Bond Fund Duration, Convexity, and Portfolio Management

Calculating the Duration of a Bond Fund

• General Principle: Calculating the duration for a bond fund or portfolio is a matter of calculating a weighted average of the individual bond durations within that fund.

• Example Scenario: A hypothetical bond fund consists of four bonds (A, B, C, and D) with a total market value of $10,000.

  • Bond A: Market Value = $4,000$, Modified Duration = $7$, Maturity = $15\,\text{years}$.

  • Bond B: Market Value = $3,000$, Modified Duration = $6$, Maturity = $10\,\text{years}$.

  • Bond C: Market Value = $1,000$, Modified Duration = $4$, Maturity = $7\,\text{years}$.

  • Bond D: Market Value = $2,000$, Modified Duration = $2$, Maturity = $5\,\text{years}$.

• Calculation Process: To find the modified duration of the portfolio ($D_P$), multiply the weight of each bond in the portfolio by its individual modified duration:

  • $(\frac{4,000}{10,000} \times 7) + (\frac{3,000}{10,000} \times 6) + (\frac{1,000}{10,000} \times 4) + (\frac{2,000}{10,000} \times 2)$

  • $Portfolio\,Modified\,Duration = 5.4$

• Note on Maturity Data: While maturity values (15, 10, 7, 5) are often provided as a given, they are not used in the calculation of modified duration for the portfolio.

The Underlying Assumption: Parallel Shifts in the Yield Curve

• Definition Applied: A portfolio duration of $5.4$ means:

  • If interest rates drop $1\,\%$, the bond fund price rises approximately $5.4\,\%$.

  • If interest rates rise $1\,\%$, the bond fund price drops approximately $5.4\,\%$.

• Parallel Shift Assumption: Duration analysis assumes a parallel shift in the yield curve. This implies that for the calculation to be accurate, interest rates must change by the exact same amount (e.g., $1\,\%$) across all maturities simultaneously (the 5-year, 7-year, 10-year, and 15-year rates in the example above).

• Real-World Context: While yield curves do not always move in perfectly parallel lines, they often move in a manner that is "not too way off," making duration a useful linear approximation for interest rate risk.

Properties of Duration (Ceteris Paribus)

These properties are largely restatements of Malkiel's Interest Rate Theorems but framed specifically through duration:

• Interest Rate Risk Correlation: Duration is effectively a synonym for interest rate risk. The higher the duration, the more interest rate risk a bond or fund carries.

• Maturity: Ceteris paribus, longer-term bonds have higher durations.

• Coupon Rate: Ceteris paribus, higher coupon rates lead to lower durations.

  • Zero-Coupon Bonds: The lowest possible coupon is zero. For a zero-coupon bond, the Macaulay duration is equal to its maturity.

• Rate of Increase: Ceteris paribus, duration increases at a decreasing rate as the maturity of the bond lengthens.

• Yield to Maturity (YTM): Ceteris paribus, the higher the YTM, the lower the duration.

  • Reinvestment Logic: To understand this, think of duration as a "payback" period. If YTM is higher, you are reinvesting coupon payments at a higher rate, which means you recover your investment (or the "payback") quicker.

Real-World Bond Fund Examples

• Vanguard Total Bond Fund Index (VPMFX):

  • This is described as the "S&P 500 for bonds."

  • It currently shows a duration of approximately $5.75$.

  • This represents the aggregate bond market duration. If rates shift down by $1\,\%$, aggregate bond funds should rise approximately $5.75\,\%$.

• JUCDX (Low Duration Bond Fund):

  • This fund specifically targets low duration to play it safe.

  • Lower duration values are preferred by investors who expect interest rates to rise and wish to avoid high price fluctuations.

• Modified vs. Effective Duration:

  • Effective Duration: This is calculated using scenario analysis and simulations. It is specifically used when bonds have embedded options, such as call provisions (common in mortgage-backed or securitized securities).

  • Comparison: For "plain vanilla" bonds without embedded options, modified duration and effective duration are essentially the same.

  • A Note on Formatting: Some financial sites (like Morningstar) list duration in "years." This is technically incorrect; modified and effective duration are numbers representing sensitivity, not a time measurement in years.

Convexity: Concepts and Applications

• Definition: Convexity is the measure of the curvature of the price-yield relationship. While duration is a straight-line (linear) approximation, the actual relationship between bond prices and yields is curved (convex).

• Graphical Representation:

  • The Y-axis represents the Bond Price ($P$).

  • The X-axis represents the Yield to Maturity ($YTM$).

  • Duration is the slope of the line tangent to the price-yield curve at a specific point. Mathematically, it is the derivative of the price with respect to the yield: $\frac{dP}{dYTM}$.

• The Superiority of Convexity: Ceteris paribus, a bond with higher convexity is better than a bond with lower convexity.

  • A more convex bond will lose less value when interest rates rise and gain more value when interest rates fall compared to a less convex bond.

• Paying for Convexity: Investors must decide if they are willing to pay a premium for higher convexity. This decision is based on volatility expectations:

  • High Volatility Expected: If interest rates are expected to fluctuate significantly (due to news regarding GDP, jobs, CPI, or PCE), convexity is valuable, and investors should pay for it.

  • Low Volatility Expected: If interest rates are expected to stay relatively stable (a "dull week"), paying extra for convexity is unnecessary as the difference between the linear duration and the actual curve is negligible for small moves.

• Duration Errors:

  • Duration overestimates the loss when interest rates go up.

  • Duration underestimates the gain when interest rates go down.

  • Duration is a decent estimator for small moves but becomes less accurate for large moves (e.g., moves of $100\,\text{basis points}$ or $1\,\%$).

Practical Market Observations

• Current Interest Rate Trends: The speaker noted that long-term yields have gone up significantly, with the 20-year rate reaching levels not seen in roughly 20 years (above $5\,\%$).

• Mortgage Impact: The 10-year yield is a critical indicator because it directly affects the 30-year mortgage rates.

• Stock Market Risk: Rising interest rates are cited as a potential catalyst that could pull the stock market down from its all-time highs despite AI-driven momentum.

Exam Details and Guidance

• Date/Attendance: No class tomorrow. The exam is Wednesday (taken remotely). Class resumes Thursday.

• Structure: 80 questions total.

• Math Component: 16 math questions worth 2 points each ($32\,\text{points}$ or approximately $30\,\%$ of the exam).

• Special Math Questions (No Practice Problems Provided):

  • 1. Find the inflation-adjusted clean price for a TIPS bond.

  • 2. Find the inflation-adjusted accrued interest for a TIPS bond.

  • Note: These are functionally identical to the examples performed in class. Adding these two values together results in the dirty price.

• General Definitions to Know:

  • Clean Price: The price of the bond excluding accrued interest.

  • Accrued Interest: Interest earned but not yet paid since the last coupon date.

  • Dirty Price: Clean Price + Accrued Interest.

Questions & Discussion

• Anecdote regarding Stevie Ray Vaughan: The speaker noted his first fight with his wife occurred on August 26, 1990, during a concert featuring Eric Clapton and Stevie Ray Vaughan. Stevie Ray Vaughan (considered one of the greatest guitarists alongside Hendrix and Eddie Van Halen) died the following day (August 27, 1990) in a helicopter crash. The speaker joked about the "butterfly effect" and whether the fight impacted history.

• Cultural Reference: The song "The Devil Went Down to Georgia" was mentioned. A student named Johnny was jokingly compared to the protagonist of the song who battles the devil with a fiddle. The speaker tied this back to the need to learn Latin and the "Lord's Prayer" for exorcising demons in his nightmares.

• Student Interaction: A student mentioned they wanted to be a doctor, prompting a discussion about learning Latin and the speaker's own aversion to blood.

• Final Exam Tip: There may be a question referencing "The Devil Went Down to Georgia" with a fake answer choice.