Malkiel's Interest Rate Theorems and Bond Duration

Course Administration and Class Schedule

• Summer Session Intensity: The instructor notes that summer classes cover significantly more material than regular semester classes due to the ninety-minute sessions that run straight through without the typical logistical delays found during the school year.

• Class Schedule and Exam:

  • There is no class on the day following this lecture due to the World Cup events; students are advised not to come in.

  • The next class meeting after the break will be the exam, scheduled for Wednesday.

  • The lecture following the exam will take place on Thursday.

• Study Period: Students have an entire day with no new material to prepare for the Wednesday exam.

Current Market Overview

• Market Performance: The market experienced extreme volatility on the preceding Friday due to higher-than-expected jobs numbers and inflation concerns, which caused interest rates to rise significantly.

• Monday Comparison: Today's market is described as a "split decision" and relatively "boring" compared to Friday's activity.

• Geopolitical Impact: Recent news involving Iran stating they would stop bombing Israel has helped settle market tensions.

Introduction to Malkiel's Interest Rate Theorems

• Origin: These five theorems are attributed to Burton Malkiel, a famous Wall Street figure and author of the book Random Walk Down Wall Street. Now $93$ years old, Malkiel published these theorems in the $1960$s as a way to dissect bond movements using math.

• Context: These theorems represent "bond math" without always requiring explicit calculations. They help predict how bond prices respond to changes in interest rates.

Theorem 1: The Inverse Relationship

• Core Principle: There is an inverse relationship between bond prices and interest rates.

• Directional Movement: When interest rates ($RD$ or Yield to Maturity) go up, bond prices go down. Conversely, when interest rates go down, bond prices go up.

Theorem 2: Curvilinearity and Convexity

• Principle: The price-yield relationship is not linear; it is curvilinear (convex to the origin).

• Mathematical Implication: For a given change in interest rates, the price increase resulting from a decrease in rates is greater than the price decrease resulting from an increase in rates of the same magnitude.

• Example Scenario:

  • If a bond trades at par ($1,000$) and interest rates increase by $1\%$, the price might drop by, for example, $20$, resulting in a price of $980$.

  • If interest rates drop by $1\%$, the price will not merely go up by $20$ ($1,020$), but rather by more (e.g., $30$, resulting in a price of $1,030$).

• Convexity Performance: Convexity is a desirable trait in a bond; the "curvier" the relationship, the more a holder "wins" on rate drops compared to what they "lose" on rate hikes.

Theorem 3: Maturity and Interest Rate Risk

• Principle: Ceteris paribus (all else being equal), for a given change in interest rates, long-term bonds exhibit more interest rate risk than short-term bonds.

• Observation: This relates to the Maturity Risk Premium. Longer-term bonds fluctuate more in price than shorter-term bonds for the same interest rate move.

Theorem 4: The Diminishing Effect of Maturity Increases

• Principle: While interest rate risk increases with maturity (Theorem 3), the relative importance of this increase diminishes as the maturities get longer.

• Slope Analysis: The curve representing the change in price over time to maturity is concave. It increases at a decreasing rate.

• Comparison Example:

  • Comparing a $3$-year bond to a $5$-year bond: The $5$-year bond has significantly more interest rate risk than the $3$-year bond.

  • Comparing a $23$-year bond to a $25$-year bond: The $25$-year bond still has more risk, but the difference between the two is much smaller than the difference in the first example.

Theorem 5: The Coupon Effect and Interest Rate Risk

• Principle: Ceteris paribus, lower coupon bonds have more interest rate risk than higher coupon bonds.

• Zero Coupon Bonds: These bonds have the maximum possible interest rate risk for a given maturity because they have the lowest possible coupon ($0$).

• Analogy to Equities:

  • Growth Stocks: Often pay no dividends (similar to zero-coupon bonds) and typically have high Betas (high systematic risk).

  • Value Stocks: Often pay high dividends and have lower Betas.

  • Similarly, high-coupon bonds typically exhibit lower price fluctuations than low-coupon bonds.

Mathematical Proof of Malkiel's Theorems

Proof of Theorem 5 (Coupon Effect)

• Comparison: Bond A ($9.5\%$ coupon, semiannual, $8$ years) vs. Bond B (Zero coupon, $8$ years).

• Scenario: Interest rates drop by $44$ basis points ($0.44\%$).

• Result:

  • Bond A price increases by $2.4\%$.

  • Bond B price increases by $3.6\%$.

• Conclusion: Even though the dollar amounts might differ, the percentage fluctuation (the measure of interest rate risk) is higher for the zero-coupon bond.

Proof of Theorems 1 through 4

• Setup: Compare bonds with a $10\%$ coupon, semiannual, at maturities of $3, 5, 20,$ and $22$ years.

• Yield Shift Example: Moving from a $10\%$ yield (par) to $11\%$

  • $3$-year bond: Price drops by $24$.

  • $5$-year bond: Price drops by $36$ (Difference of $12$).

  • $20$-year bond: Price drops by $72$.

  • $22$-year bond: Price drops by $74$ (Difference of only $2$).

• Theorems Verified:

  • Theorem 1: Price dropped as yield rose.

  • Theorem 3: The $22$-year bond dropped more ($74$) than the $3$-year bond ($24$).

  • Theorem 4: The move from $3$ to $5$ years added $12$ in risk, whereas the move from $20$ to $22$ years only added $2$ in risk, demonstrating an "increasing at a decreasing rate" pattern.

Bond Comparison Case Study: Bond 1 vs. Bond 2

• Bond 1 Data:

  • Coupon: $10\%$ (Semiannual)

  • Maturity: $3$ years

  • Yield to Maturity ($RD$): $10\%$

  • Price: $1,000$ (Par, because $Coupon = Yield$)

• Bond 2 Data:

  • Coupon: $15\%$ (Semiannual)

  • Maturity: $4$ years

  • Yield to Maturity ($RD$): $15\%$

  • Price: $1,000$ (Par, because $Coupon = Yield$)

• Analysis Challenge:

  • Bond 1 has a lower coupon, which suggests more risk (Theorem 5).

  • Bond 2 has a longer maturity, which suggests more risk (Theorem 3).

  • Because multiple factors are changing simultaneously, ceteris paribus does not apply, and one cannot determine which bond has more risk simply by looking at them. This necessitates the use of Duration.

Macaulay Duration

• Definition: The weighted average time (measured in years) it takes to receive the cash flows from a bond, taking into account the time value of money and reinvestment.

• Interpretation: It is similar to a "compounding payback" period.

• Zero Coupon Bond Property: For a zero-coupon bond, Macaulay duration is exactly equal to its time to maturity.

• Coupon Bond Property: For any bond paying a coupon, Macaulay duration will always be less than its maturity.

• Calculation Results for Case Study:

  • Bond 1 Macaulay Duration: $2.665$ years.

  • Bond 2 Macaulay Duration: $3.148$ years.

  • Conclusion: Bond 2 has more interest rate risk because its Macaulay duration is higher.

Modified Duration

• Formula:     $Modified\,Duration = \frac{Macaulay\,Duration}{1 + y}$

  • For annual coupons, $y = YTM$.

  • For semiannual coupons, $y = \frac{YTM}{2}$.

• Case Study Calculations:

  • Bond 1: $\frac{2.665}{1 + 0.05} = 2.538$

  • Bond 2: $\frac{3.148}{1 + 0.075} = 2.929$

• Interpretation: Modified duration provides a linear approximation of the percentage price change for a $1\%$ change in interest rates.

  • For Bond 1: If rates drop by $1\%$, the price rises by approximately $2.538\%$. If rates rise by $1\%$, the price drops by approximately $2.538\%$.

• Comparison to Beta: While it looks like a stock's Beta, it measures a bond's sensitivity to its own yield change, not an index. It is usually expressed as a positive number, though the actual relationship is inverse (negative slope).

Approximate Modified Duration

• Purpose: Used to calculate modified duration when Macaulay duration is unknown.

• Formula:     $Approx.\,Mod.\,Duration = \frac{V_{-} - V_{+}}{2 \times V_{0} \times \Delta Yield}$

  • $V_{0}$: Current price of the bond.

  • $V_{-}$: Price if yield drops by a specific amount (e.g., $100$ basis points or $1\%$).

  • $V_{+}$: Price if yield rises by the same amount.

  • $\Delta Yield$: The change in yield (expressed as a decimal, e.g., $0.01$ for $1\%$).

• Case Study (Bond 1):

  • $V_{0} = 1,000$

  • $V_{-} (9\%\,yield, 4.5\%\,semiannual) = 1,025.79$

  • $V_{+} (11\%\,yield, 5.5\%\,semiannual) = 975.02$

  • $Result = \frac{1,025.79 - 975.02}{2 \times 1,000 \times 0.01} = 2.538$

Strategic Use of Duration

• Managerial Role: A bond fund manager's job is to manage interest rate (duration) risk based on macroeconomic predictions.

• Bullish on Rates (Rates expected to drop): A manager will increase the fund's duration (moving into lower coupon, longer-term bonds) to maximize price gains.

• Bearish on Rates (Rates expected to rise): A manager will decrease the fund's duration (moving into higher coupon, shorter-term bonds) to minimize price losses.

• Macro Factors: Managers look for "deflationary events" (IP dropping) such as lower-than-expected CPI/PCE, lower crude oil prices, better productivity, or higher unemployment to justify increasing duration.

Dollar Duration (DB01)

• Terminology: Also known as Dollar Value of a Basis Point (DV01), Money Duration, or Price Value of a Basis Point (PVBP).

• Formula:     $DB01 = \frac{Modified\,Duration \times V_{0}}{10,000}$

• Case Study Calculations:

  • Bond 1: $\frac{2.538 \times 1,000}{10,000} = 0.25$ (or $25$ cents)

  • Bond 2: $\frac{2.929 \times 1,000}{10,000} = 0.29$ (or $29$ cents)

• Interpretation: This value represents the dollar change in the bond price for a single basis point ($0.01\%$) change in yield.

  • Example: If rates rise by $10$ basis points, Bond 1 will drop in price by $10 \times 0.25 = 2.50$.

Personal Anecdotes and Cultural Context

• The Pink and White Striped Shirt: The professor wears a pink and white striped shirt to commemorate his first date with his wife, which occurred $36$ years ago today. The date took place at the Chicago Blues Fest in Grant Park, meeting in front of the "South Lion" statue at the Art Institute of Chicago.

• The Ticket Sports Radio: A specific sports talk radio station in Dallas-Fort Worth ($1310$ AM/$96.7$ FM). The professor mentions "drops" (audio clips used for comedic effect).

• Mike Tyson and "Wisconsinin": The professor references an interview where Mike Tyson associated Wisconsin with Jeffrey Dahmer, pronouncing it "Wisconsin-in" followed by dramatic music. This became a family joke where his children believed the state was actually named "Wisconsinin."

• SpaceX IPO Rumor: Reference to a potential SpaceX IPO at $135$ per share, which would value the company at approximately $1.7$ to $1.8$ trillion dollars, potentially making it the fifth-largest company immediately upon hitting the secondary market.