Advanced Fixed Income — Duration, Convexity, and Yield Curve Strategy

Why This Matters

At the institutional level, bond portfolio management is a precision discipline. Duration (duração), convexity (convexidade), and yield curve analysis determine whether a Kz 1 billion pension fund or sovereign wealth portfolio gains or loses hundreds of millions of Kwanza when rates move. These tools are not just academic — they are the daily working language of Angola’s bond traders at BAI, BFA, and the Ministry of Finance.

Duration: Measuring Interest Rate Sensitivity

Duration measures a bond’s price sensitivity to changes in interest rates. It answers: “If yields change by 1%, how much does my bond’s price change?”

Macaulay Duration

The weighted-average time to receive all cash flows from a bond, where weights are the present value of each cash flow divided by the bond’s price.

Example: A 5-year Angolan treasury bond, 21% coupon (semi-annual), priced at par:

Each semi-annual coupon of Kz 105,000 per Kz 1,000,000 face value is discounted back at the yield rate. The time-weighted average of these present values gives Macaulay duration of approximately 3.7 years.

Interpretation: Although this is a 5-year bond, you effectively recover your investment in 3.7 years because the high 21% coupon front-loads cash flows.

Angola context: High coupons (20%+) mean Angolan bonds have shorter durations relative to maturity than bonds in low-rate markets. A 10-year Angola OT at 22% has a Macaulay duration of roughly 5.5 years — much less than a 10-year US Treasury at 4% (duration ~8 years). This is significant: Angola bonds are less sensitive to rate changes per year of maturity.

Modified Duration

Modified Duration = Macaulay Duration / (1 + Yield/Frequency)

This converts Macaulay duration into a direct price sensitivity measure.

For our 5-year bond: Modified Duration = 3.7 / (1 + 0.105) = 3.35

Interpretation: A 1% (100 basis point) increase in yields causes the bond price to fall approximately 3.35%. On a Kz 10,000,000 position, that is a Kz 335,000 loss.

Dollar Duration (Duração Kwanza)

Modified Duration × Market Value = Kwanza sensitivity per 1% rate move.

Kz 10,000,000 × 3.35% = Kz 335,000 per 100bp move.

This is how institutional portfolio managers measure and manage their total interest rate exposure.

Convexity: The Curvature of Price Sensitivity

Duration is a linear approximation — it assumes price changes are proportional to yield changes. In reality, the relationship is curved (convex): bond prices rise more when rates fall than they decline when rates rise by the same amount.

Positive convexity is beneficial — you gain more on rate declines than you lose on rate increases. Standard fixed-rate bonds have positive convexity. The higher the convexity, the better the risk-return profile.

For Angola’s high-coupon bonds, convexity is moderate but beneficial. A portfolio optimized for convexity will outperform a duration-matched portfolio with lower convexity over time.

Yield Curve Strategies for Institutions

Bullet Strategy

Concentrate all holdings at a single maturity point (e.g., all 5-year bonds). Maximum exposure to that specific yield and maximum benefit if the curve moves favorably at that point.

Barbell Strategy (Revisited)

Short-end + long-end. If the yield curve flattens (long rates fall relative to short rates), the barbell outperforms as long bonds appreciate.

Butterfly Trade

Long the middle of the curve, short the wings (or vice versa). This is a pure yield curve shape bet with minimal duration exposure.

Example: Angola’s curve is currently steep (3.5% from 91-day to 10-year). If you expect the curve to flatten because the BNA cuts short rates while long rates stay stable:

• Buy 3-year OTs (middle)
• Fund with short-end bills and long-end bonds

This captures the flattening without taking a large outright rate bet.

Worked Example: Duration-Matched Portfolio Rebalancing

An institutional portfolio holds Kz 50 billion in government bonds with target duration of 4.0 years:

Current portfolio duration: 3.72 years — below the 4.0 target. The portfolio is underweight duration.

Rebalancing: Sell Kz 3B of 2-year OTs and buy Kz 3B of 10-year OTs.

New duration contribution: 0.54 - (3/50 × 1.8) + (3/50 × 6.8) = 0.54 - 0.108 + 0.408 = 0.84 from adjustments. New total: 3.72 + 0.30 = 4.02 — on target.

Key Takeaways

• Macaulay duration is the weighted-average time to cash flows — Angola’s high coupons create shorter effective durations
• Modified duration quantifies price sensitivity to yield changes (% price change per 1% yield change)
• Convexity captures the curvature — positive convexity is always desirable
• Institutional strategies (bullet, barbell, butterfly) express views on yield curve shape changes
• Duration management is the primary risk tool for bond portfolios — match portfolio duration to investment horizon
• In Angola’s steep yield curve environment, curve strategies can add 1-3% annually to returns

Common Mistakes

Ignoring duration when rates are changing — A 10-year bond with duration 5.5 can lose 5.5% on a mere 1% rate hike. Know your exposure.

Confusing duration with maturity — A high-coupon Angola bond has much shorter duration than its stated maturity. Duration is the risk metric; maturity is just the final date.

Over-concentrating on one part of the curve — All-5-year portfolios have concentrated risk at one point. Spread exposure across the curve.

What’s Next

Duration tells you about interest rate risk. But what about the risk that the borrower cannot pay? The next lesson covers credit analysis — assessing Angola’s sovereign creditworthiness and the credit quality of potential corporate issuers.

Next Lesson: Credit Analysis — Assessing Sovereign and Corporate Risk

Calculate duration and yields with the Bond Yield Calculator. Monitor the yield curve on the Bond Dashboard.