# Bond’s Holding Period Return, Duration, and Investment Horizon

Although short-term interest rate risk is a concern to some investors, other investors have a long-term horizon. Day-to-day changes in bond prices cause unrealized capital gains and losses. A long-term investor is concerned mostly with the total return over the investment horizon. The investor considers both coupon reinvestment risk and market price risk in case the bond needs to be sold before maturity.

The buy-and-hold investor has a higher total return if interest rates rise and a lower total return if rates fall. When interest rates rise, duration measures the immediate drop in value or price. Thereafter, as time goes by, the bond price is “pulled to par.” At some point in the bond’s lifetime, those two effects offset each other, and the gain on reinvested coupons is equal to the loss on the sale of the bond. That point is the Macaulay duration statistic.

## Duration Gap

The duration gap is the difference between the Macaulay duration and the investment horizon. Mathematically:

$$\text{Duration gap = MacDur – Investment quad horizon}$$

• when the investment horizon is greater than the Macaulay duration of a bond, coupon reinvestment risk dominates market price risk. The investor’s risk is to lower interest rates. The duration gap is negative;
• when the investment horizon is equal to the Macaulay duration of a bond, coupon reinvestment risk offsets market price risk. As such, the investor is hedged against interest rate risk, and the duration gap is zero;
• when the investment horizon is less than the Macaulay duration of the bond, market price risk dominates coupon reinvestment risk. The investor’s risk is to higher interest rates, and the duration gap is positive.

## Question

Assume that an investor plans to retire in 10 years. The investor buys a newly issued, 10-year, 8% annual coupon payment bond. Macaulay duration is 8.6 years. What is the duration gap at the time of purchase?

1. -1.525
2. -1.425
3. -1.400

Solution

Duration gap = MacDur – Investment horizon

= 8.6 – 10 = -1.400

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