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Eduzan / MODULE 07 — Fixed Income

FI 10 Interest Rate Risk and Return

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

How the three sources of a bond’s return interact, and why holding a bond for its Macaulay duration neutralizes interest rate risk. Check-yourself items are study aids.

Earlier yield work showed that an investor earns exactly the yield-to-maturity (YTM) quoted at purchase only when a specific set of conditions holds. Here we open up the return and look at what actually drives it. A holder of a fixed-rate bond collects return from three distinct places: (1) the promised coupon and principal payments received on schedule, (2) the reinvestment of each coupon as it arrives, and (3) any capital gain or loss if the bond is sold before it matures. The first source is contractual and, ignoring default, certain. The other two move with market interest rates, and that is where the risk lives.

The realized return over a holding period is called the horizon yield: the single annualized rate that links what an investor pays today to everything received later. To compute it, add the future value of the reinvested coupons to the sale or redemption amount, divide by the purchase price, and annualize over the holding period T.

That reinvested-coupon total is itself an annuity accumulation: each coupon PMT is rolled forward at the prevailing reinvestment rate r across the N periods that remain until the horizon date.

When the horizon yield equals the original YTM

To see the benchmark case, follow two investors who each buy the same instrument: BRWA’s new 10-year, 6.2 percent annual coupon eurobond, priced at par. The first, Viswan Family Office (VFO), holds to maturity. The second, Baywhite Financial, plans to sell after four years. We start with interest rates unchanged, then shock them.

Example 1 · Worked

VFO buys BRWA’s 10-year, 6.2 percent annual coupon eurobond at par (settlement 15 October 2025, maturity 15 October 2035) and holds it for the full ten years. Coupons are reinvested at 6.2 percent.

Find the future value of the reinvested coupons and VFO’s horizon yield.

Solution. Accumulating ten annual coupons of 6.2 at 6.2 percent gives a future value of 82.493 per 100 of par. Of this, 62 is the plain sum of coupons (6.2 times 10) and the remaining 20.493 is interest-on-interest, the gain from reinvesting and compounding the coupons. Since the bond is held to maturity, it redeems at par, so

The horizon yield equals the original YTM. This holds under three assumptions: the bond is held to maturity, the issuer does not default, and every coupon is reinvested at that same 6.2 percent.

Example 2 · Worked

Baywhite buys the identical BRWA bond at par but sells immediately after collecting the fourth coupon. Rates stay at 6.2 percent throughout.

Find Baywhite’s horizon yield over its four-year holding period.

Solution. Four coupons reinvested at 6.2 percent accumulate to 27.203 per 100 of par, of which 2.403 is interest-on-interest (27.203 minus the 24.800 of plain coupons). With six years still to run and the YTM unchanged at 6.2 percent, the bond sells at par, 100. So

Baywhite earns 6.2 percent even though it did not hold to maturity. The horizon yield matches the original YTM whenever coupons are reinvested at that YTM and the bond is sold on its constant-yield price trajectory, so that no capital gain or loss arises on the sale.

Interest-on-interest and the constant-yield trajectory

Interest income is the return earned simply from the passage of time: it covers the coupons, the reinvestment of those coupons, and the gradual amortization of any purchase discount or premium back toward par. A capital gain or loss is separate; it comes from a change in the YTM (the implied market discount rate). The constant-yield price trajectory traces the bond’s carrying value over time at the original yield. For a par bond, that trajectory is a flat line at 100, so selling on it produces no gain or loss. A sale above the trajectory is a capital gain; a sale below it is a capital loss.

Shocking the interest rate

Now raise rates by 100 basis points (bps) the instant after purchase, lifting the yield and the reinvestment rate from 6.2 percent to 7.2 percent, then lower them by 100 bps to 5.2 percent. The two investors respond very differently because one sells and one does not.

Example 3 · Worked

Rates jump to 7.2 percent immediately after purchase.

Recompute the horizon yields for VFO (holds ten years) and Baywhite (sells after four years).

Solution. VFO. Coupons now reinvest at 7.2 percent, so their future value rises to 86.475. VFO holds to maturity, so it redeems at 100 and takes no capital loss:

VFO ends up 23 bps better off than in Example 1 purely from higher reinvestment income. Baywhite. Its reinvested coupons grow to 27.609, but it must sell with six years left at the higher 7.2 percent yield, so the sale price falls to 95.263:

For Baywhite the small reinvestment gain is swamped by the capital loss on the early sale, so its return falls below the 6.2 percent of Example 2.

Example 4 · Worked

Instead, rates fall to 5.2 percent immediately after purchase.

Recompute the horizon yields for VFO and Baywhite.

Solution. VFO. Coupons reinvest at the lower 5.2 percent, so their future value drops to 78.715; redemption is still 100:

VFO now earns 22 bps less than in Example 1 and 45 bps less than in Example 3, all from weaker reinvestment. For a buy-and-hold investor the entire interest rate risk is reinvestment risk. Baywhite. Reinvested coupons fall to 26.802, but selling with six years left at the lower 5.2 percent yield lifts the sale price to 105.043:

Here the capital gain on the sale more than offsets the weaker reinvestment, so Baywhite beats both Example 2 and Example 3.

The two forces at work are the two faces of interest rate risk. The future value of reinvested coupons rises when rates rise and falls when rates fall; that is reinvestment risk. The sale price of a bond that still has life left after the horizon falls when rates rise and rises when rates fall; that is price risk. They pull in opposite directions. Someone who exits even before collecting a single coupon carries price risk alone; a buy-and-hold investor carries reinvestment risk alone. Two investors holding the very same bond can therefore carry entirely different interest rate risk, decided by their investment horizon.

Check yourself

A family office buys a six-year, 5.8 percent annual coupon eurobond at par. Rates rise to 6.5 percent right after purchase, and it sells after three years. What is the annualized total return?

About 5.28 percent. Reinvested coupons grow to 18.556; the sale price with three years remaining at 6.5 percent is 98.146; the annualized return is ((18.556 + 98.146) / 100) to the power one-third, minus 1, which is 5.28 percent. It sits below the 5.8 percent coupon because the capital loss outweighs the extra reinvestment income.

An investor buys a five-year, 5.6 percent annual coupon bond at par. Rates rise to 5.9 percent immediately, and the bond is sold after two years. What is the total return?

About 5.23 percent. Reinvested coupons total 11.530; the sale price with three years left at 5.9 percent is 99.197; the return is ((11.530 + 99.197) / 100) to the power one-half, minus 1, which is 5.23 percent, below the 5.6 percent coupon.

If interest rates rise after purchase, what happens to the future value of an investor’s reinvested coupons, and what is the carrying value of a bond?

The future value of reinvested coupons rises, because the coupons are now reinvested at a higher rate; future value of coupons moves directly with interest rates. The carrying value is the original purchase price adjusted for amortization of any discount or premium: purchase price plus accumulated discount amortization for a bond bought below par, or minus accumulated premium amortization for one bought above par.

If reinvestment risk and price risk always offset, is there a horizon at which they cancel exactly? To find it, add a third investor to the same BRWA bond: Hightest Capital, with an eight-year horizon. This is long enough that reinvestment matters, but short enough that a sale still happens with two years of bond life left, so both effects are present.

Example 5 · Worked

Hightest buys BRWA’s 10-year, 6.2 percent bond at par and sells after eight years, when two years remain to maturity.

Compute Hightest’s horizon yield when rates stay at 6.2 percent, when they rise to 7.2 percent, and when they fall to 5.2 percent.

Solution. Rates stable at 6.2 percent. Reinvested coupons reach 61.807 and the bond sells at par:

Rates rise to 7.2 percent. Reinvested coupons climb to 64.071, but the sale price falls to 98.197:

The capital loss of 1.803 on the sale is nearly matched by 2.265 of extra reinvestment income. Rates fall to 5.2 percent. Reinvested coupons drop to 59.630 while the sale price rises to 101.854:

The capital gain of 1.854 roughly offsets the weaker reinvestment. Across all three scenarios Hightest earns close to 6.2 percent. Its eight-year horizon has very nearly neutralized interest rate risk.

Collecting the three investors together makes the pattern sharp. VFO and Baywhite each swing several basis points as rates move, but in opposite directions; Hightest barely moves at all.

Horizon yields on BRWA’s 10-year bond, by investor and scenario

Investor (horizon)	Rates stable 6.20%	Rates rise to 7.20%	Rates fall to 5.20%
VFO (10 years, buy and hold)	6.20%	6.43%	5.98%
Baywhite (sells after 4 years)	6.20%	5.28%	7.16%
Hightest (sells after 8 years)	6.20%	6.24%	6.17%

The long-horizon investor gains from rising rates through reinvestment; the short-horizon investor gains from falling rates through price. The eight-year investor is close to indifferent.

Macaulay duration as the balancing horizon

The eight-year horizon was not arbitrary. It is close to the Macaulay duration of this bond: the holding period at which the reinvestment gain (loss) from a one-time parallel shift in the yield curve is exactly cancelled by the price loss (gain) on the sale. It takes its name from Frederick Macaulay, a Canadian economist who set out the idea back in 1938. As time passes after a rate rise, the bond price is pulled back to par while the extra reinvestment income builds; at the Macaulay duration the two are equal in size and opposite in sign, and the same balancing holds when rates fall. For this BRWA bond the figure works out to 7.7429 years (computed in the next section), which is exactly why Hightest, at eight years, sits so close to fully hedged.

This gives clean general statements about the relationship among horizon, duration, and interest rate risk.

Macaulay duration, investment horizon, and interest rate risk

Relationship	Dominant risk	Source of interest rate risk
Investment horizon > Macaulay duration	Reinvestment risk	Falling interest rates
Investment horizon = Macaulay duration	Price risk = reinvestment risk	None
Investment horizon < Macaulay duration	Price risk	Rising interest rates

The gap between the two quantities has a name. The duration gap is the Macaulay duration minus the investment horizon.

VFO, at ten years, has a negative duration gap, so its main threat is falling rates. Baywhite, at four years, has a positive gap and fears rising rates. Hightest, at eight years, has a gap near zero and is close to hedged. One caution: both quantities shrink as time elapses, since the remaining horizon and the bond’s duration both fall, so the match does not stay perfect without rebalancing.

Figure 1: Reinvestment gain and price loss offset at the Macaulay duration (rates rise)

After a rate rise, the price loss is largest at the start and is pulled to par over time, while the reinvestment gain starts small and compounds. They cross at the Macaulay duration, where the horizon is hedged. When rates fall, the roles reverse: an early price gain fades while a reinvestment shortfall grows, meeting again at the duration.

The same logic on a longer bond

The offset is a general property, not a quirk of one bond. Take three investors who each buy, at par, a 5.61 percent Romanian eurobond maturing in 30 years and carrying a 15.16-year Macaulay duration. Rook Point holds for five years (well below the duration), Fyleton fifteen years (almost equal to it), and Amy thirty years (well above it). Watch the middle row: Fyleton’s horizon yield stays near 5.61 percent in every scenario, while the mismatched investors swing.

Horizon yields on the 5.61%, 30-year bond (Macaulay duration 15.16 years)

Investor (horizon)	FV of coupons, stable	Sale price, stable	Yield, stable 5.61%	Yield, rise to 6.11%	Yield, fall to 5.11%
Rook Point (5 years)	31.379	100.000	5.610%	4.626%	6.659%
Fyleton (15 years)	126.765	100.000	5.610%	5.613%	5.624%
Amy (30 years)	414.223	100.000	5.610%	5.861%	5.367%

Under the rate-rise scenario the sale prices are 93.675, 95.179, and 100.000; under the rate-fall scenario they are 106.970, 105.152, and 100.000. When the horizon nearly equals the Macaulay duration (Fyleton), the coupon reinvestment gain or loss offsets the price loss or gain, so the return holds steady.

Check yourself

Hightest Capital buys a 15-year, 6.8 percent annual coupon bond with a Macaulay duration of 9.85 years and has a seven-year investment horizon. What is the duration gap, and which rate move threatens it?

The duration gap is 9.85 minus 7.0, which is 2.85 years. A positive gap means the Macaulay duration exceeds the horizon, so price risk dominates and rising interest rates are the threat.

An investor buys a seven-year, 6.4 percent bond with a Macaulay duration of 5.86 years and a four-year horizon. Rates rise 50 bps right after purchase. What risk dominates?

Price risk. The four-year horizon is shorter than the 5.86-year duration (a positive duration gap), so market price risk dominates reinvestment risk, and the rate rise produces a capital loss on the eventual sale.

Match each duration gap to its dominant risk source: negative gap and positive gap.

A negative gap (duration below horizon) means reinvestment risk dominates, so the threat is falling interest rates. A positive gap (duration above horizon) means price risk dominates, so the threat is rising interest rates.

The balancing horizon has a precise definition. It measures, as a present-value-weighted average, when the payments on a bond land: each receipt date is weighted by the slice of the full price (the present value) that the flow arriving then contributes. Cash flows further out carry more time but, once discounted, a smaller weight, so the duration lands below the time-to-maturity for any coupon-paying bond.

Here i indexes the periods, PV_i is the present value of the cash flow in period i, and PV_Full sums those present values into the full price. The term t counts the days elapsed since the last coupon, while T gives the length of the whole coupon period in days. On an issuance or coupon date, t is zero, so every flow arrives exactly at its period number.

Example 6 · Worked

Compute the Macaulay duration of BRWA’s 10-year, 6.2 percent annual eurobond at issuance, priced at par with a 6.2 percent YTM, on a 30/360 basis (settlement 15 October 2025, maturity 15 October 2035).

Build the present-value-weighted table and read off the duration.

Solution. Each coupon is 6.2 and the final period adds the 100 principal. Discount every flow at 6.2 percent, convert each present value to a weight, and multiply by the time to receipt.

Macaulay duration of BRWA’s 10-year, 6.2% eurobond at issuance

Period	Time to receipt	Cash flow	Present value	Weight	Time × weight
1	1	6.2	5.8380	0.0584	0.0584
2	2	6.2	5.4972	0.0550	0.1099
3	3	6.2	5.1763	0.0518	0.1553
4	4	6.2	4.8741	0.0487	0.1950
5	5	6.2	4.5895	0.0459	0.2295
6	6	6.2	4.3216	0.0432	0.2593
7	7	6.2	4.0693	0.0407	0.2849
8	8	6.2	3.8317	0.0383	0.3065
9	9	6.2	3.6080	0.0361	0.3247
10	10	106.2	58.1942	0.5819	5.8194
Total			100.0000	1.0000	7.7429

The present values sum to 100.0000 (the par price), and the last column sums to 7.7429 years, the Macaulay duration. Notice the final payment of 106.2 alone accounts for about 58 percent of the price. Because the coupon is annual, no annualization step is needed; for a semiannual bond the periodic figure would be divided by 2, and the units are years. As expected, 7.7429 is below the 10-year maturity, and it is close to Hightest’s eight-year horizon, which is why that investor was nearly hedged.

Two shortcuts: the spreadsheet function and the closed form

Beyond building the table, Macaulay duration can be read straight from the DURATION function in Excel or Google Sheets, using the settlement date, maturity date, coupon, yield, coupon frequency, and a day-count basis (0 or omitted is 30/360). It can also be computed in one closed-form expression derived with algebra and calculus, which is handy between coupon dates:

Here r is the periodic yield, N the number of periods to maturity from the start of the current period, c the periodic coupon rate, and t/T the fraction of the coupon period elapsed. Applied to the BRWA bond at issuance (r = c = 0.062, N = 10, t/T = 0), this returns 7.7429 years, matching the table.

Duration falls as time passes

Macaulay duration is not fixed. Each day that elapses shortens every cash flow’s time to receipt, so the duration drifts lower. Recomputing the BRWA bond just 57 days after issuance makes the drift visible.

Example 7 · Worked

Recompute the BRWA bond’s Macaulay duration on 11 December 2025, which is 57 days after issuance, on a 30/360 basis, with the YTM still 6.2 percent. Now t/T equals 57/360, so every payment sits 57 days closer than at issuance.

Build the table and compare with the issuance figure.

Solution. The first flow now arrives in 1 minus 57/360, which is 0.8417 periods, and each later flow follows one period behind that.

Macaulay duration of BRWA’s bond, 57 days after issuance

Period	Time to receipt	Cash flow	Present value	Weight	Time × weight
1	0.8417	6.2	5.8939	0.0584	0.0491
2	1.8417	6.2	5.5498	0.0550	0.1012
3	2.8417	6.2	5.2258	0.0518	0.1471
4	3.8417	6.2	4.9207	0.0487	0.1872
5	4.8417	6.2	4.6335	0.0459	0.2222
6	5.8417	6.2	4.3630	0.0432	0.2525
7	6.8417	6.2	4.1082	0.0407	0.2784
8	7.8417	6.2	3.8684	0.0383	0.3005
9	8.8417	6.2	3.6426	0.0361	0.3190
10	9.8417	106.2	58.7511	0.5819	5.7273
Total			100.9570	1.0000	7.5845

Being 57 days later raises every present value slightly (the full price is now 100.9570, reflecting accrued interest), while the weights are unchanged. Because the times to receipt are all smaller, the Macaulay duration falls to 7.5845 years, down from 7.7429 at issuance. Duration shrinks steadily as the receipt dates draw nearer.

Annualizing a semiannual bond

When a bond pays more than once a year, the table is built in periods and then annualized by dividing by the number of periods per year. The next example shows the two-period-per-year case.

Example 8 · Worked

A bond has three years to maturity, a 4 percent coupon paid semiannually, and is priced at 100 (so the periodic YTM is 2 percent).

Find the annualized Macaulay duration.

Solution. There are six semiannual periods, each paying a 2 coupon, with 100 principal added at period 6. Discount at 2 percent per period and weight by present value.

Macaulay duration of a 3-year, 4% semiannual bond

Period	Time to receipt	Cash flow	Present value	Weight	Time × weight
1	1	2	1.961	0.0196	0.01961
2	2	2	1.922	0.0192	0.03845
3	3	2	1.885	0.0189	0.05654
4	4	2	1.848	0.0185	0.07391
5	5	2	1.811	0.0181	0.09057
6	6	102	90.573	0.9057	5.43438
Total			100.00000	1.00000	5.71346

The weighted sum in periods is 5.71346. Dividing by the 2 periods per year gives an annualized Macaulay duration of 2.857 years. A zero-coupon bond is the limiting case: with a single cash flow, its Macaulay duration equals its time-to-maturity (unless the calculation falls between coupon dates).

Check yourself

A five-year, zero-coupon bond is valued on its issuance date. Without building a table, what is its Macaulay duration?

5.0000 years. A zero-coupon bond has one cash flow at maturity, so its present-value-weighted time to receipt is simply its time-to-maturity, here five years.

A bond has two years to maturity, a 4 percent coupon paid semiannually, and a 4.60 percent YTM. The valuation date falls 63 days into the opening coupon period, on a 30/360 basis. Roughly what is the annualized Macaulay duration?

About 1.7666 years. The opening payment arrives in (180 minus 63) / 180 periods, that is 0.65, with each subsequent flow one period further out; the period-weighted sum of 3.5331, halved, gives 1.7666 years.

An investor plans to hold for four years a seven-year, 6.7 percent annual coupon bond bought at par. Rates fall to 6.3 percent right after purchase. Is the horizon yield above or below 6.7 percent?

Above, at about 6.89 percent. Reinvested coupons grow to 29.441 and the sale price with three years left rises to 101.063, so the capital gain outweighs the weaker reinvestment and the return exceeds the coupon.

End of lesson.

Back to MODULE 07 — Fixed Income

FI 10 Interest Rate Risk and Return

Overview

Sources of Return from a Fixed-Rate Bond

When the horizon yield equals the original YTM

Shocking the interest rate

Investment Horizon and Interest Rate Risk

Macaulay duration as the balancing horizon

The same logic on a longer bond

Calculating Macaulay Duration

Duration falls as time passes

Annualizing a semiannual bond