A bond's duration estimates how far its price moves when yields change, as if price followed a straight line. On Singapore's 2.25% government bond due 2036, at its closing bid yield on 2 November 2026, that line says a one-point rise in yield cuts the price by 8.66%. Exact repricing gives 8.25%. On the 3% bond due 2072 it says 26.00%, against 21.84%. Both are our arithmetic on MAS's published yields. Modified duration is the percentage version; Macaulay duration tells the same fact as an average wait in years. The further out a bond's money sits, the more the line misses: it overstates losses and understates gains.
Duration alone predicts a reopening price about 1.2 points low
MAS reopened the 2072 green bond at auction in September 2024 and again in September 2025: at a cut-off yield of 2.62% and price 110.823 per 100 of face the first time, and at 2.16% and 125.166 the second.
Removing the year between, we priced the bond at 2.62% on the 2025 settlement date: 110.71, with modified duration 26.02. Duration alone then predicts 123.97 at 2.16%, about 1.2 points below MAS's price. Adding convexity, the second-order term that bends the line, gives 125.09, under 0.1 short. Competitive bids at these auctions are made as yields to two decimal places, so both cut-off yields are exact, and our formula reproduces both of MAS's prices to three decimals. One bond and one 46 basis point fall make an illustration and test nothing.
Macaulay weighted each payment's wait by its present value
Frederick Macaulay defined duration in a 1938 NBER study. Chapter II treats each payment as a separate loan and weights the time to each "by the ratio of the present value of the individual future payment to the sum of all the present values" (p. 48). For two loans of $1,000 each, one repaid after a year and one after 100 years, present-value weighting gives a duration of 50½ years; weighting by the sums repaid gives more than 99, which he called absurd. He concluded: "The argument for present value weighting seems strong" (pp. 46–47).
Modified duration divides Macaulay's figure by 1 + y/2 for semi-annual coupons, on a coupon date, turning years into a percentage per point of yield. Who first stated that link we could not establish.
On our bond duration calculator's default bond, $1,000 face with a 5% semi-annual coupon, 10 years left and a 6% yield, Macaulay duration is 7.895 years and modified 7.665. Duration predicts −7.67% for a one-point rise; the exact reprice is −7.32%. Until 3 November 2026 the calculator derived Macaulay from modified duration, which is exact only on a coupon date, so a zero-coupon bond with 10.3 years left showed 10.304. The revision of that date computes the weighted time directly and gives 10.300.
Rising yields cost less than the straight line says
When yields rise, the line predicts a bigger price fall than happens; when they fall, a smaller gain. We took the 23 Singapore Government bonds with an auction record and a closing bid on 2 November 2026, moved each yield by one point and repriced exactly. Four of them:
| Bond | Modified duration | Rise, exact | Fall, exact |
|---|---|---|---|
| 2.875%, 2028 | 1.69 | −1.67% | +1.71% |
| 2.25%, 2036 | 8.66 | −8.25% | +9.10% |
| 3.25%, 2054 | 18.87 | −16.78% | +21.37% |
| 3%, 2072 | 26.00 | −21.84% | +31.47% |
Duration alone gives the modified-duration figure in both directions, with the sign flipped. At a one-point rise the median miss across the 23 was about 0.2 points of price, and five bonds missed by a full point or more. The rise needed for a one-point miss shrinks from 758 basis points on the 2028 bond to 48 on the 2072 bond. For scale, MAS's 10-year benchmark yield rose 1.91 points in the year to September 2022, the largest rise over any 365 days in MAS's daily series, which starts in November 2014.
The longest bond has the duration of a far shorter zero
A zero-coupon bond pays everything at maturity, so its Macaulay duration is its term. Coupons pull duration below term: as a share of years left it falls from 0.976 on the 2028 bond to 0.575 on the 2072 bond, which has the duration of a zero due in 26.3 years. The likeness ends there. On our arithmetic the bond's convexity is 1.38 times the zero's, so at a two-point rise the bond loses 37.26% and the zero 40.39%.
Longer does not always mean more duration. Macaulay found the turn: "if the bond be selling below par, D will attain a maximum value before n reaches infinity" (p. 50). By our arithmetic a 2% bond at a 10% yield peaks at 26.5 years of term, with a duration of 13.60 years.
Yield to maturity assumes every coupon is reinvested at that yield
MAS's page on calculating SGS returns counts in yield to maturity "returns from reinvesting all coupon payments". Take the calculator's default bond, bought at 6% and held ten years: on our arithmetic, coupons reinvested at 0% give a realised 4.89%, and at 12% they give 7.43%.
We also jumped the yield once, to anywhere from 3% to 9%, and sold at different horizons. After one year the realised yields spanned 41.5 points; at maturity, 1.3. At 7.9 years, the Macaulay duration, they spanned under 0.1, as price loss and reinvestment gain nearly cancel. F. M. Redington built immunization on that cancelling in 1952, assuming one rate for every term; only exact matching of cash flows, he wrote, is immune to every change.
Our arithmetic finds accrued interest inside MAS's auction prices
We repriced 198 reopenings of coupon-paying bonds among MAS's 324 bond auctions since 1987. In 33, settlement fell on a coupon date and nothing had accrued. In all 165 others, MAS's cut-off price sat nearer our price with accrued interest than our clean price. No MAS text we opened says so; it is our inference.
MAS's Rules and Market Practices for SGS, in the March 2011 edition we opened, use "the US method of semi-annual interest compounding" and an Actual/Actual day count, but compound the part-period to the next coupon when more than six months remain, where the US Treasury formula and our calculator use simple interest. With MAS's version our prices came within 0.0015 of MAS's for 181 of the 198, against 132 with the US one. Bank Negara Malaysia gives the same semi-annual, Actual/Actual basis for Malaysian Government Securities.
Savings Bonds and callable bonds sit outside the duration model
MAS's Singapore Savings Bonds FAQ says "changes in the interest rate environment will have no effect on the bond's value", and that holders can always redeem for the amount invested. A callable bond can change its cash flows when yields move, and Robert Alps's actuarial study note places it outside a model that also assumes "a flat yield curve". Macaulay granted the same of his own measure: assuming each future half-year's rate equals the bond's yield "may easily be contrary to fact" (p. 52).
Contains information from MAS's SGS auction results and SGS prices and yields datasets, accessed on 3 November 2026 from MAS's bonds and bills statistics on eservices.mas.gov.sg, which is made available under the terms of the Monetary Authority of Singapore's Terms of Use of Datasets.
