Fixed Income Term Structures and Strategies
Key Takeaways
- Yield curves visually represent how bond interest rates vary across different timeframes, showing normal, inverted, or flat patterns that signal economic conditions .
- Spot rates, forward rates, and YTM are interconnected; forward rates indicate market expectations for future interest rates, while YTM calculates total return if held to maturity .
- Portfolio strategies like barbell vs bullet balance risk: barbells (short + long bonds) offer higher convexity for volatile markets, while bullets (single maturity) maximize yield in stable rate environments .
- Duration and convexity measure bond sensitivity to rate changes; duration estimates price volatility, while convexity refines that estimate, especially for large rate swings .
- Embedded options (call/put/convert features) alter bond risk profiles, introducing negative convexity or equity upside, impacting yields and pricing .
What Exactly Is a Fixed Income Term Structure?
When we talk about fixed income term structures, we’re really describing the relationship between interest rates and time. Imagine locking money away for 1 year versus 10 years—you’d expect different returns, right? That’s the term structure in action: it’s a snapshot of rates for various maturities at a single point in time.
Most times, longer maturities mean higher yields—that’s a "normal" curve. Think lenders demanding extra compensation for the uncertainty of lending over a decade versus a year. But sometimes, things flip. Short-term rates climb above long-term ones, creating an "inverted" curve. Historically, this inversion often signals a recession might be brewing, ’cause investors are piling into long bonds anticipating rate drops ahead .
And then there’s the "flat" curve. Not very exciting, but it happens when the economy’s in transition—maybe growth’s slowing, but inflation hasn’t cooled yet. Central banks might be hesitating, and the market’s kinda unsure.
Why should you care? Whether you’re a pension manager or just starting with bonds, this curve tells you where the smart money thinks rates—and the economy—are headed. Corporates use it to decide when to issue debt; traders use it to price everything from mortgages to complex derivatives. It’s kinda the backbone of debt markets.
The Building Blocks: Spot Rates, Forward Rates, and YTM
Spot Rates: Today’s Cost of Money
A spot rate is simply the yield on a zero-coupon bond maturing at a specific date. No coupons, just one payment at the end. These rates are foundational ’cause they let you calculate the "true" value of any bond by discounting each cash flow at its matching spot rate. For example, if the 2-year spot rate is 3%, you’d discount a bond’s year-2 coupon by 3% .
Forward Rates: The Market’s Future Bets
Forward rates answer: "What’s next?" If today’s 1-year rate is 2% and the 2-year rate is 3%, the 1-year forward rate (i.e., the rate for a loan starting next year) isn’t just 4%—it’s derived mathematically:
(1 + z₂)² = (1 + z₁) × (1 + f₁,₂)
Where:
z₂= 2-year spot ratez₁= 1-year spot ratef₁,₂= 1-year forward rate, one year from now
This stuff matters for things like FRA contracts or deciding whether to lock in a future rate.
Yield to Maturity (YTM): The Bond’s "Average" Return
YTM is trickier than it looks. It’s the single rate that discounts all a bond’s cash flows to its current price. But here’s the catch: it assumes you can reinvest coupons at that same YTM—which rarely happens. So if rates rise after you buy, your actual return might lag the YTM. Still, it’s a handy yardstick for comparing bonds.
Table: Key Rate Differences
Yield Curve Shapes and What They Scream About the Economy
Normal Curves: Up and Right
This is the default shape. Short rates low, long rates higher. Why? Term premium. Investors want extra yield for the risk of lending long—inflation could spike, or the issuer might weaken. A steep normal curve often means the market expects growth and inflation ahead.
Inverted Curves: When Short Rates Beat Long
Rare and scary. Why would anyone accept less yield for lending 10 years versus 2? Usually ’cause they’re betting on a downturn. They expect central banks to slash rates soon to stimulate things, so they’re grabbing long bonds now before prices surge. Recent inversions (like 2019) preceded recessions—it’s a watched metric for a reason .
Flat/Humped Curves: Transition Times
A flat curve says uncertainty. Maybe growth’s slowing, but inflation’s sticky. Or the Fed’s hiking short rates, but long-term inflation expectations stay anchored. Humped curves (rates peak mid-term) are rarer—sometimes signaling mid-cycle slowdowns.
Real-World Insight: In 2023, the ECB held rates steady despite inflation. Short yields rose on hike fears, but long yields barely budged—flattening the curve. Why? Investors figured aggressive hikes now would hurt growth later. And yep, 2024 saw slower Eurozone growth .
Traditional Theories: Why Do Long Rates Differ From Short?
Pure Expectations Theory: "It’s All About Future Rates"
This theory’s dead simple: long rates are just averages of expected future short rates. If markets think rates will rise, the curve slopes up; if they’ll fall, it slopes down. Problem? It ignores risk. Investors aren’t robots—they demand extra yield for uncertainty.
Liquidity Preference: "Long Money Deserves a Bonus"
Here’s the fix: yes, forward rates reflect expected futures rates, but plus a premium for holding longer bonds. That’s the liquidity premium. Makes sense—tying money up for 30 years should pay more than 1 year, even if future rates stay flat.
Segmented Markets: "Different Folks, Different Maturities"
Pension funds need 30-year bonds. Hedge funds live in the 2-year space. So rates are set supply-and-demand within maturity "buckets", not some grand expectation. Ever seen long rates fall while shorts rise? Segmentation might explain it—like insurers hoarding long bonds.
Which theory wins? Honestly, all play a role. Expectations drive much of it, but premiums matter, and institutional demand creates distortions. For bond geeks, that’s what makes it fun.
Quantifying Interest Rate Risk: Duration & Convexity Demystified
Duration: Your Bond’s Sensitivity Gauge
Duration measures how much a bond’s price swings when rates move. For example:
- Bond A: 5-year maturity, 5% coupon → duration ≈ 4.55 years
- Bond B: 10-year maturity, 5% coupon → duration ≈ 8.11 years
If rates rise 1%, Bond B drops ~8.11%; Bond A only ~4.55%. Longer maturities and lower coupons both boost duration .
Convexity: The Curve in the Price/Rate Relationship
Duration assumes a linear bond/rate relationship—but it’s actually curved. That’s convexity. Positive convexity (most vanilla bonds) means:
- When rates fall, prices rise faster than duration predicts
- When rates rise, prices fall slower than duration predicts
It’s like a free bonus—you gain more than you lose from equal rate moves. But callable bonds have negative convexity: if rates plunge, the issuer calls the bond, capping your gains. That’s why callables yield more—investors demand compensation for that risk .
Table: Convexity Effects in Practice
Barbell vs. Bullet Strategies: Structuring for Rate Uncertainty
The Barbell Approach: Short + Long Bonds
Picture barbell portfolios: you hold, say, 2-year and 30-year bonds, but skip the intermediates. Why? Convexity arbitrage. Long bonds have crazy-high convexity—if rates swing, you win big. Short bonds stabilize things. Together, they often outperform bullets with the same duration when rates move sharply .
But—there’s always a "but"—barbells usually yield less than bullets. Normal curves slope up, so mid-bonds (like 10-year) pay more than a mix of shorts and longs. You’re paying for that convexity safety net.
The Bullet Strategy: Concentrating on One Maturity
Bullets are simpler: all bonds mature around the same date (e.g., 10 years). If the curve is steep and stable, you harvest that juicy mid-term yield. Great for known future liabilities—like funding college in 2030 .
Which Wins?
- Rates volatile? Barbell likely shines—its convexity pays off.
- Curve steepening? Bullet might win—long yields rise faster, hurting barbells.
- Curve flattening? Barbell often triumphs—long bonds rally hard.
Real institutional example: In May 2010, a barbell of 70.95% 5-year Treasuries + 29.05% 30-year Treasuries had identical duration to a 10-year bullet. But its convexity was 129.8 vs. the bullet’s 78.3—making it resilient if rates whipsawed .
Embedded Options: How Calls, Puts & Converts Warp Yields
Callable Bonds: Issuers Hold the Cards
Call features let issuers redeem bonds early, usually when rates fall. Investors hate this—you get your cash back just as reinvestment options worsen. So callables yield more than vanilla bonds to compensate. Pricing-wise:
Price_{callable} = Price_{straight} - Value_{call option}
The issuer’s option literally cuts your bond’s value .
Puttable Bonds: Your Escape Hatch
Put options let you sell the bond back early—handy if rates rise and you’re stuck with a low coupon. Naturally, these bonds yield less than vanilla bonds; you’re paying for flexibility. Their pricing adds the put’s value:
Price_{puttable} = Price_{straight} + Value_{put option}
Convertible Bonds: Betting on the Stock
Converts let bondholders swap debt for stock at preset ratios. They’re hybrids:
- Conversion ratio: Shares per bond (e.g., 25 shares per $1k bond)
- Conversion premium: How much pricier the bond is vs. its stock value now
Companies issue these when stock prices are low—they offer lower yields, betting shares will rise and make conversion appealing. If the stock soars, your bond behaves like equity. If it tanks, you still have bond downside protection .
Real-World Bond Math: Pricing, YTM, and Accrued Interest
Calculating a Bond’s Full Price
Bonds trade between coupon dates, so buyers compensate sellers for accrued interest. The full ("dirty") price is:
Full Price = Flat Price + Accrued Interest
Where:
Accrued Interest = \frac{\text{Days since last coupon}}{\text{Days in coupon period}} \times \text{Coupon payment}
Example: That Apple 1.4% 2028 bond... bought June 14, 2024, at $88.44 (flat). Last coupon was Feb 5, so 130 days accrued.
AI = \frac{130}{184} \times (1.4\% \times \$1,000) ≈ \$4.96
So you’d pay $884.40 + $4.96 = $889.36 per $1k bond. At maturity, you get $1,000 back—that price discount ($111.60) boosts your yield beyond the tiny coupon .
Yield to Maturity (YTM) Calculation
YTM solves for the discount rate making all future cash flows’ PV equal the current price. For a semi-annual bond:
Price = \sum_{t=1}^{2T} \frac{Coupon/2}{(1+YTM/2)^t} + \frac{Face Value}{(1+YTM/2)^{2T}}
It’s trial-and-error, but calculators handle it. That Apple bond’s YTM was ~4.5%—much higher than its 1.4% coupon, thanks to buying below par .
FAQs: Fixed Income Term Structures
Why does an inverted yield curve signal recession?
It shows investors expect lower future rates—typically because they foresee economic weakness forcing central banks to cut. Historically, inversions preceded recessions (e.g., 2007, 2019) .
What’s the difference between Z-spread and OAS?
- Z-spread: Constant spread over spot rates to price a bond. Measures credit/liquidity risk.
- OAS (Option-Adjusted Spread): Z-spread minus the value of embedded options. Crucial for comparing callables/puttables .
Can bullet portfolios have embedded options?
Technically yes, but classic bullet bonds are non-callable—that lump-sum maturity repayment is key. Callables would undermine the liability-matching purpose .
How do repo markets affect short-term rates?
Repos (repurchase agreements) are collateralized overnight loans. High repo demand pushes short rates down; stress (like 2008 or 2019) sends them soaring. The repo rate directly anchors the short end of the term structure .
Why do long bonds have higher convexity?
Convexity ≈ curvature of the price/yield relationship. Long bonds’ prices bend sharply as rates move—math-wise, it’s a second-order effect of duration. A 30-year bond’s price jumps more from a 1% rate drop than a 5-year bond’s does .
Comments
Post a Comment