Fixed Income Securities
Understand bonds, duration, and yield curve dynamics
⏱️ 24 min⚡ 8 interactions
1. The Original Financial Instrument
Before stocks, before crypto, there were bonds. For centuries, governments and corporations have borrowed money by issuing bonds—a promise to pay back principal plus interest. The bond market is twice the size of the stock market, yet most people don't understand how it works.
📜 Core Concept
A bond is a loan you make to a borrower (government or company). You receive regular coupon payments (interest) and get your principal back at maturity. Bond prices move inversely to interest rates: when rates rise, bond prices fall. The key metrics are yield (return), duration (interest rate sensitivity), and credit risk (default probability).
🏛️ Interactive: Choose Bond Type
Bond Type
treasury
Default Risk
None
Tax Status
Taxable
Liquidity
Highest
2. How Bonds Are Priced
🎓 The Math Behind Bond Prices
Present Value: The Foundation
A bond is just a stream of future cash flows: periodic coupon payments plus the face value at maturity. To find its price today, we calculate the present value (PV) of all these cash flows, discounted at the current market interest rate.
Bond Pricing Formula:
Bond Price = PV(Coupons) + PV(Face Value)
PV(Coupons) = Σ [C / (1+r)ⁿ] for each period n
PV(Face Value) = F / (1+r)ᴺ
Where:
C = Coupon payment per period
F = Face value (usually $1,000)
r = Market discount rate per period
N = Total number of periods
Example:
$1,000 face, 5% coupon, 10 years
C = $50/year (semi-annual = $25)
N = 20 periods (10 years × 2)
If market rate = 5%, price = $1,000
Key insight: The formula sums the present value of each coupon payment plus the lump sum at maturity. Each payment is discounted more heavily the further in the future it occurs. This is why time value of money matters!
Premium, Par, and Discount Bonds
The relationship between a bond's coupon rate (fixed at issuance) and the market rate (changes daily) determines whether it trades at a premium (above $1,000), at par (exactly $1,000), or at a discount (below $1,000).
Three Bond Pricing Scenarios:
1. Premium Bond (Coupon > Market Rate)
Scenario: You own 6% coupon bond, market rate drops to 4%
Logic: Your bond pays $60/year, new bonds only pay $40/year
Result: Your bond is MORE valuable, trades at $1,170 (premium +$170)
Example: 10-year bond, $1,000 face, 6% coupon, 4% market = $1,162 price
⚠️ Current yield (6% / 1.162 = 5.16%) < coupon rate (6%)
2. Par Bond (Coupon = Market Rate)
Scenario: 5% coupon bond, market rate is also 5%
Logic: Your bond pays same as new bonds—no advantage/disadvantage
Result: Bond trades at exactly $1,000 (par value)
Example: Any maturity, $1,000 face, 5% coupon, 5% market = $1,000 price
Current yield (5%) = coupon rate (5%) = YTM (5%)
3. Discount Bond (Coupon < Market Rate)
Scenario: You own 3% coupon bond, market rate rises to 5%
Logic: Your bond pays $30/year, new bonds pay $50/year
Result: Your bond is LESS valuable, trades at $845 (discount -$155)
Example: 10-year bond, $1,000 face, 3% coupon, 5% market = $845 price
⚠️ Current yield (3% / 0.845 = 3.55%) > coupon rate (3%)
📊 Why This Matters for Investors:
• Premium bonds: Great income (high coupon), but price will fall toward par as maturity approaches (capital loss)
• Par bonds: Fair deal—yield equals market rate, price stays stable (assuming rates don't change)
• Discount bonds: Lower income (low coupon), but price will rise toward par as maturity approaches (capital gain)
Total return is the same for all three (yield to maturity), just different mix of income vs capital gain!
Market reality: When the Fed raises rates, existing bonds (with lower coupons) fall in price. In 2022, Fed hiked from 0% to 5%, and 10-year Treasury bonds fell 16% despite being "risk-free"—they had interest rate risk! The inverse price-rate relationship is NON-NEGOTIABLE in fixed income.
Yield Metrics: Three Ways to Measure Returns
Bonds have multiple yield measures, each telling you something different. Coupon yield is fixed. Current yield changes with price. Yield to maturity (YTM) is the true return if you hold to maturity.
Comparing Yield Measures:
Coupon Yield (Nominal Yield)
Formula:Coupon Rate = (Annual Coupon / Face Value)
Example:$50 coupon / $1,000 face = 5.0%
This never changes! Set at issuance and stays fixed for life of bond.
Current Yield
Formula:Current Yield = (Annual Coupon / Current Price)
Example:$50 coupon / $950 price = 5.26%
Changes daily with price! Useful for comparing income across bonds, but ignores capital gains/losses.
Yield to Maturity (YTM) ⭐ Most Important
Total return if held to maturity (coupon income + capital gain/loss)
Formula:Solve: Price = Σ[Coupon/(1+YTM)ⁿ] + Face/(1+YTM)ᴺ
Example (discount bond):$950 price, 5% coupon, 10Y → YTM = 5.59%
Example (premium bond):$1,050 price, 5% coupon, 10Y → YTM = 4.45%
YTM is the discount rate that makes PV of all cash flows equal current price. It's the IRR of the bond!
⚠️ Common Mistake:
Many investors confuse current yield with total return. A bond yielding 6% sounds great, but if you bought it at a $200 premium and it matures at par, you'll lose $200 over the bond's life. YTM accounts for this—a premium bond's YTM is LOWER than its coupon because you'll eat the capital loss.
Rule: For premium bonds, YTM < Current Yield < Coupon. For discount bonds, YTM > Current Yield > Coupon.
Pro tip: YTM assumes you reinvest all coupons at the YTM rate (rarely happens in real life). If rates fall, your reinvestment return is lower (reinvestment risk). If rates rise, you're happy—you can reinvest at higher rates! This is why laddering (staggered maturities) reduces reinvestment risk.
💰 Interactive: Bond Price Calculator
$1000
5%
10 years
5%
Bond Price
$1000.00
Premium +$0.00
Annual Coupon
$50.00
Current Yield
5.00%
Premium/Discount
+0.0%
💡 Key Rule: When market rates equal the coupon rate, bonds trade at a par. This is the inverse relationship between rates and prices!
📅 Interactive: Payment Frequency
Payment Amount
$25.00
Payments/Year
2
Total Payments
20
Total Interest
$500
3. Duration: Measuring Interest Rate Risk
🎓 Duration: The #1 Risk Metric for Bonds
What is Duration? (Not Just Time to Maturity!)
Duration is the weighted average time until you receive a bond's cash flows. But more importantly, it measures interest rate sensitivity: how much the bond's price changes when rates move. A duration of 7 means a 1% rate increase causes ~7% price decline.
Two Types of Duration:
1. Macaulay Duration (Weighted Average Time)
Formula:
Duration = Σ [(t × PV of Cash Flow_t) / Bond Price]
Example: 10-year bond, 5% coupon, 5% yield
• Year 1: $50 coupon, PV = $47.62, weight = 4.8%
• Year 5: $50 coupon, PV = $39.18, weight = 3.9%
• Year 10: $50 coupon + $1,000 face, PV = $644.61, weight = 64.5%
Macaulay Duration = ~8.11 years
Think of it as the "center of gravity" of the bond's cash flows. Higher coupons pull duration DOWN (more cash up front). Longer maturity pulls duration UP (cash flows further out).
2. Modified Duration ⭐ (Price Sensitivity)
Formula:
Modified Duration = Macaulay Duration / (1 + Yield/n)
(n = payment frequency, typically 2 for semi-annual)
Example: Same bond as above
• Macaulay Duration = 8.11 years
• Yield = 5%, semi-annual (n=2)
• Modified Duration = 8.11 / (1 + 0.05/2) = 7.91
Interpretation: 1% rate ↑ → 7.91% price ↓
Price Change Formula:
ΔPrice ≈ -Modified Duration × ΔYield × Price
Bond: $1,000, Mod Duration 7.91, rates ↑ 1%
ΔPrice = -7.91 × 0.01 × $1,000 = -$79.10
New price: $920.90 (down 7.91%)
Why modified duration? Macaulay tells you TIME (years), modified tells you RISK (% price change). Portfolio managers use modified duration to measure and hedge interest rate risk. A $100M bond portfolio with duration 6.0 will lose ~$6M if rates rise 1%.
Duration Drivers: What Makes Duration High or Low?
Duration isn't random—it's determined by three factors: time to maturity, coupon rate, and yield level. Understanding these helps you predict which bonds are risky when rates rise.
Duration Rules of Thumb:
Rule 1: Maturity ↑ → Duration ↑ (But Not 1:1!)
2-year bond (5% coupon, 5% yield):Duration = 1.95
10-year bond (5% coupon, 5% yield):Duration = 8.11
30-year bond (5% coupon, 5% yield):Duration = 17.29
Note: 30Y bond has 15× maturity of 2Y, but only 8.9× duration! Distant cash flows matter less (heavy discounting).
Rule 2: Coupon Rate ↑ → Duration ↓
Same 10-year maturity, 5% yield, varying coupons:
0% coupon (zero-coupon bond):Duration = 10.0
3% coupon bond:Duration = 8.75
5% coupon bond:Duration = 8.11
8% coupon bond:Duration = 7.25
Higher coupons = more cash early = less sensitive to rate changes. Zero-coupon bonds have duration = maturity (most risky!).
Rule 3: Yield ↑ → Duration ↓ (Slightly)
10-year bond, 5% coupon, varying yields:
2% yield environment:Duration = 8.72
5% yield environment:Duration = 8.11
10% yield environment:Duration = 7.15
Higher yields discount future cash flows more heavily, reducing duration. Effect is smaller than maturity/coupon, but matters for long bonds.
📊 Practical Application:
If you expect rates to RISE (2023-2024 Fed tightening): Buy SHORT duration bonds (2-3 years) or HIGH coupon bonds. Minimize pain!
If you expect rates to FALL (2008 crisis, 2020 COVID): Buy LONG duration bonds (20-30 years) or ZERO coupon bonds. Maximum gain!
Example: 2022, everyone "knew" Fed would hike. Smart money moved to 2-year Treasuries (duration ~1.9). Long bond investors got crushed—30-year Treasuries fell 39% (duration ~17)!
Zero-coupon bond special case: No coupons means ALL cash comes at maturity. Duration = maturity exactly. A 20-year zero has duration 20.0—incredibly sensitive to rates. Great for speculation (huge gains/losses) or immunizing liabilities (pension funds match duration of liabilities with zero-coupon bonds).
Convexity: Duration's Bigger Brother
Duration is a LINEAR approximation of price changes. But bond price-yield relationships are CURVED (convex). Convexity measures this curvature—and it's always GOOD for bondholders! Higher convexity = less downside when rates rise, more upside when rates fall.
Why Duration Alone Isn't Enough:
Duration Error Example (10-year bond, 5% coupon, 5% yield)
Current price: $1,000, Modified duration: 7.91
Rate Change
Duration Est.
Actual Price
Rates -1%:
$1,079.10
$1,081.15 (+$2.05)
Rates -2%:
$1,158.20
$1,169.47 (+$11.27)
Rates +1%:
$920.90
$922.78 (+$1.88)
Rates +2%:
$841.80
$850.61 (+$8.81)
See the pattern? Duration UNDERESTIMATES gains when rates fall, OVERESTIMATES losses when rates rise. The error grows with larger rate changes. This is convexity at work!
Better Formula (Duration + Convexity):
ΔPrice ≈ [-Duration × ΔYield + ½ × Convexity × (ΔYield)²] × Price
Convexity measures the RATE OF CHANGE of duration. It's the second derivative of price with respect to yield.
Higher convexity bonds:
• Long maturity (>10 years)
• Low coupon rate (zeros have highest convexity)
• Bonds with embedded options (callable bonds have negative convexity!)
All else equal, you WANT high convexity—it's free insurance against rate moves!
⚠️ Negative Convexity: The Callable Bond Trap
Callable bonds (issuer can redeem early) have NEGATIVE convexity when rates fall. Why? If rates drop, issuer calls the bond and refinances at lower rates. You get par back, capping your upside!
Example: Rates fall 2%, normal bond gains 16%, callable bond gains only 5% (gets called at $1,050). Rates rise 2%, callable bond falls 15% (no call protection). You lose both ways! This is why callable bonds yield MORE (compensation for negative convexity).
Portfolio management: Pension funds and insurance companies obsess over duration matching (assets = liabilities duration) to hedge interest rate risk. But they also seek positive convexity—if rates move EITHER direction, convexity helps! Modern portfolios use options, futures, and swaps to adjust duration and convexity cheaply without selling bonds.
⏱️ Interactive: Duration Analysis
Macaulay Duration
7.99
years
Modified Duration
7.79
% change per 1% rate change
Years to Maturity
10
years
0%
Current Bond Price:$1000.00
Duration-Based Estimate:$1000.00
Actual New Price:$1000.00
Price Change:$0.00
(0.00%)
⚠️ Duration estimates work best for small rate changes. For larger changes, the estimate error is 0.00(0.0%). This is why we also need convexity!
📊 Interactive: Maturity Comparison
2Y
-1.9%
7Y
-5.6%
20Y
-11.6%
Price change when rates rise by 1%
4. Yield Curves & Credit Risk
🎓 Reading the Economy Through Yield Curves
The Yield Curve: Market's Crystal Ball
The yield curve plots interest rates across different maturities (3-month to 30-year). Its shape reveals what bond investors collectively believe about future growth, inflation, and Fed policy. It's one of the most reliable recession predictors in finance.
Three Yield Curve Shapes:
1. Normal Curve (Upward Sloping) 📈
3M
2.5%
2Y
3.2%
10Y
4.0%
30Y
4.5%
What it means: Healthy economy, investors want compensation for lending long-term
Why it happens: Inflation + growth expected to continue, longer bonds have more risk
Historical frequency: ~70% of the time (normal state)
Investor action: Balanced portfolio, slight tilt to long bonds for yield
Example: 2017-2018, economy growing 2-3%, Fed slowly raising rates, curve stayed normal. S&P 500 up 20%+, no recession fears.
2. Inverted Curve (Downward Sloping) 📉 ⚠️ DANGER
3M
5.2%
2Y
4.8%
10Y
4.2%
30Y
4.0%
What it means: Recession likely within 6-18 months
Why it happens: Fed hiking kills growth → investors expect rate CUTS later → long bonds rally (yields fall)
Historical accuracy: Predicted 7 of last 7 recessions (1990, 2001, 2008, 2020*)
Investor action: Move to short bonds, reduce stock exposure, prepare for recession
*2020 recession was COVID-driven, but curve inverted in 2019. 2022-2023 inversion (most severe since 1980!) preceded 2023 banking crisis and recession fears.
Example: 2006-2007, curve inverted. Everyone ignored it ("this time is different!"). 2008 financial crisis followed—worst recession since 1930s.
3. Flat Curve (No Slope) ➡️
3M
3.8%
2Y
3.9%
10Y
3.9%
30Y
4.0%
What it means: Uncertainty, transition period (normal → inverted OR inverted → normal)
Why it happens: Fed at terminal rate (done hiking), market unsure about next move
Historical frequency: ~15% of the time (brief periods)
Investor action: Wait and watch, keep duration neutral, preserve capital
Example: Late 2018, Fed paused hiking at 2.5%, curve flattened. Market couldn't decide if recession coming or growth continuing. Fed pivoted to cuts in 2019.
Why inversions predict recessions: Short rates rise (Fed tightening) while long rates stay lower (investors expect Fed will CUT rates later due to recession). It's a self-fulfilling prophecy—tight monetary policy eventually kills growth, forcing Fed to cut. The curve inverts 6-18 months BEFORE recession starts, giving you time to prepare!
Credit Spreads: The Price of Default Risk
Corporate bonds pay MORE than Treasuries because companies can default (government can print money, companies can't). The extra yield is the credit spread—compensation for bearing default risk. Spreads widen in recessions (panic) and tighten in booms (complacency).
Credit Rating Scale (S&P/Fitch):
Investment Grade (Safe) ✅
AAA(Prime)
+0.3-0.5%
Microsoft, J&J
AA(High grade)
+0.5-0.8%
JPMorgan, Walmart
A(Upper medium)
+0.8-1.2%
Boeing, Intel
BBB(Lower medium)
+1.2-2.0%
Ford, Macy's
Default rate: 0.5-2% over 10 years. Institutional investors (pensions, insurance) can only buy these.
Junk / High Yield (Risky) ⚠️
BB(Speculative)
+2.5-4.0%
Uber, Tesla (2019)
B(Highly speculative)
+4.0-6.0%
Troubled retailers
CCC(Substantial risk)
+6.0-10%
Near-default companies
CC, C, D(Default imminent/occurred)
+10-20%+
Distressed debt
Default rate: 20-40% over 10 years for B/CCC. High yield, high risk. "Fallen angels" (downgraded from IG) often good buys.
📊 Historical Default Rates (1981-2020):
AAA: 0.0% (none defaulted)
AA: 0.05%
A: 0.2%
BBB: 0.8%
BB: 4.2%
B: 12.8%
CCC: 35.6%
CC/C: 65%+
The spread compensates for expected defaults. BBB +150bps means if 1% default and you recover 40¢ on dollar, you break even vs Treasuries.
Credit spread widening = fear: 2008 crisis, BBB spreads went from +150bps to +650bps (5% extra yield!). Investors panicked, refused to buy corporates. Those who bought BBB bonds at +600bps in Dec 2008 earned 40%+ in 2009 as spreads normalized. Credit crises create opportunities for brave investors!
Using Curves and Spreads Together
The Treasury yield curve tells you where interest rates are heading. Credit spreads tell you how much risk investors are willing to take. Combine them for a complete market picture.
Four Market Regimes:
1. Goldilocks (Normal Curve + Tight Spreads)
🟢
Setup: 10Y-2Y spread +100bps, BBB spread +150bps
Meaning: Growth strong, no recession fears, credit healthy
Action: Buy stocks, extend bond duration, take credit risk (BBB/BB)
Example: 2017, economy humming, spreads compressed, S&P 500 +21%
2. Late Cycle (Flat Curve + Tight Spreads)
🟡
Setup: 10Y-2Y spread +20bps, BBB spread +140bps
Meaning: Growth peaking, Fed done hiking, credit still OK
Action: Reduce duration, rotate to quality (AAA/AA), trim stocks
Example: Q4 2018, curve nearly flat, volatility spiked, stocks fell 20%
3. Warning (Inverted Curve + Widening Spreads)
🟠
Setup: 10Y-2Y spread -50bps, BBB spread +300bps
Meaning: Recession coming, credit stress emerging
Action: Buy Treasuries, sell credit, hedge equity exposure
Example: 2007, curve inverted, spreads widening, smart money exited before crisis
4. Crisis (Any Curve + Blowout Spreads)
🔴
Setup: BBB spread +600bps, High Yield +1000bps
Meaning: Credit markets frozen, defaults rising, panic selling
Action: Hold cash/Treasuries initially, then BUY distressed credit at bottom
Example: March 2020, BBB +400bps, HY +1100bps. Those who bought made 50%+ as Fed intervened
💡 Pro Strategy: Barbell Portfolio
When curve is inverted (2Y > 10Y), use a barbell: 50% short-term Treasuries (earning high yield) + 50% long-term Treasuries (gain when Fed cuts rates). Avoid the middle (5-7Y bonds) which has worst risk/reward.
Example: 2023, 2Y = 5%, 10Y = 4%. Barbell: 50% in 2Y T-bills (5% yield) + 50% in 30Y bonds (duration 20). When Fed cuts in 2024, 30Y bonds rally 30%+. Total return beats straight 10Y bonds!
The big picture: Yield curves and credit spreads are forward-looking. They aggregate the views of millions of investors. When BOTH flash red (inverted + wide spreads), listen! The market is rarely wrong for long. 2000, 2007, 2019 all showed these warning signs months before crashes. Use them as your financial early warning system.
📈 Interactive: Yield Curve Shapes
3M1Y5Y10Y30Y
Normal: Long-term yields higher than short-term. Investors want more compensation for lending longer.
Inverted: Short-term yields higher. Predicts recession 70% of the time within 18 months.
Flat: Little difference across maturities. Uncertainty about future rates and growth.
🏆 Interactive: Credit Rating
Treasury Yield (Risk-Free):5.00%
Credit Spread:+0.3%
Corporate Bond Yield:5.30%
Annual Interest on $10,000 bond:
Treasury
$500
Extra Yield
$30
Total
$530
💨 Interactive: Real vs Nominal Yield
2%
Nominal Yield
5.0%
What you see
Inflation
2.0%
Purchasing power loss
Real Yield
3.0%
What you actually earn
✅
Positive Real Yield
Your purchasing power grows
⚠️
Negative Real Yield
You lose purchasing power even with interest!
5. Key Takeaways
↔️
Inverse Price-Rate Relationship
Bond prices and interest rates move in opposite directions. When the Fed raises rates, existing bonds lose value because new bonds pay higher coupons. This is the fundamental law of fixed income.
⏱️
Duration = Price Sensitivity
Duration measures how much a bond's price changes when rates move. A duration of 7 means a 1% rate increase causes a ~7% price drop. Longer maturities = higher duration = more volatility.
📈
Yield Curve Predicts Recessions
An inverted yield curve (short-term rates higher than long-term) has predicted the last 7 recessions. It signals that investors expect the Fed to cut rates in the future due to economic weakness.
⚖️
Credit Risk = Extra Yield
Corporate bonds pay more than Treasuries because of default risk. The credit spread (extra yield) varies with ratings: AAA +0.3%, BBB +1.5%, BB (junk) +3.5% or more. Higher yield = higher risk.
💨
Real Yield Matters Most
A 5% bond yield sounds good until you subtract 4% inflation—you're only earning 1% in real terms. When inflation exceeds bond yields, bondholders lose purchasing power despite collecting interest.
🎯
Laddering Reduces Risk
Don't put all your money in one maturity. Build a bond ladder: 20% maturing each year for 5 years. This gives you liquidity, reduces reinvestment risk, and smooths out interest rate fluctuations.