Interest Rate Risk — CFA Level I (Fixed Income)

"If bonds were celebrities, interest rates would be the paparazzi." — your friendly, slightly dramatic TA

You already know how to read yields (we covered yield measures earlier) and how to sniff out credit issues (credit risk analysis). Now let’s stare into the heart of the bond market’s most existential dread: interest rate risk — the reason fixed-income managers drink so much coffee and own calculators like religious relics.

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Why this matters (and why equities were not invited)

• Equities: theoretically infinite life, price driven by cash flows + growth expectations and a mess of behavioral chaos.
• Bonds: fixed cash flows and set maturity — which sounds comforting until rates move and your bond’s price pirouettes.

Interest rate risk = the sensitivity of a bond's price (or portfolio value) to changes in market interest rates. For CFA candidates, mastering this is non-negotiable: it's the backbone of immunization, hedging, risk limits, and portfolio construction.

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The basic toolkit: Duration, Modified Duration, DV01, Convexity

1) Macaulay duration (D)

• Definition: Weighted average time (in years) until a bond's cash flows are received, weighted by present value.
• Intuition: think of duration as the "center of gravity" of cash flows.

2) Modified duration (D_mod)

• Definition: Approximate percentage price change for a 1% (100 bps) change in yield.

Code block (formulas):

Macaulay duration: D = sum( t * PV(CF_t) ) / Price
Modified duration: D_mod = D / (1 + y/k)
Approximate price change: DeltaP/P ≈ -D_mod * DeltaY

• y = yield to maturity; k = compounding frequency.

3) Dollar duration / DV01

• Dollar duration: D_mod * Price — gives the absolute dollar change in price per 1% change in yield.
• DV01 (dollar value of a 1 basis point): ≈ Dollar duration / 100

4) Convexity

• Duration gives a linear approximation. Bonds are not linear.
• Convexity adjusts for curvature, improving the estimate for larger yield moves.

Code:

Approximate price change with convexity:
DeltaP/P ≈ -D_mod * DeltaY + 0.5 * Convexity * (DeltaY)^2

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Quick example — the theater where math performs

Imagine a 5-year annual coupon bond, par = 1000, coupon = 6% (60/year), YTM = 5%.

• Compute price, Macaulay duration (say 4.4 years), modified duration (4.19).

If yields rise by 100 bps (1%):

• Approx % price change ≈ -4.19 * 0.01 = -4.19%.
• If price was 1,045, new price ≈ 1,045 * (1 - 0.0419) ≈ 1,001 — ouch.

Add convexity and the drop is slightly less severe (convexity cushions price moves).

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Types of yield-curve moves (and why they feel different)

Move type	What it is	Price sensitivity nuance
Parallel shift	Entire curve up/down by same bps	Duration-based estimates work well
Twist (steepening/flattening)	Short and long rates move differently	Key-rate durations matter
Butterfly	Middle of curve moves vs wings	Portfolio may gain/lose depending on convexity

"Not all rate moves are created equal." If rates rise everywhere, your duration number is a pretty decent map. If the curve bends like a gymnast, you need key-rate durations.

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Key rate duration (KRD)

• Why: Duration assumes a parallel shift. Real world = not parallel.
• What: Sensitivity of bond price to a 1 bps move at a specific point (key rate) on the yield curve, holding other key rates constant.

Use KRDs to decompose portfolio exposure across maturities (e.g., 2y, 5y, 10y). This helps diagnose which part of the curve will wreck you when it tilts.

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Special cases: embedded options and effective duration

• Callable and putable bonds change cash-flow timing when rates move — the bondholder/issuer may exercise options.
• Use effective duration, which measures price sensitivity by shocking yields and re-pricing using an option-adjusted valuation.

Short version: option bonds = duration that’s rate-dependent. That’s messy and delightful for quants.

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Hedging & Immunization — a mini-strategy guide

Two common aims:

1. Immunization: Match the duration of assets and liabilities so small parallel yield moves leave the surplus roughly unchanged.
   • Requires periodic rebalancing (duration drifts as coupons/reinvestments occur).
   • Watch for convexity mismatch — parallel immunization ignores curvature.
2. Hedging: Use futures, swaps, or interest-rate derivatives to offset DV01.
   • Example: If portfolio DV01 = +$50k per bp, short interest rate futures with DV01 ≈ $50k to neutralize.

Practical checklist for hedging:

• Compute portfolio DV01.
• Choose instrument (Treasury futures, IRS) and compute instrument DV01.
• Trade size = portfolio DV01 / instrument DV01 (adjust signs).

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Common interview/training questions (aka "trickiness")

• Q: If yields fall, what happens to a callable bond's price vs a non-callable bond?

  • A: Callable bond has less upside because issuer will likely call; effective duration is lower.

• Q: Which is larger, price change predicted by duration alone, or duration+convexity, for a fall in yields?

  • A: Duration alone underestimates price increase. Convexity adds positive adjustment.

• Q: Equities vs bonds: which has longer duration?

  • A: Equities can be thought of as having extremely long/undefined duration; but their sensitivity to rates is indirect (discount rate changes) and compound by growth expectations.

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Quick mental anchors (so you don’t panic on exam day)

• Higher coupon → shorter duration (cash returns faster).
• Longer maturity → longer duration (cash is further away).
• Higher yield → shorter duration (present values shift weights toward earlier cash flows).
• Convexity is your friend for big rate moves (it reduces losses when rates rise and increases gains when rates fall relative to linear estimate).

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Final mic-drop: summary and study checklist

• Interest rate risk = bond price sensitivity to rate changes. Main tools: duration and convexity.
• Use modified duration for small parallel shifts; key-rate durations for curve shape risk; effective duration for optionality.
• Hedging is about matching DV01; immunization is about matching duration (and ideally convexity).

Actionable CFA prep checklist:

1. Memorize formulas and be able to compute Macaulay → modified duration.
2. Practice a few numeric examples (compute price change with and without convexity).
3. Understand the intuition behind key-rate duration and effective duration.
4. Know DV01 and how to size a hedge using futures/swaps.

"Treat duration like a compass, convexity like a seatbelt, and key-rate durations like the weather forecast. Together they stop your bond portfolio from getting lost in the rate jungle."

Tags: ["intermediate", "humorous", "visual", "finance"]