (click) Semi-annual convention → =RATE(10*2, 0, -250, 1000)*2.
  3. Zero coupon bond: 10 years, yield 8% → price? ($456.39)
    🧩 Excel hint (click) Semi-annual PV of $1,000: =ABS(PV(8%/2, 10*2, 0, 1000)).
  4. Annual coupon 5%, 15 years. Draw price–yield profile.
    🧩 Excel hint (click) In A2:A50 put yields 1%…12%. In B2 use: =ABS(PV(A2, 15, 5%*1000, 1000)) (annual) or semi-annual: =ABS(PV(A2/2, 15*2, 5%*1000/2, 1000)). Fill down → Insert → Scatter with Smooth Lines.
  5. IBM 5-year 2% annual coupon priced $950 → YTM? (3.09%)
    🧩 Excel hint (click) Annual coupons: =RATE(5, 2%*1000, -950, 1000).
  6. IBM 10-year 4% semi-annual coupon priced $950 → YTM? (4.63%)
    🧩 Excel hint (click) Semi-annual: =RATE(10*2, 4%*1000/2, -950, 1000)*2.
  7. IBM 10-year 5% annual coupon; required return 8% → price? (798.70)
    🧩 Excel hint (click) Price from yield: =ABS(PV(8%, 10, 5%*1000, 1000)).
  8. IBM 5-year 5% semi-annual; required return 8% → price? ($878.34)
    🧩 Excel hint (click) =ABS(PV(8%/2, 5*2, 5%*1000/2, 1000)).
  9. IBM 20-year zero; required return 8% → price? (208.29)
    🧩 Excel hint (click) =ABS(PV(8%/2, 20*2, 0, 1000)).
  10. Collingwood Homes: 8.5% coupon, 18.5 years, price $964.20, semi-annual → YTM? (8.90%)
    🧩 Excel hint (click) =RATE(18.5*2, 8.5%*1000/2, -964.20, 1000)*2.
  11. Grand Adventure: 9.5% annual coupon, YTM 11.2%, 11 years → price? ($895.43)
    🧩 Excel hint (click) =ABS(PV(11.2%, 11, 9.5%*1000, 1000)).
  12. D&L Movers zero: price $319.24, par $1,000, YTM 9.17% → years? (12.73 years)
    🧩 Excel hint (click) Semi-annual periods then ÷2: =NPER(9.17%/2, 0, -319.24, 1000)/2.
  13. Zero: face $1,000, price $212.56, 25 years → YTM? (6.29%)
    🧩 Excel hint (click) =RATE(25*2, 0, -212.56, 1000)*2.
  14. Stainless Tubs: 6% coupon, semi-annual, 11 years, price $989 → YTM? (6.14%)
    🧩 Excel hint (click) =RATE(11*2, 6%*1000/2, -989, 1000)*2.

Videos — homework help

Part I    Q1-Q2    Q3-Q4    Q5-Q8    Q9-Q14

Quiz 2

Help Video    Practice Quiz

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