fixed income

Macaulay Duration

• Definition: Macaulay Duration is a measure of the weighted average time until cash flows are received, aiding in understanding how interest rate changes affect bond pricing.

• Formula: Duration gap is calculated using the formula:
  \text{Duration gap} = \text{Macaulay duration} - \text{Investment horizon}

Relationship Between Investment Horizon and Duration

• Investment Horizon > Macaulay Duration

  • Dominant Risk: Reinvestment Risk

  • Source of Interest Rate Risk: Falling Interest Rates

• Investment Horizon = Macaulay Duration

  • Risk Equivalence: Price Risk = Reinvestment Risk

• Investment Horizon < Macaulay Duration

  • Dominant Risk: Price Risk

  • Source of Interest Rate Risk: Rising Interest Rates

Overview of Duration Gaps

• Negative Duration Gap: For investors with an investment horizon longer than the Macaulay duration, indicates a preference for reinvestment risk.

• Zero Duration Gap: Suggests a perfect balance between the bond's duration and the investment horizon, minimizing interest rate risk.

• Positive Duration Gap: Indicates a higher risk of price volatility due to rising interest rates, as the investment horizon is shorter than the bond's duration.

Implications Over Time

• As time progresses, both the investment horizon and the Macaulay duration of a bond decrease, affecting the associated risks for the investors.

Modified duration = Mac Duration / (1+r)

Mac duration of non callable perpetual bond = 1+r/ r

money duration = mod duration * full price

Full price includes accrued interest

Bonds with the lowest coupon will be the most sesentive to change in yield

Duration and convexity of portfolio use market value of bond for weight average weight

Re purchase agreement (repo) - sale of security with a simultaneous agreement to buy the same security back at a set price and purchase date

Repurchase price = bonds face value X ( 1+ ( repo rate x days / years)

Convexity effect - the percentage price change is greater when the yield goes down vs up

Macaulay Duration

• Definition:

  • Average time to receipt of promised cash flows, weighted by shares of the full price corresponding to each promised future cash flow.

Modified Duration

• Definition:

  • First derivative of price with respect to yield.

  • Calculated as Macaulay duration divided by (1 + yield per period).

Money Duration

• Definition:

  • Holding period that would balance reinvestment and price risks for an investor.

Price Value of a Basis Point (PVBP)

• Definition:

  • Estimates the percentage price change for a bond given a change in its yield-to-maturity.

  • Can be calculated as:

  • Modified duration multiplied by the full price of the bond.

  • Alternatively, it can be estimated as the difference in price of a 1 basis point (bp) yield decrease and a 1 bp yield increase, divided by 2.

Yield Duration of Zero-Coupon and Perpetual Bonds

• Key Point:

  • Zero-coupon bonds have a single payment, which is the face value at maturity.

  • The present weighted value of this single cash flow is crucial for calculating duration in these types of bonds.