A bond's price is not an opinion. Given a yield, it is the present value of a schedule - and the size of the move when yields shift is a number you can compute in advance.
14 min read · Level 2
The retail conversation about bonds is almost entirely about credit - who the issuer is, how sound it looks, what the rating says - and for high-quality bonds credit is not where the money moves. A government bond of impeccable standing lost more than a quarter of its market value in the table above with the borrower in perfect health, on a rate move of two percentage points, and no amount of credit analysis would have anticipated a penny of it. The dominant variable was maturity, which is printed on the instrument, requires no judgement to read, and is the one thing nobody asks about when they are told a bond is safe. This is worth stating baldly because the industry's vocabulary actively works against it: bonds are sorted, sold and discussed by credit quality, so the reader arrives with a mental model in which safe means will not lose money, when what safe actually means in that sentence is will probably pay. Those are different guarantees. The practical consequence is that the first question to ask about any fixed income holding is not who owes the money but when it comes back - and if the answer is thirty years, you are holding an instrument whose price behaviour has far more in common with a volatile asset than with the cash it is often shelved beside.
Start with the mechanism rather than the formula, because the formula is only bookkeeping once the mechanism is clear.
You hold the anchor bond: par 1,000, 4% coupon, five years to run, bought at par. The morning after you buy it, prevailing five-year yields move to 6%. New issues of the same maturity and the same credit now pay 60 a year for the same 1,000.
Nobody will pay you 1,000 for a claim on 40 a year when 60 a year is on sale next to it for the same money. Your bond is not defective and its issuer has not changed. It is simply competing against a better-paying alternative, and there is exactly one lever left: the price.
How far does the price fall? Precisely far enough that a buyer paying the new price earns 6% overall. Two routes to the same answer, and doing both is the best way to see that the result is arithmetic rather than sentiment.
Route one - price the shortfall. Your bond pays 40 a year where the market now pays 60. The buyer is short 20 a year for five years. The present value of 20 a year for five years at 6% is
20 x [(1 - 1.06^-5) / 0.06] = 20 x 4.212364 = 84.25
so the price must fall by 84.25, from 1,000 to 915.75.
Route two - discount the actual cash flows. Coupons: 40 x 4.212364 = 168.49 Principal: 1,000 x 1.06^-5 = 747.26 Price 915.75
Identical to the cent. The discount to par IS the present value of the coupon shortfall; those are two descriptions of one thing.
Run it the other way. If yields fall to 2%, your 40 a year is now the generous one and the price rises to 1,094.27. At 3% it is 1,045.80. At 5% it is 956.71. There is nothing to negotiate at any of these points: fix the yield and the price follows, fix the price and the yield follows. The yield is doing the job a discount rate does in any present-value calculation, and it is the only input - which is why bond pricing is arithmetic where a discounted cash flow valuation of a company is a set of estimates. What the market actually argues about is the yield.
This is the sharpest difference between this lesson and its counterpart in public equities. An equity's quote is a fact about two people who agreed a moment ago; there is no formula that turns a company into a price. A bond's price, given a yield, is a calculation any two people will agree on. All of the judgement has moved into the yield, and the yield is where credit, maturity and liquidity all get argued out.
A note on units. Bond yields move in basis points - one basis point is one hundredth of a percentage point, so a move from 4.00% to 4.25% is 25 basis points. The convention exists because differences of a few hundredths of a percent are the substance of the market.
And a note on the phrase "rates rose". There is no single rate. Yields differ by maturity, and the set of them across maturities is the yield curve. A curve can shift up in parallel, steepen, flatten or twist, and a bond is exposed only to the part of the curve where its own cash flows sit.
Because knowing the direction is worth very little and knowing the magnitude is worth a great deal - and the magnitude is not obvious from the coupon, the credit or the name.
The measure that gives it to you is duration. Macaulay duration is the weighted average time until you receive the bond's money, weighting each payment by its present value as a fraction of the price. For the anchor bond at a 4% yield:
t cash flow present value weight t x weight 1 40 38.4615 0.038462 0.038462 2 40 36.9822 0.036982 0.073964 3 40 35.5599 0.035560 0.106680 4 40 34.1922 0.034192 0.136769 5 1,040 854.8042 0.854804 4.274021 -------------------------------------------------------- 1,000.0000 1.000000 4.6299
Macaulay duration is 4.629895 years, the total of that last column. Note what that table shows: 85.48% of the bond's present value sits in the final payment. A coupon bond behaves far more like its maturity date than its interim payments suggest, and the lower the coupon the more true that is - which is why a zero-coupon bond's duration equals its maturity exactly.
Modified duration converts that into a price sensitivity by dividing by one plus the yield:
4.629895 / 1.04 = 4.451822
and the working approximation is that the price moves by minus the modified duration times the change in yield:
price change for +100 basis points = -4.451822 x 0.01 x 1,000 = -44.52
Now check it against the exact answer, because the approximation is an approximation and this lesson would be dishonest if it stopped here.
+100bp approximate price 955.48 exact price 956.71 exact change -4.3295% -100bp approximate price 1,044.52 exact price 1,045.80 exact change +4.5797%
The approximation overstates the loss by 1.22 per 1,000 on the way up and understates the gain by 1.28 per 1,000 on the way down. That asymmetry is convexity: the true price-yield relationship is a curve and modified duration is the straight line tangent to it. The curve lies above the line in both directions, so the straight line is pessimistic about a rise in yields and pessimistic about a fall too - and the error grows with the size of the move.
+25bp approximate change -11.13 exact -11.05 error 0.08 +200bp approximate change -89.04 exact -84.25 error 4.79
Twenty-five basis points and the approximation is fine. Two hundred and it is out by nearly 5 per 1,000, all of it in the direction of frightening you more than the arithmetic warrants.
Educational, not personalised financial advice. Nothing in the Investor Academy is a recommendation to buy, sell or hold anything — what is right for you depends on your own circumstances and the rules where you live.
The reason any of this matters in practice is that duration varies enormously across bonds that all look alike on a screen. Four bonds, all with a 4% coupon, all trading at par when yields are 4%, differing only in maturity. Yields then move from 4% to 6%:
maturity modified duration price at 4% price at 6% change 2 years 1.8861 1,000.00 963.33 -3.6668% 5 years 4.4518 1,000.00 915.75 -8.4247% 10 years 8.1109 1,000.00 852.80 -14.7202% 30 years 17.2920 1,000.00 724.70 -27.5297%
Same issuer, same coupon, same credit, same day. The thirty-year bond lost more than seven times as much as the two-year one, and the only difference between them is when the money comes back. Anyone who describes a government bond as low-risk without saying which maturity has said almost nothing.
Now the extreme case. A ten-year zero-coupon bond has a Macaulay duration of exactly ten years, because there is only one payment and it arrives at year ten. At a 4% yield it costs 1,000 / 1.04^10 = 675.56. Move the yield to 5% and it costs 1,000 / 1.05^10 = 613.91, a fall of 9.1258%. Its modified duration is 10 / 1.04 = 9.6154, so the approximation predicts a 9.6154% fall - overstating the actual loss by half a percentage point on a single 100 basis point move, which is the convexity effect again, larger here because the duration is larger.
The zero is worth holding in mind as a limiting case. It is the purest form of the instrument - no reinvestment risk at all, as the previous lesson showed - and simultaneously the most sensitive to rates of any bond of that maturity. Removing one risk did not remove risk.
The most familiar mistake is the sentence "I am holding to maturity, so the price does not matter." It is half true and the half that is false is expensive. What is true: if the issuer pays, your own cash flows are unaffected by anything the price did in between. What is false: the fall in price is a real measure of what you gave up, because you are now locked into 4% while 6% is available, and the 84.25 is precisely the value of that. It is also false if you might need to sell, and false if you hold through a fund, which has no maturity at which anything is pulled back to par.
The second is using modified duration for a large move. It is a first derivative. At 25 basis points it is accurate to a rounding error; at 200 it is out by 4.79 per 1,000 on the anchor bond and by considerably more on a long one. When the move is large, discount the cash flows again rather than scaling the estimate.
The third is assuming the curve moves in parallel. Short and long yields routinely move by different amounts and occasionally in different directions. A portfolio whose duration is right in aggregate can still be badly positioned if all of its exposure sits at one end of the curve.
The fourth is confusing duration with maturity. They coincide only for a zero-coupon bond. The anchor bond matures in five years and has a duration of 4.63 years, and that gap is entirely made up of the coupons arriving before the end.
The fifth is treating a rate move as news about the issuer. Nothing in this lesson involved the company at all. A high-quality bond can fall 27% on a rate move with the borrower in perfect health, and holders who go looking for a credit explanation will not find one.
Write the payments out, choose the yield, and compute. Two lines of arithmetic settle what would otherwise be a guess, and the discipline matters because a plausible-sounding bond price is very easy to produce and very hard to spot as wrong. If two routes to the answer agree - the shortfall method and the direct discounting - you have also caught your own errors.
Given a price and a schedule there is exactly one yield that reconciles them, and it is found numerically rather than by formula. Solving it yourself, once, for a bond somebody has quoted to you is the fastest way to discover whether the figure on the sheet is the yield to maturity, the current yield or the coupon - three different numbers that are routinely presented under the same word.
Duration answers the question how much would this holding move if yields moved by X, which is a question about how large a position you should take. It does not answer whether yields will move, and treating a duration figure as a prediction is how a risk measure becomes a bet. If you find yourself choosing a duration because of where you think rates are going, you have changed activity without noticing.
Above roughly a hundred basis points the linear estimate drifts, always in the direction of overstating losses and understating gains. Re-pricing the actual schedule takes the same two lines it took the first time, and the difference between the estimate and the exact figure is itself informative: a large gap is telling you the position is highly convex.
List your holdings by maturity, not just by total duration. Two portfolios with the same average duration - one built from five-year bonds, the other from a mix of one-year and thirty-year - behave completely differently when the curve steepens or flattens, and only one of them is what you thought you owned.
Before buying any bond or bond fund, whenever you are choosing between two maturities, and whenever you want to know in advance what a plausible rate move would do to a holding rather than discovering it on a statement.
Do not use these tools to position for a forecast of rates - the whole market is doing that with better information - and do not apply duration arithmetic to a bond whose cash flows are not actually fixed, such as a callable or floating-rate issue, where the schedule you are discounting is not the schedule you will receive.
Long-dated bonds usually yield a little more than short ones, and the extra is visible, quotable and easy to justify to yourself. What is not visible on the same screen is that the additional sensitivity is not a little more, it is several times more. Extending from five years to thirty in the table above bought whatever the yield difference happened to be that day, and took modified duration from 4.45 to 17.29 - so a two-point move in yields cost 27.53% of principal instead of 8.42%. Holders who made that switch for the yield frequently describe the subsequent loss as a market event. It was a position they chose, and its size was computable on the day they chose it.
Extra yield from a longer maturity is payment for extra price sensitivity, and the exchange rate is poor. Compute the duration before deciding the yield pickup is generous.
When a bond portfolio falls, a common response is that the holdings will be held to maturity and therefore nothing has been lost. As a statement about contractual cash flows this is correct. It becomes a problem in two specific circumstances that tend to arrive together: when the money is needed earlier than expected, and when the holding is a fund rather than an individual bond and there is no maturity date to hold to. The phrase is also frequently applied to positions the holder had no intention of keeping until maturity when they bought them, which is the tell - a plan adopted after the price fell is a description of the price fall rather than a plan.
Hold to maturity is only an answer if the instrument has a maturity, the money has no earlier use, and the intention predates the loss. Check all three before using it.
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