Unit 7 introduces fixed income securities — instruments that pay a predetermined stream of cash flows. Bonds are the world's largest financial market and the primary tool governments and corporations use to raise long-term capital. Understanding bonds means understanding the inverse relationship between prices and yields, how to price any fixed-income security using TVM, and how bond ratings translate directly into the cost of borrowing. For BBYM community members, bonds also represent accessible, low-risk savings vehicles — particularly I Bonds and municipal bonds that offer tax advantages.
Part 1 — Core Topics Explained
Every major concept tested on the Unit 7 assessment
📋 Learning Objectives
- Identify and define the key features of a bond: par value, coupon rate, maturity, and required return
- Price a bond using the present value of its future cash flows (coupons + par)
- Explain why bond prices and yields move in opposite directions — and give a numerical example
- Define yield to maturity (YTM) and distinguish it from current yield and coupon rate
- Explain interest rate risk and how duration measures a bond's sensitivity to rate changes
- Interpret bond ratings and explain how ratings affect borrowing costs
- Compare Treasury, corporate, municipal, and I Bonds — including the tax advantages of municipal bonds
- Connect bonds to BBYM community finance: evaluating savings bonds, municipal bonds, and CDFI bond issuances
1. Bond Anatomy — The Five Key Features
A bond is a long-term debt instrument in which a borrower (issuer) promises to pay a lender (bondholder) a series of interest payments plus the return of principal at maturity. Every bond has five defining features:
| Feature | Definition | Example | Where It Appears in the Pricing Formula |
|---|---|---|---|
| Par Value (Face Value) | The principal amount the issuer will repay at maturity. Almost always $1,000 for corporate bonds. | $1,000 | M in the bond price formula — the lump sum received at maturity |
| Coupon Rate | The stated annual interest rate as a percentage of par value. Set at issuance and fixed for the life of the bond (for fixed-rate bonds). | 5% on $1,000 = $50/year | Determines INT (coupon payment) = Par × Coupon Rate |
| Coupon Payment (INT) | The actual dollar interest paid each period. For semiannual bonds (standard US), paid every 6 months. | $50/year = $25 every 6 months | INT in the numerator of each period's PV calculation |
| Maturity Date | The date on which the issuer repays par value and the bond expires. Can range from 1 to 30+ years. | 10 years from issuance | N = number of periods in the pricing formula |
| Required Return (rₛ) | The market interest rate for bonds of similar risk and maturity — the discount rate used to price the bond. Changes constantly as market conditions change. | 7% when coupon is 5% | rₛ in the denominator of each period's PV calculation |
2. The Fundamental Inverse Relationship — The Most Important Bond Concept
The single most tested concept in bond analysis: when interest rates rise, bond prices fall; when rates fall, bond prices rise. This is mathematically inevitable, not coincidental.
Why the Inverse Relationship Is Unavoidable:
A bond's coupon payment is fixed forever at issuance. If market rates rise after the bond is issued, new bonds pay higher coupons — making the old bond's fixed coupon less attractive. The only way the old bond can compete with new higher-yielding bonds is for its price to fall, so that the same fixed coupon represents a higher yield on the lower purchase price.
Conversely, if rates fall, the old bond's fixed coupon is now better than new bonds — so buyers bid up its price until the yield equals the new lower market rate.
A bond's coupon payment is fixed forever at issuance. If market rates rise after the bond is issued, new bonds pay higher coupons — making the old bond's fixed coupon less attractive. The only way the old bond can compete with new higher-yielding bonds is for its price to fall, so that the same fixed coupon represents a higher yield on the lower purchase price.
Conversely, if rates fall, the old bond's fixed coupon is now better than new bonds — so buyers bid up its price until the yield equals the new lower market rate.
Discount Bond
Required Return > Coupon Rate
(rₛ = 7%, coupon = 5%)
(rₛ = 7%, coupon = 5%)
Price < $1,000
Bond trades below par. Market demands more yield than the coupon provides.
Par Bond
Required Return = Coupon Rate
(rₛ = 5%, coupon = 5%)
(rₛ = 5%, coupon = 5%)
Price = $1,000
Bond trades at face value. Market rate equals the coupon rate exactly.
Premium Bond
Required Return < Coupon Rate
(rₛ = 3%, coupon = 5%)
(rₛ = 3%, coupon = 5%)
Price > $1,000
Bond trades above par. The coupon exceeds the market rate — buyers pay extra for it.
Assessment Q7 Answer — Explained:
"A bond with a 5% coupon is trading in a market where required returns rise to 7%. The bond's price will: Decrease below par value."
The bond was issued paying 5%. Now the market requires 7%. No rational investor pays $1,000 for a bond earning 5% when they can buy a new bond earning 7% for the same price. The 5% bond's price must fall until its yield (coupon ÷ price) rises to approximately 7%. That price will be below $1,000 — it trades at a discount.
"A bond with a 5% coupon is trading in a market where required returns rise to 7%. The bond's price will: Decrease below par value."
The bond was issued paying 5%. Now the market requires 7%. No rational investor pays $1,000 for a bond earning 5% when they can buy a new bond earning 7% for the same price. The 5% bond's price must fall until its yield (coupon ÷ price) rises to approximately 7%. That price will be below $1,000 — it trades at a discount.
3. Yield to Maturity (YTM) — The True Return
Yield to maturity is the total annualized return an investor earns if they buy the bond today at its current market price and hold it until maturity, receiving all coupons and the par value at maturity. It is the bond's internal rate of return (IRR).
| Yield Measure | Formula | What It Tells You | Limitation |
|---|---|---|---|
| Coupon Rate | Annual Coupon ÷ Par Value | The interest rate set at issuance; never changes | Ignores the market price — useless for evaluating bonds trading above or below par |
| Current Yield | Annual Coupon ÷ Current Market Price | Approximate annual income return at today's price | Ignores capital gain/loss at maturity — understates YTM for discount bonds, overstates for premium |
| Yield to Maturity (YTM) | The rₛ that makes PV of all cash flows = current price (requires calculator/iteration) | The true total annualized return if held to maturity | Assumes all coupons are reinvested at the YTM rate — which may not happen in practice |
YTM vs. Current Yield — Why the Difference Matters:
A bond with a 5% coupon trading at $857 (discount bond, market rates = 7%):
Current Yield = $50 ÷ $857 = 5.84%
YTM = ~7.0% (accounts for the $143 capital gain when $1,000 par is returned at maturity)
The current yield understates the true return by 1.16 percentage points because it ignores the fact that you paid $857 but receive $1,000 at maturity. YTM captures both the coupon income AND the price appreciation — always use YTM for accurate comparisons.
A bond with a 5% coupon trading at $857 (discount bond, market rates = 7%):
Current Yield = $50 ÷ $857 = 5.84%
YTM = ~7.0% (accounts for the $143 capital gain when $1,000 par is returned at maturity)
The current yield understates the true return by 1.16 percentage points because it ignores the fact that you paid $857 but receive $1,000 at maturity. YTM captures both the coupon income AND the price appreciation — always use YTM for accurate comparisons.
4. Interest Rate Risk and Duration
Interest rate risk is the risk that a bond's value will fall due to rising interest rates. Duration measures how sensitive a bond's price is to changes in interest rates — the higher the duration, the more the price moves for a given rate change.
| Factor | Effect on Duration / Rate Sensitivity | Why |
|---|---|---|
| Longer maturity | Higher duration — more rate sensitive | Cash flows extend further into the future, where discounting has a larger effect on PV |
| Lower coupon rate | Higher duration — more rate sensitive | More of the bond's value comes from the distant par payment (less from near-term coupons) |
| Zero-coupon bond | Maximum duration = maturity | 100% of value comes from the single par payment at maturity — maximally sensitive to rate changes |
| Higher coupon rate | Lower duration — less rate sensitive | More value is received early via coupons, reducing sensitivity to the distant par payment |
Practical Rule — Duration and Rate Changes:
Modified Duration ≈ % change in bond price for a 1% change in yield.
A bond with duration of 8 years loses approximately 8% in price if yields rise 1%.
A bond with duration of 3 years loses approximately 3% in price for the same 1% rate rise.
This is why long-term bonds are far riskier during rising rate environments than short-term bonds. Retirees and conservative investors hold shorter-duration bonds specifically to reduce this risk.
Modified Duration ≈ % change in bond price for a 1% change in yield.
A bond with duration of 8 years loses approximately 8% in price if yields rise 1%.
A bond with duration of 3 years loses approximately 3% in price for the same 1% rate rise.
This is why long-term bonds are far riskier during rising rate environments than short-term bonds. Retirees and conservative investors hold shorter-duration bonds specifically to reduce this risk.
5. Municipal Bonds and Tax Advantages
Municipal bonds (munis) are issued by state and local governments to finance public projects — schools, roads, hospitals, water systems. Their defining feature: interest income is exempt from federal income tax (and often state/local taxes too).
Tax-Equivalent Yield — Comparing Munis to Taxable Bonds
Taxable Equivalent Yield = Muni Yield ÷ (1 − Tax Rate)
Example: 4% muni yield for a 25% tax-bracket investor:
TEY = 4% ÷ (1 − 0.25) = 4% ÷ 0.75 = 5.33%
This investor needs a taxable bond yielding 5.33% or more to beat the after-tax return of the 4% muni.
TEY = 4% ÷ (1 − 0.25) = 4% ÷ 0.75 = 5.33%
This investor needs a taxable bond yielding 5.33% or more to beat the after-tax return of the 4% muni.
BBYM Community Connection — Municipal Bonds Finance Local Projects:
The Birmingham Water Works, Jefferson County Schools, and Birmingham airport improvements have all been financed in part through municipal bond issuances. When Birmingham issues muni bonds, it borrows from investors (including community members) and repays them with tax-free interest. For BBYM families in higher tax brackets, municipal bonds from their own community can offer competitive after-tax returns while directly financing local infrastructure. This is community-aligned investing at its most direct.
The Birmingham Water Works, Jefferson County Schools, and Birmingham airport improvements have all been financed in part through municipal bond issuances. When Birmingham issues muni bonds, it borrows from investors (including community members) and repays them with tax-free interest. For BBYM families in higher tax brackets, municipal bonds from their own community can offer competitive after-tax returns while directly financing local infrastructure. This is community-aligned investing at its most direct.
6. I Bonds — Inflation-Protected Savings for Community Members
Series I Savings Bonds (I Bonds) are US Treasury instruments that pay a combined fixed rate plus an inflation adjustment (tied to CPI). They are designed specifically to protect savers from inflation.
| Feature | I Bond Detail | Comparison to Regular Savings Account |
|---|---|---|
| Interest Rate | Fixed rate + CPI inflation rate, adjusted every 6 months | Savings account rates may not keep pace with inflation |
| Purchase Limit | $10,000/year per person electronically via TreasuryDirect.gov | No limit on savings account deposits |
| Minimum Hold | Must hold for 12 months; penalty of 3 months interest if redeemed before 5 years | Full liquidity from savings accounts |
| Tax Treatment | Federal tax only; exempt from state/local tax; can be deferred until redemption | Savings account interest is fully taxable each year |
| Safety | Backed by full faith and credit of US Treasury — zero default risk | FDIC insured up to $250,000 |
| Best For | Emergency funds beyond the immediate 6-month cash reserve; medium-term savings where inflation protection matters | Immediate emergency fund (first 3–6 months of expenses) |
BBYM Savings Strategy — Layering Safety and Growth:
Tier 1: Keep 3–6 months of expenses in a high-yield savings account or credit union account (liquid, FDIC/NCUA insured).
Tier 2: Hold $5,000–$10,000 in I Bonds (inflation protection, federal guarantee, slightly less liquid).
Tier 3: Invest the remainder in a diversified portfolio for long-term growth (covered in later units).
This layered approach gives BBYM families liquidity, inflation protection, and growth potential without unnecessary risk at any tier.
Tier 1: Keep 3–6 months of expenses in a high-yield savings account or credit union account (liquid, FDIC/NCUA insured).
Tier 2: Hold $5,000–$10,000 in I Bonds (inflation protection, federal guarantee, slightly less liquid).
Tier 3: Invest the remainder in a diversified portfolio for long-term growth (covered in later units).
This layered approach gives BBYM families liquidity, inflation protection, and growth potential without unnecessary risk at any tier.
Part 2 — Bond Pricing: Formulas & Worked Examples
Pricing a bond is TVM applied to a stream of coupon payments plus a lump-sum par value
The Bond Pricing Formula
A bond's fair market price equals the present value of all future coupon payments (an annuity) plus the present value of the par value (a lump sum), both discounted at the required return rₛ.
Bond Price Formula
Vₛ = INT × [1 − 1/(1+rₛ)ⁿ] ÷ rₛ + M ÷ (1+rₛ)ⁿ
= PV of Coupon Annuity + PV of Par Value (lump sum)
INT = annual coupon payment (Par × Coupon Rate) | M = par value (usually $1,000)
rₛ = required return per period | N = number of periods to maturity
For semiannual bonds (standard): divide INT by 2, divide rₛ by 2, multiply N by 2
rₛ = required return per period | N = number of periods to maturity
For semiannual bonds (standard): divide INT by 2, divide rₛ by 2, multiply N by 2
Current Yield
Current Yield = Annual Coupon Payment ÷ Current Market Price
Quick approximation of yield — ignores capital gain/loss at maturity. Always less accurate than YTM.
Worked Example 1 — Discount Bond (Annual Coupons)
Setup: 5% coupon bond, $1,000 par, 10-year maturity. Market required return = 7%.
INT = $1,000 × 5% = $50/year | M = $1,000 | rₛ = 7% = 0.07 | N = 10
PV of coupons (annuity):
= $50 × [1 − 1/(1.07)¹⁰] ÷ 0.07
= $50 × [1 − 0.5083] ÷ 0.07
= $50 × 7.0236 = $351.18
PV of par value (lump sum):
= $1,000 ÷ (1.07)¹⁰ = $1,000 ÷ 1.9672 = $508.35
Bond Price = $351.18 + $508.35 = $859.53
The bond trades at $859.53 — a discount of $140.47 below par, because the market requires 7% but the bond only pays 5%. This confirms the Assessment Q7 answer: price decreases below par when required return rises above coupon rate.
INT = $1,000 × 5% = $50/year | M = $1,000 | rₛ = 7% = 0.07 | N = 10
PV of coupons (annuity):
= $50 × [1 − 1/(1.07)¹⁰] ÷ 0.07
= $50 × [1 − 0.5083] ÷ 0.07
= $50 × 7.0236 = $351.18
PV of par value (lump sum):
= $1,000 ÷ (1.07)¹⁰ = $1,000 ÷ 1.9672 = $508.35
Bond Price = $351.18 + $508.35 = $859.53
The bond trades at $859.53 — a discount of $140.47 below par, because the market requires 7% but the bond only pays 5%. This confirms the Assessment Q7 answer: price decreases below par when required return rises above coupon rate.
Worked Example 2 — Premium Bond (Annual Coupons)
Setup: 5% coupon bond, $1,000 par, 10-year maturity. Market required return = 3%.
INT = $50 | M = $1,000 | rₛ = 3% = 0.03 | N = 10
PV of coupons:
= $50 × [1 − 1/(1.03)¹⁰] ÷ 0.03
= $50 × [1 − 0.7441] ÷ 0.03
= $50 × 8.5302 = $426.51
PV of par:
= $1,000 ÷ (1.03)¹⁰ = $1,000 ÷ 1.3439 = $744.09
Bond Price = $426.51 + $744.09 = $1,170.60
The bond trades at $1,170.60 — a premium of $170.60 above par, because the 5% coupon exceeds the 3% market rate. Buyers pay extra for the above-market income stream.
INT = $50 | M = $1,000 | rₛ = 3% = 0.03 | N = 10
PV of coupons:
= $50 × [1 − 1/(1.03)¹⁰] ÷ 0.03
= $50 × [1 − 0.7441] ÷ 0.03
= $50 × 8.5302 = $426.51
PV of par:
= $1,000 ÷ (1.03)¹⁰ = $1,000 ÷ 1.3439 = $744.09
Bond Price = $426.51 + $744.09 = $1,170.60
The bond trades at $1,170.60 — a premium of $170.60 above par, because the 5% coupon exceeds the 3% market rate. Buyers pay extra for the above-market income stream.
Worked Example 3 — Semiannual Coupons (Standard US Bonds)
Setup: 6% coupon bond, $1,000 par, 5-year maturity, semiannual coupons. Required return = 8%.
Adjust for semiannual: INT = $30 per period | rₛ = 4% per period | N = 10 periods
PV of coupons:
= $30 × [1 − 1/(1.04)¹⁰] ÷ 0.04
= $30 × [1 − 0.6756] ÷ 0.04
= $30 × 8.1109 = $243.33
PV of par:
= $1,000 ÷ (1.04)¹⁰ = $1,000 ÷ 1.4802 = $675.56
Bond Price = $243.33 + $675.56 = $918.89
Discount bond: 6% coupon in an 8% required-return environment → price below par at $918.89.
Adjust for semiannual: INT = $30 per period | rₛ = 4% per period | N = 10 periods
PV of coupons:
= $30 × [1 − 1/(1.04)¹⁰] ÷ 0.04
= $30 × [1 − 0.6756] ÷ 0.04
= $30 × 8.1109 = $243.33
PV of par:
= $1,000 ÷ (1.04)¹⁰ = $1,000 ÷ 1.4802 = $675.56
Bond Price = $243.33 + $675.56 = $918.89
Discount bond: 6% coupon in an 8% required-return environment → price below par at $918.89.
Price-Yield Relationship Summary
| 5% Coupon Bond, $1,000 Par, 10 Years | Bond Price | Type | Current Yield |
|---|---|---|---|
| Required Return = 3% | $1,170.60 | Premium | $50 ÷ $1,170.60 = 4.27% |
| Required Return = 5% | $1,000.00 | Par | $50 ÷ $1,000 = 5.00% |
| Required Return = 7% | $859.53 | Discount | $50 ÷ $859.53 = 5.82% |
| Required Return = 10% | $692.77 | Deep Discount | $50 ÷ $692.77 = 7.22% |
As required return doubles from 5% to 10%, the bond price drops by $307 — a 30.7% loss of value. This is interest rate risk in action.
Part 3 — Bond Ratings & Types
How credit ratings work, what they mean for borrowing costs, and the major bond categories
Bond Credit Ratings — The DRP in Letter Form
Bond ratings are letter grades assigned by agencies (S&P, Moody's, Fitch) that summarize the issuer's creditworthiness and default risk. They directly determine the DRP component of the interest rate — and therefore the borrowing cost for every corporation, city, and school district that issues bonds.
| S&P / Moody's | Category | Meaning | Typical Yield Spread Over Treasury | BBYM Implication |
|---|---|---|---|---|
| Investment Grade | Highest quality — extremely strong capacity to meet financial commitments | +0.5 – 1.0% | Apple, Microsoft-level. Minimal default risk. Lowest borrowing cost. | |
| Investment Grade | Very high quality — very strong capacity to meet commitments | +1.0 – 1.5% | Major banks, stable governments. Still very safe. | |
| Investment Grade | Upper-medium grade — strong capacity but somewhat susceptible to economic changes | +1.5 – 2.0% | Most large S&P 500 companies. Solid but not the strongest. | |
| Investment Grade (minimum) | Adequate protection but adverse conditions could weaken capacity to pay | +2.0 – 3.0% | The investment-grade floor. Many institutional investors cannot hold below BBB. | |
| Speculative / "Junk" | Speculative — meaningful default risk; rated bonds are called "high yield" | +4.0%+ | Small companies, distressed issuers. Higher return potential but real default risk. |
The Investment-Grade Cliff — Why BBB to BB Is a Big Deal:
Falling from BBB (lowest investment grade) to BB (highest junk) is not a small step — it is a cliff. Many large institutional investors (pension funds, insurance companies, money market funds) are legally or contractually prohibited from holding below-investment-grade bonds. When a bond is downgraded to junk, these institutions must sell immediately, flooding the market with supply and crashing the bond's price. The issuer's borrowing costs can jump 2–4 percentage points overnight. This is called a "fallen angel" downgrade and can trigger a corporate crisis.
Falling from BBB (lowest investment grade) to BB (highest junk) is not a small step — it is a cliff. Many large institutional investors (pension funds, insurance companies, money market funds) are legally or contractually prohibited from holding below-investment-grade bonds. When a bond is downgraded to junk, these institutions must sell immediately, flooding the market with supply and crashing the bond's price. The issuer's borrowing costs can jump 2–4 percentage points overnight. This is called a "fallen angel" downgrade and can trigger a corporate crisis.
Major Bond Types — Side-by-Side Comparison
| Feature | US Treasury | Corporate | Municipal | I Bond |
|---|---|---|---|---|
| Issuer | US Federal Government | Corporations | State & local governments | US Treasury |
| Default Risk | Zero (DRP = 0) | Varies by rating (AAA to junk) | Very low to low (most munis are investment grade) | Zero (DRP = 0) |
| Federal Tax | Taxable | Taxable | Tax-exempt | Taxable (deferrable) |
| State/Local Tax | Exempt | Taxable | Usually exempt (own state) | Exempt |
| Typical Yield | Lowest (benchmark) | Treasury + DRP spread | Lower than corporate (offset by tax benefit) | Fixed rate + CPI adjustment |
| Inflation Protection | None (TIPS only) | None | None | Yes — CPI-linked rate |
| Best For | Safety, liquidity, benchmarking | Higher yield investors comfortable with credit risk | High-tax-bracket investors seeking tax-free income | Inflation-protected savings up to $10K/year |
| Purchase Access | TreasuryDirect.gov, brokerages | Brokerages, bond markets | Brokerages, financial advisors | TreasuryDirect.gov only |
Bond Ratings and Community Finance — The BBYM Connection
Why Bond Ratings Matter for Birmingham-Bessemer:
Jefferson County, Birmingham, and the Birmingham school system all issue municipal bonds. Their credit ratings directly determine the interest rate taxpayers pay on public debt:
Jefferson County was famously downgraded to junk status during its 2011 bankruptcy (the largest US municipal bankruptcy at the time) over sewer system financing. This made it significantly more expensive to borrow for public projects — costs ultimately borne by residents through higher taxes and reduced services.
A community that understands bond ratings can advocate for responsible public financial management — and recognize when political decisions are creating long-term borrowing cost problems. This is civic financial literacy at its most consequential.
Jefferson County, Birmingham, and the Birmingham school system all issue municipal bonds. Their credit ratings directly determine the interest rate taxpayers pay on public debt:
Jefferson County was famously downgraded to junk status during its 2011 bankruptcy (the largest US municipal bankruptcy at the time) over sewer system financing. This made it significantly more expensive to borrow for public projects — costs ultimately borne by residents through higher taxes and reduced services.
A community that understands bond ratings can advocate for responsible public financial management — and recognize when political decisions are creating long-term borrowing cost problems. This is civic financial literacy at its most consequential.
Part 4 — Key Terms Defined
Master these 16 terms for the Unit 7 assessment and all future fixed-income discussions
Bond
A long-term debt instrument in which an issuer (borrower) promises to pay a bondholder (lender) a series of interest payments (coupons) plus the return of principal (par value) at a specified maturity date. The most common form of long-term corporate and government debt.
Par Value (Face Value)
The principal amount printed on a bond — the amount the issuer promises to repay at maturity. Almost always $1,000 for US corporate bonds. The reference point for coupon rate calculations and for determining whether a bond trades at a discount, par, or premium.
Coupon Rate
The stated annual interest rate as a percentage of par value, set at issuance and fixed for the life of a standard bond. A 5% coupon on a $1,000 par bond pays $50/year. The coupon rate never changes — but the bond's market price (and therefore its yield) changes constantly as market rates move.
Coupon Payment (INT)
The actual dollar interest paid each period. Calculated as Par Value × Coupon Rate. For standard US bonds, paid semiannually (every 6 months). Even if the bond's market price changes, the coupon payment in dollars never changes — it is fixed at issuance.
Maturity Date
The date on which the bond expires and the issuer repays par value to the bondholder. Short-term bonds mature in 1–5 years; intermediate in 5–10 years; long-term in 10–30+ years. Longer maturities mean greater interest rate risk (higher duration).
Required Return (rₛ)
The market interest rate appropriate for bonds of similar risk, maturity, and liquidity — the discount rate used to price the bond. Changes continuously as market conditions, inflation expectations, and credit perceptions change. When rₛ > coupon rate, the bond trades at a discount; when rₛ < coupon rate, it trades at a premium.
Discount Bond
A bond whose current market price is below par value ($1,000). Occurs when the required market return exceeds the bond's coupon rate — the bond must sell at a lower price so that buyers earn the market rate on their investment. The bond will "pull to par" as it approaches maturity.
Premium Bond
A bond whose current market price is above par value. Occurs when the required market return is below the coupon rate — the bond's above-market coupon is valuable, so buyers bid up its price. Also "pulls to par" at maturity.
Yield to Maturity (YTM)
The total annualized return earned if a bond is purchased at today's market price and held until maturity, with all coupons reinvested at the YTM rate. The bond's IRR. The most complete measure of bond yield — accounts for coupon income, capital gain or loss at maturity, and the time value of both. The rₛ that makes the PV of all cash flows equal the current price.
Current Yield
Annual coupon payment divided by current market price. A quick, approximate measure of a bond's income return. Does NOT account for the capital gain or loss at maturity — underestimates YTM for discount bonds and overestimates it for premium bonds. Useful for quick comparisons but always inferior to YTM for accurate analysis.
Interest Rate Risk
The risk that a bond's market value will decline due to rising interest rates. All fixed-rate bonds carry interest rate risk. The risk is greater for longer-maturity and lower-coupon bonds. Cannot be eliminated for bonds held before maturity — but can be managed by matching bond maturities to investment horizons (immunization).
Duration
A measure of a bond's interest rate sensitivity — the weighted average time to receive all cash flows, expressed in years. Modified duration approximates the percentage price change for a 1% change in yield. Higher duration = more price-sensitive to rate changes. Longer maturity and lower coupon rate both increase duration.
Zero-Coupon Bond
A bond that pays no periodic interest — issued at a deep discount and redeems at par at maturity. The entire return comes from price appreciation. Has the highest duration of any bond type (equal to its maturity). Used for precise duration-matching in liability immunization strategies. US Savings Bonds (EE series) work similarly.
Investment-Grade Bond
A bond rated BBB/Baa or higher by major credit rating agencies — considered suitable for conservative investors. Many institutional investors (pension funds, insurance companies) are restricted to investment-grade bonds. Falling below BBB to "junk" status triggers forced selling and sharply higher borrowing costs.
Municipal Bond (Muni)
A bond issued by a state or local government to finance public projects. Interest is exempt from federal income tax (and often state/local taxes). Lower stated yields than comparable taxable bonds — but after-tax returns can be superior for investors in higher tax brackets. Tax-Equivalent Yield = Muni Yield ÷ (1 − Tax Rate).
I Bond (Series I Savings Bond)
A US Treasury savings bond whose interest rate combines a fixed rate plus a variable inflation adjustment (tied to CPI), updated every 6 months. Protects purchasing power against inflation. Capped at $10,000/year per person; must hold 12 months minimum; 3-month interest penalty if redeemed before 5 years. Purchase only through TreasuryDirect.gov.
Part 5 — Practice Questions
Show all work — these mirror the Unit 7 assessment format exactly
Conceptual Questions
Q1A bond with a 5% coupon is trading in a market where required returns rise to 7%. The bond's price will: A) Increase above par. B) Remain at par. C) Decrease below par. D) Become worthless. (This is the Unit 7 curriculum assessment question.)▼
Answer: C — Decrease below par value.
When required returns rise from 5% to 7%, the existing bond paying 5% becomes less attractive than new bonds paying 7%. To remain competitive, the bond's price must fall until its effective yield equals 7%. Since the $50 annual coupon is fixed, the price must drop to approximately $859 (on a 10-year, $1,000 par bond) so that $50/$859 ≈ 5.82% current yield, with the additional return coming from the $141 capital gain at maturity when the investor receives $1,000 for the $859 bond — bringing the total YTM to approximately 7%.
When required returns rise from 5% to 7%, the existing bond paying 5% becomes less attractive than new bonds paying 7%. To remain competitive, the bond's price must fall until its effective yield equals 7%. Since the $50 annual coupon is fixed, the price must drop to approximately $859 (on a 10-year, $1,000 par bond) so that $50/$859 ≈ 5.82% current yield, with the additional return coming from the $141 capital gain at maturity when the investor receives $1,000 for the $859 bond — bringing the total YTM to approximately 7%.
Q2Explain in plain language why bond prices and interest rates move in opposite directions. Use an analogy to make it intuitive.▼
The inverse relationship is mathematically inevitable because a bond's coupon is fixed at issuance while market rates constantly change.
Plain language: Imagine you buy a bond paying $50/year. Then the market rate rises and new bonds now pay $70/year. No one will pay full price ($1,000) for your $50-paying bond when they can buy a new $70-paying bond for $1,000. Your bond's price must fall until it becomes a bargain at its lower price — specifically, until the $50 coupon represents approximately 7% on the discounted price.
Analogy — Used Car Market: You paid $30,000 for a car. Now the manufacturer releases a better model for the same $30,000. Your used car's value immediately drops — not because it got worse, but because buyers have a better alternative. Bond prices work the same way: new bonds are always competing with existing bonds, and price adjusts to equalize returns.
Plain language: Imagine you buy a bond paying $50/year. Then the market rate rises and new bonds now pay $70/year. No one will pay full price ($1,000) for your $50-paying bond when they can buy a new $70-paying bond for $1,000. Your bond's price must fall until it becomes a bargain at its lower price — specifically, until the $50 coupon represents approximately 7% on the discounted price.
Analogy — Used Car Market: You paid $30,000 for a car. Now the manufacturer releases a better model for the same $30,000. Your used car's value immediately drops — not because it got worse, but because buyers have a better alternative. Bond prices work the same way: new bonds are always competing with existing bonds, and price adjusts to equalize returns.
Q3Which bond carries more interest rate risk — Bond A (5% coupon, 2-year maturity) or Bond B (5% coupon, 20-year maturity)? Explain using duration concepts. Which bond would you prefer if you expected rates to fall?▼
Bond B (20-year) carries far more interest rate risk.
Both bonds have the same 5% coupon, so the difference is entirely in maturity. Duration increases with maturity — Bond B's cash flows extend 20 years into the future, where the discounting effect of a rate change is dramatically amplified. Bond A returns most of its value quickly (in 2 years), so a rate change has little time to compound its effect on price.
A rough approximation using modified duration: if Bond B has a duration of ~13 years, a 1% rate increase drops its price by approximately 13%. Bond A with a duration of ~1.9 years drops by only ~1.9% for the same rate increase.
If expecting rates to fall: You want Bond B. When rates fall, bond prices rise — and higher-duration bonds rise more. If rates fall 1%, Bond B gains ~13% while Bond A gains only ~1.9%. Bond B is the better choice for investors anticipating rate cuts — it amplifies the price appreciation from falling rates.
Both bonds have the same 5% coupon, so the difference is entirely in maturity. Duration increases with maturity — Bond B's cash flows extend 20 years into the future, where the discounting effect of a rate change is dramatically amplified. Bond A returns most of its value quickly (in 2 years), so a rate change has little time to compound its effect on price.
A rough approximation using modified duration: if Bond B has a duration of ~13 years, a 1% rate increase drops its price by approximately 13%. Bond A with a duration of ~1.9 years drops by only ~1.9% for the same rate increase.
If expecting rates to fall: You want Bond B. When rates fall, bond prices rise — and higher-duration bonds rise more. If rates fall 1%, Bond B gains ~13% while Bond A gains only ~1.9%. Bond B is the better choice for investors anticipating rate cuts — it amplifies the price appreciation from falling rates.
Q4What is the difference between yield to maturity and current yield? Give an example where they differ significantly and explain which measure is more useful and why.▼
Current Yield = Annual Coupon ÷ Current Price. It measures only the income return — what the coupon pays relative to today's price.
YTM is the total annualized return including both coupon income AND any capital gain or loss at maturity. It is the bond's internal rate of return.
Example where they differ significantly — Deep Discount Bond:
5% coupon, $1,000 par, 10-year maturity, trading at $700 (deep discount):
Current Yield = $50 ÷ $700 = 7.14%
YTM = approximately 9.2% (includes the $300 capital gain over 10 years)
The current yield understates the true return by over 2 percentage points because it ignores the fact that you paid $700 but receive $1,000 at maturity — a $300 gain spread over 10 years.
YTM is more useful for virtually all bond analysis because it captures the complete return picture. Current yield is only useful as a quick approximation when discussing a bond's income characteristics — never for comparing bonds with different maturities or price levels.
YTM is the total annualized return including both coupon income AND any capital gain or loss at maturity. It is the bond's internal rate of return.
Example where they differ significantly — Deep Discount Bond:
5% coupon, $1,000 par, 10-year maturity, trading at $700 (deep discount):
Current Yield = $50 ÷ $700 = 7.14%
YTM = approximately 9.2% (includes the $300 capital gain over 10 years)
The current yield understates the true return by over 2 percentage points because it ignores the fact that you paid $700 but receive $1,000 at maturity — a $300 gain spread over 10 years.
YTM is more useful for virtually all bond analysis because it captures the complete return picture. Current yield is only useful as a quick approximation when discussing a bond's income characteristics — never for comparing bonds with different maturities or price levels.
Calculation Questions
Q5Price a bond with: 6% annual coupon, $1,000 par value, 8-year maturity, required return = 8%. Is it a discount or premium bond?▼
INT = $60 | M = $1,000 | rₛ = 8% = 0.08 | N = 8
PV of coupons:
= $60 × [1 − 1/(1.08)⁸] ÷ 0.08
= $60 × [1 − 0.5403] ÷ 0.08
= $60 × 5.7466 = $344.80
PV of par:
= $1,000 ÷ (1.08)⁸ = $1,000 ÷ 1.8509 = $540.27
Bond Price = $344.80 + $540.27 = $885.07
This is a discount bond — price ($885) is below par ($1,000) because the required return (8%) exceeds the coupon rate (6%). The investor earns the full 8% through a combination of the $60 coupon AND the $115 price appreciation to par at maturity.
PV of coupons:
= $60 × [1 − 1/(1.08)⁸] ÷ 0.08
= $60 × [1 − 0.5403] ÷ 0.08
= $60 × 5.7466 = $344.80
PV of par:
= $1,000 ÷ (1.08)⁸ = $1,000 ÷ 1.8509 = $540.27
Bond Price = $344.80 + $540.27 = $885.07
This is a discount bond — price ($885) is below par ($1,000) because the required return (8%) exceeds the coupon rate (6%). The investor earns the full 8% through a combination of the $60 coupon AND the $115 price appreciation to par at maturity.
Q6A $1,000 par bond with a 7% annual coupon has 5 years to maturity and is currently priced at $1,082.00. (a) Is this a discount or premium bond? (b) Calculate its current yield. (c) Is the YTM higher or lower than the coupon rate—and why?▼
(a) Price ($1,082) > Par ($1,000) → Premium bond. Required return is below the 7% coupon rate.
(b) Current Yield = $70 ÷ $1,082 = 6.47%
(c) YTM is lower than the coupon rate (7%).
Reason: This is a premium bond — the investor pays $1,082 today but only receives $1,000 at maturity. That $82 capital loss partially offsets the above-market coupon income. The YTM (which accounts for this loss) will be lower than both the coupon rate (7%) and the current yield (6.47%).
To verify: The YTM is approximately 5.0–5.5% — the rₛ that prices the bond at $1,082. This makes sense because a premium bond means the market only requires a return below 7% for this bond's risk level.
(b) Current Yield = $70 ÷ $1,082 = 6.47%
(c) YTM is lower than the coupon rate (7%).
Reason: This is a premium bond — the investor pays $1,082 today but only receives $1,000 at maturity. That $82 capital loss partially offsets the above-market coupon income. The YTM (which accounts for this loss) will be lower than both the coupon rate (7%) and the current yield (6.47%).
To verify: The YTM is approximately 5.0–5.5% — the rₛ that prices the bond at $1,082. This makes sense because a premium bond means the market only requires a return below 7% for this bond's risk level.
Q7A municipal bond yields 3.8%. Calculate the taxable-equivalent yield for investors in the following tax brackets: (a) 22%, (b) 32%, (c) 37%. At which bracket does the muni become more attractive than a comparable 5.5% corporate bond?▼
Taxable Equivalent Yield = Muni Yield ÷ (1 − Tax Rate)
(a) 22% bracket: 3.8% ÷ (1 − 0.22) = 3.8% ÷ 0.78 = 4.87%
(b) 32% bracket: 3.8% ÷ (1 − 0.32) = 3.8% ÷ 0.68 = 5.59%
(c) 37% bracket: 3.8% ÷ (1 − 0.37) = 3.8% ÷ 0.63 = 6.03%
Comparison to 5.5% corporate bond:
• 22% bracket: TEY = 4.87% < 5.5% corporate → Corporate bond wins after tax
• 32% bracket: TEY = 5.59% > 5.5% corporate → Municipal bond wins
• 37% bracket: TEY = 6.03% > 5.5% corporate → Municipal bond wins by a larger margin
The muni becomes superior somewhere between the 22% and 32% tax brackets — approximately at the 26–27% marginal rate (solve: 3.8% ÷ (1−t) = 5.5% → t = 1 − 3.8/5.5 = 30.9%). Investors in the 31%+ brackets should generally prefer municipal bonds over comparable taxable bonds at these yield levels.
(a) 22% bracket: 3.8% ÷ (1 − 0.22) = 3.8% ÷ 0.78 = 4.87%
(b) 32% bracket: 3.8% ÷ (1 − 0.32) = 3.8% ÷ 0.68 = 5.59%
(c) 37% bracket: 3.8% ÷ (1 − 0.37) = 3.8% ÷ 0.63 = 6.03%
Comparison to 5.5% corporate bond:
• 22% bracket: TEY = 4.87% < 5.5% corporate → Corporate bond wins after tax
• 32% bracket: TEY = 5.59% > 5.5% corporate → Municipal bond wins
• 37% bracket: TEY = 6.03% > 5.5% corporate → Municipal bond wins by a larger margin
The muni becomes superior somewhere between the 22% and 32% tax brackets — approximately at the 26–27% marginal rate (solve: 3.8% ÷ (1−t) = 5.5% → t = 1 − 3.8/5.5 = 30.9%). Investors in the 31%+ brackets should generally prefer municipal bonds over comparable taxable bonds at these yield levels.
Q8The Birmingham school district issues a 20-year general obligation bond rated AA with a 4.5% coupon. Two years later, the district is downgraded to BBB. How does this affect: (a) the required return on the bond, (b) the bond's market price, (c) the district's future borrowing costs?▼
(a) Required return increases. A downgrade from AA to BBB means the DRP component of the interest rate rises — investors now require higher compensation for the increased default risk. If AA bonds carried a 1.0–1.5% spread over Treasuries and BBB carries a 2.0–3.0% spread, the required return on the district's existing bonds might increase by 0.75–1.5 percentage points.
(b) The bond's market price falls. Higher required return (higher discount rate) means lower present value for the fixed future cash flows. The existing 4.5% coupon bond, originally priced near par, now trades at a discount as the market reprices it to reflect the higher required return. On a 20-year bond, this price drop could be substantial — potentially 8–15% depending on the magnitude of the rate increase.
(c) Future borrowing costs increase significantly. When the district next issues bonds, it must offer higher coupon rates to attract investors. At BBB vs. AA, the district might pay 1–2% more per year on all new debt — costing taxpayers millions in additional interest over multi-decade bond terms. This is the direct fiscal cost of credit rating deterioration — and a compelling argument for sound public financial management that BBYM students can use to evaluate local government decisions.
(b) The bond's market price falls. Higher required return (higher discount rate) means lower present value for the fixed future cash flows. The existing 4.5% coupon bond, originally priced near par, now trades at a discount as the market reprices it to reflect the higher required return. On a 20-year bond, this price drop could be substantial — potentially 8–15% depending on the magnitude of the rate increase.
(c) Future borrowing costs increase significantly. When the district next issues bonds, it must offer higher coupon rates to attract investors. At BBB vs. AA, the district might pay 1–2% more per year on all new debt — costing taxpayers millions in additional interest over multi-decade bond terms. This is the direct fiscal cost of credit rating deterioration — and a compelling argument for sound public financial management that BBYM students can use to evaluate local government decisions.
Q9Compare a 4% coupon, 30-year bond to a 4% coupon, 5-year bond. If interest rates rise by 2%, which bond loses more value in percentage terms? Explain using duration logic without calculating exact prices.▼
The 30-year bond loses dramatically more value — potentially 15–20% of its price vs. 4–5% for the 5-year bond — for the same 2% rate increase.
Duration logic:
The 30-year bond has a duration of approximately 17–19 years (still less than maturity because of coupon payments). A modified duration of ~17 means a 2% rate rise causes approximately 34% price decline.
The 5-year bond has a duration of approximately 4.4 years. A 2% rate rise causes approximately 8.8% price decline.
Why the difference? The 30-year bond's cash flows are spread over three decades. When you discount those distant cash flows at a higher rate, the PV reduction is enormous — especially the $1,000 par payment 30 years away, which shrinks dramatically as the discount rate rises. The 5-year bond returns its par value in just 5 years, so rate changes have far less time to compound their discounting effect.
Practical implication: A BBYM community fund investing in bonds to fund near-term obligations (within 5 years) should hold short-duration bonds. Only long-horizon investors who can ride out price fluctuations should hold long-duration bonds.
Duration logic:
The 30-year bond has a duration of approximately 17–19 years (still less than maturity because of coupon payments). A modified duration of ~17 means a 2% rate rise causes approximately 34% price decline.
The 5-year bond has a duration of approximately 4.4 years. A 2% rate rise causes approximately 8.8% price decline.
Why the difference? The 30-year bond's cash flows are spread over three decades. When you discount those distant cash flows at a higher rate, the PV reduction is enormous — especially the $1,000 par payment 30 years away, which shrinks dramatically as the discount rate rises. The 5-year bond returns its par value in just 5 years, so rate changes have far less time to compound their discounting effect.
Practical implication: A BBYM community fund investing in bonds to fund near-term obligations (within 5 years) should hold short-duration bonds. Only long-horizon investors who can ride out price fluctuations should hold long-duration bonds.
Q10A BBYM family member has $10,000 to invest for 3 years. They are considering: (a) a high-yield savings account at 4.5%, (b) a 3-year Treasury Note at 4.8%, (c) an I Bond at a current composite rate of 5.2%, (d) a BBB-rated corporate bond at 6.0%. Rank these options from safest to highest-yielding, and explain who should choose each.▼
Safety ranking (safest to least safe):
1. I Bond = Treasury Note (both are US government — zero default risk)
3. High-yield savings account (FDIC insured up to $250,000 — essentially zero risk for this amount)
4. BBB Corporate Bond (low but real default risk; also has interest rate risk if sold before maturity)
Yield ranking (lowest to highest): Savings (4.5%) < T-Note (4.8%) < I Bond (5.2%) < Corporate Bond (6.0%)
Who should choose each:
(a) High-yield savings account — Best for anyone who might need the money before 3 years. Full liquidity with no lock-up period. Good for the emergency fund layer of a financial plan. Worst after-tax return of the four options.
(b) Treasury Note — Best for investors who want guaranteed returns with no state tax (Treasury interest is exempt from state tax) and who are comfortable locking in for exactly 3 years. Marketable — can be sold before maturity if needed (with potential price risk).
(c) I Bond — Best if inflation is a concern and the investor can commit for at least 12 months (minimum hold). The inflation adjustment protects purchasing power better than the fixed-rate alternatives. Limit: only $10,000/year through TreasuryDirect. Slight inconvenience of early redemption penalty (3 months interest) before 5 years.
(d) BBB Corporate Bond — Best for investors comfortable with moderate credit risk who want the highest yield. The additional 0.8–1.5% over Treasuries compensates for the real (if small) default risk and potential price volatility. Not appropriate if the money is needed before maturity.
1. I Bond = Treasury Note (both are US government — zero default risk)
3. High-yield savings account (FDIC insured up to $250,000 — essentially zero risk for this amount)
4. BBB Corporate Bond (low but real default risk; also has interest rate risk if sold before maturity)
Yield ranking (lowest to highest): Savings (4.5%) < T-Note (4.8%) < I Bond (5.2%) < Corporate Bond (6.0%)
Who should choose each:
(a) High-yield savings account — Best for anyone who might need the money before 3 years. Full liquidity with no lock-up period. Good for the emergency fund layer of a financial plan. Worst after-tax return of the four options.
(b) Treasury Note — Best for investors who want guaranteed returns with no state tax (Treasury interest is exempt from state tax) and who are comfortable locking in for exactly 3 years. Marketable — can be sold before maturity if needed (with potential price risk).
(c) I Bond — Best if inflation is a concern and the investor can commit for at least 12 months (minimum hold). The inflation adjustment protects purchasing power better than the fixed-rate alternatives. Limit: only $10,000/year through TreasuryDirect. Slight inconvenience of early redemption penalty (3 months interest) before 5 years.
(d) BBB Corporate Bond — Best for investors comfortable with moderate credit risk who want the highest yield. The additional 0.8–1.5% over Treasuries compensates for the real (if small) default risk and potential price volatility. Not appropriate if the money is needed before maturity.
Q11Price a semiannual coupon bond: 8% annual coupon, $1,000 par, 6-year maturity, required return = 6% annually. Show the semiannual adjustment and determine if it is discount, par, or premium.▼
Semiannual adjustments: INT = $40/period | rₛ = 3%/period | N = 12 periods
PV of coupons:
= $40 × [1 − 1/(1.03)¹²] ÷ 0.03
= $40 × [1 − 0.7014] ÷ 0.03
= $40 × 9.9540 = $398.16
PV of par:
= $1,000 ÷ (1.03)¹² = $1,000 ÷ 1.4258 = $701.38
Bond Price = $398.16 + $701.38 = $1,099.54
This is a premium bond — price ($1,099.54) exceeds par ($1,000) because the coupon rate (8%) exceeds the required return (6%). Investors are willing to pay $99.54 above par for the privilege of receiving above-market coupon payments over 6 years.
PV of coupons:
= $40 × [1 − 1/(1.03)¹²] ÷ 0.03
= $40 × [1 − 0.7014] ÷ 0.03
= $40 × 9.9540 = $398.16
PV of par:
= $1,000 ÷ (1.03)¹² = $1,000 ÷ 1.4258 = $701.38
Bond Price = $398.16 + $701.38 = $1,099.54
This is a premium bond — price ($1,099.54) exceeds par ($1,000) because the coupon rate (8%) exceeds the required return (6%). Investors are willing to pay $99.54 above par for the privilege of receiving above-market coupon payments over 6 years.
Part 6 — Quick Reference Summary
Read this the night before the assessment
Unit 7 in 5 Essential Sentences
Sentence 1
A bond's fair price equals the present value of its coupon annuity plus the present value of its par value, both discounted at the required return rₛ: Vₛ = INT × [1−1/(1+rₛ)ⁿ]÷rₛ + M÷(1+rₛ)ⁿ.
Sentence 2
Bond prices and interest rates always move inversely: when required returns rise above the coupon rate, the bond trades at a discount (below par); when required returns fall below the coupon rate, it trades at a premium (above par) — this is the Assessment Q7 answer.
Sentence 3
Yield to maturity (YTM) is the complete total return if held to maturity — it captures both coupon income and capital gain/loss — and is always more accurate than current yield (Annual Coupon ÷ Price) for comparing bonds.
Sentence 4
Duration measures interest rate sensitivity: longer maturity and lower coupon both increase duration; a bond with 10-year duration loses approximately 10% of its price for every 1% rise in rates — making long-term bonds riskier in rising-rate environments.
Sentence 5
Municipal bonds offer federal tax-exempt income (use Taxable Equivalent Yield = Muni Yield ÷ (1−t) to compare); I Bonds provide inflation protection up to $10,000/year; bond ratings (AAA to junk) directly determine the DRP and borrowing cost for every issuer.
Must-Know Facts for the Assessment
| Concept / Formula | Answer |
|---|---|
| Bond price formula | Vₛ = INT × [1−1/(1+rₛ)ⁿ]÷rₛ + M÷(1+rₛ)ⁿ |
| Current yield | Annual Coupon ÷ Current Market Price |
| Tax-equivalent yield | Muni Yield ÷ (1 − Tax Rate) |
| Assessment Q7 answer | 5% coupon, market rises to 7% → price decreases below par |
| Discount bond condition | Required return > coupon rate → price < par |
| Premium bond condition | Required return < coupon rate → price > par |
| Par bond condition | Required return = coupon rate → price = $1,000 |
| YTM vs. current yield | YTM is more complete — includes capital gain/loss at maturity; current yield ignores this |
| Higher duration → | More price-sensitive to rate changes; longer maturity & lower coupon both increase duration |
| Investment-grade floor | BBB/Baa — below this = junk/speculative; falling from BBB triggers forced institutional selling |
| Municipal bond tax benefit | Interest exempt from federal income tax (and usually state/local of same state) |
| I Bond key facts | $10,000/year limit; 12-month minimum hold; 3-month interest penalty before 5 years; inflation-linked; TreasuryDirect.gov only |
| Semiannual bond adjustment | Divide annual coupon by 2, divide annual rate by 2, multiply years by 2 |
| US Treasury DRP | Zero — the risk-free benchmark; all other bonds priced as Treasury rate + spread |