Investment horizon, price risk vs reinvestment risk, and duration gap

Why do two investors in the same bond face different risks?

Interest rates act on a bond investor through two channels at once. The coupons are reinvested at the new rate, and a bond sold before maturity is priced at the new yield. Rising rates raise the first and cut the second. Falling rates do the reverse.

Reinvestment risk is the risk that coupons must be reinvested at lower rates, which hurts when rates fall. Price risk is the risk that the sale price falls, which hurts when rates rise. Both come from the same rate move, with opposite signs.

A parallel shift in yields right after purchasethe same change at every maturity, once

• Rates rise Memorize good for coupons, bad for the price
  • Reinvestment gain small at first, builds with time
  • Price loss immediate, then shrinks as the bond is pulled to par
• Rates fall Memorize bad for coupons, good for the price
  • Reinvestment loss small at first, builds with time
  • Price gain immediate, then shrinks as the bond is pulled to par

The weight of each channel depends on the investor’s investment horizon , the date the cash is needed. A long horizon gives reinvestment income time to build. A short horizon leaves little of it, and the sale price decides the result. So one bond can carry opposite risks for two holders.

Where do the two risks cancel?

Take the 8-year, 5% annual bond from the previous lesson, bought at par. Move yields by one percentage point right after purchase, the kind of parallel shift where every yield changes by the same amount, once. The horizon yields are:

Horizon	Rates to 6%	Rates to 4%	Dominant risk
3 years	3.76%	6.28%	Price risk
7 years	5.03%	4.97%	Almost balanced
8 years (maturity)	5.15%	4.85%	Reinvestment risk

At 3 years the investor ends up far from 5% in both directions, and rising rates hurt. At 8 years the sign has flipped: rising rates help. Somewhere between them, near 7 years, the shift barely matters. That turning point is the bond’s Macaulay duration , which for this bond is 6.79 years (you calculate it in the next lesson).

So the Macaulay duration is the holding period at which a one-time, immediate, parallel shift in yields has about no net effect on the realized return. It is not the maturity, and the match is only approximate.

Go deeper Why the offset is only approximate

The balance is exact only for tiny shifts. The price of a bond is curved in yield (convexity, covered in a later module), so larger moves break the symmetry. A real yield curve also twists instead of shifting in parallel, and rates keep moving after purchase. Treat the Macaulay duration as the best single guess at the balance point, not a guarantee.

How does the duration gap tell you which rate move to fear?

The duration gap compares the bond’s Macaulay duration with the investor’s horizon, and its sign names the dangerous direction of rates.

$$\text{Gap} = \text{MacDur} - H$$

Here $H$ is the investment horizon in years. Maturity does not appear.

Duration gap = Macaulay duration − investment horizon

• Positive gap (horizon below MacDur) Memorize price risk dominates; rising rates hurt
• Zero gap (horizon equals MacDur) Memorize the two risks roughly offset
• Negative gap (horizon above MacDur) Memorize reinvestment risk dominates; falling rates hurt

Horizon vs MacDur	Gap	Dominant risk	Rates that hurt
Horizon below MacDur	Positive	Price risk	Rising
Horizon equals MacDur	About zero	Balanced	Neither, roughly
Horizon above MacDur	Negative	Reinvestment risk	Falling

The gap is not fixed. As time passes the investor’s remaining horizon shortens and so does the bond’s Macaulay duration, so a position that was balanced can drift and should be monitored.

Exam tip Positive gap, short horizon, fear rising rates. Negative gap, long horizon, fear falling rates. If you can say the horizon first, the rest follows.

Go deeper Matching a liability: a preview

Choosing a bond whose Macaulay duration equals the date a liability falls due is the simplest case of immunization, a liability-driven approach you meet again at Level III. At this level the point is the direction: a duration gap near zero removes most of the exposure to a one-off parallel shift.