Relationship among a Bond’s Holding Period, Macaulay Duration, and Investment Horizon

Holding Period Return (Horizon Yield)

This represents the total return an investor anticipates from holding a bond over a specific duration. It’s influenced by the coupon payments received and any change in the bond’s price due to interest rate movements.

Formula:

\[r = \left( \frac{FV + F}{PV} \right)^{\frac{1}{T}} – 1\]

Where:

• \(r =\) Realized rate of return or Horizon Yield.
• \(FV =\) Future value of the reinvested coupons.
• \(F =\) Face value of the bond (often considered as 100).
• \(PV =\) Present value or the bond’s current price.
• \(T\) = Investment horizon.

The previous learning objective provided an example of calculating the horizon yield.

Macaulay Duration

Introduced by Frederick Macaulay in 1938, the Macaulay duration provides a measure of the weighted average time until a bond’s cash flows are received. It serves as an indicator of the bond’s price sensitivity to interest rate changes. When the investment horizon matches the Macaulay duration of a bond, the bond is nearly hedged against interest rate risk. Any loss from price risk due to rising rates is approximately offset by gains from reinvestment risk and vice versa. We will delve deeper into the Macaulay Duration on the next learning objective.

Investment Horizon

This is the period an investor plans to hold onto a bond. The relationship between the investment horizon and the Macaulay duration determines the bond’s dominant source of interest rate risk:

• Investment Horizon > Macaulay Duration: The bond faces a dominant risk from reinvestment, arising due to the reinvestment of coupons, possibly at a less favorable rate.
• Investment Horizon < Macaulay Duration: Price risk is the prevailing concern. The bond’s price might be adversely affected by rising interest rates.
• Investment Horizon = Macaulay Duration: The bond is nearly immune to interest rate risk. Here, price risk and reinvestment risk balance out.

Duration Gap

This represents the difference between a bond’s Macaulay duration and the investor’s investment horizon. Duration gap=Macaulay duration−Investment horizon

A negative duration gap indicates that the bond’s Macaulay duration is lower than the investment horizon of the investor. In this scenario, the primary concern is reinvestment risk, predominantly arising due to falling interest rates. On the other hand, a positive duration gap suggests that the bond’s Macaulay duration is higher than the investor’s investment horizon. Here, the main risk stems from potential price fluctuations, mainly driven by increasing interest rates.

Overall, the duration gap helps in identifying the primary source of interest rate risk a bond faces, be it from reinvestment or price changes.

Question

Which of the following statements about a bond with a Macaulay duration higher than the investment horizon is the most accurate?

1. The bond predominantly faces reinvestment risk.
2. The bond is primarily exposed to price risk.
3. The bond is nearly hedged against interest rate risk.

Solution

The correct answer is B.

When the Macaulay duration of a bond is higher than the investment horizon, the bond is primarily exposed to price risk, especially from rising interest rates.

A is incorrect: This would be the case when the investment horizon is higher than the Macaulay duration.

C is incorrect: The bond is nearly hedged against interest rate risk when the investment horizon matches the Macaulay duration.