Rates · 10 lessons

Revision sheet

Every Rates lesson’s essentials, formula and takeaway on one page. Each heading opens the full lesson.

R1Government bonds and how they’re quoted

Government bond: A tradable loan to a government: it pays interest along the way and repays its face value, which prices treat as 100, on a set date. UK bonds are , US ones Treasuries and German ones .
Coupon and maturity: The coupon is the interest, a percentage of face value. Gilts and Treasuries pay it in two halves, every six months: a 4% coupon pays £2 per £100 twice a year. Maturity is the date the bond is repaid.
Bills, bonds and linkers: last a year or less and pay no coupon: they are sold below 100 and repaid at 100. Bonds run out to 30 years and beyond. and raise their payments with inflation.
Clean and dirty price: Screens show the , per 100 of face value. The buyer pays the : clean plus . A 4¼% gilt 73 days into a 181-day period carries 0.857 per 100.
Decimals and 32nds: Gilts and Bunds are quoted in decimals. Treasuries are quoted in 32nds of a point: 101-08 is 101.25, and a “+” adds half a 32nd. Gilts and Treasuries normally settle the next business day (T+1).

What a buyer pays

Dirty = clean + (C ÷ 2) × d ÷ D; 99-16+ = 99 + 16.5 ÷ 32

C: annual coupon per 100 of face value
d: days since the last coupon
D: days in the coupon period (181–184 for a half-year)
+: half a 32nd (1/64) in a Treasury quote

Remember: Dirty = clean + accrued; 99-16+ = 99.515625. Bonds are quoted clean, but you pay the dirty price: the clean price plus the coupon the seller has already earned.

R2Price, yield and the curve

Yield: The single annual rate that discounts a bond’s payments back to its price: the return locked in by buying and holding to maturity. The payments are fixed, so when yields rise, prices fall.
Premium, par and discount: A coupon above the yield puts the price above 100, a premium; a coupon below it, under 100, a discount; equal, exactly 100, par. A 3% two-year bond yielding 4.5% is worth about 97.19.
The yield curve: Yields plotted against maturity. It usually slopes upwards, because lenders want extra to lend for longer. When short yields sit above long ones the curve is , which usually means cuts are expected.
Forward rate: The future rate that today’s curve locks in. With the 1-year rate at 4% and the 2-year at 4.5%, the one-year rate a year from now is about 5%.
Naming a curve move: Bull: yields fell. Bear: yields rose. Steepening: the gap between long and short yields, such as 2s10s (10-year minus 2-year), widened; flattening: it narrowed. Expected cuts usually bring a bull .

Price of a two-year bond from its yield

P = C ÷ (1 + y) + (100 + C) ÷ (1 + y)²

P: price per 100 of face value
C: the yearly coupon per 100
y: the yield, as a decimal (4.5% = 0.045)

Remember: 1y1y ≈ 2 × the 2-year rate − the 1-year rate. Prices and yields move in opposite directions, and two points on the curve imply the rate in between.

R3Duration, DV01 and convexity

Price and yield: A bond’s price and its move in opposite directions. Long-dated bonds move further than short-dated ones for the same change in yield.
Duration: The percentage a bond’s price moves for a one-point (100bp) change in yield. A of 7.7 means a 1-point rise in yield costs about 7.7% of the price.
What lengthens duration: A later maturity and a lower coupon. A has the most duration for its maturity, because all of its value arrives at the end.
DV01: The profit or loss on a position for a 1bp move in yield: about notional × duration × 0.0001. £10m of a bond with a duration of 7.7 has a DV01 of about £7,700.
Long and short: Owning bonds (long) loses money when yields rise and gains when they fall. Having sold bonds you don’t own (short) is the reverse.
Convexity: The price–yield line is a curve, so a bond gains more when yields fall than it loses when they rise by the same amount. It matters most for large moves.

P&L from a yield move

DV01 ≈ notional × D × 0.0001

DV01: profit or loss for a 1bp move in yield
notional: face value of the bonds held
D: modified duration

Remember: DV01 ≈ notional × duration × 0.0001. Long bonds lose when yields rise, and the longer the bond, the more each basis point costs.

R4Money markets, repo and the expected policy path

Bank Rate and SONIA: The sets , paid on banks’ overnight money at the Bank of England. , the average overnight sterling borrowing rate, tracks it; and are the dollar and euro rates.
Treasury bills: Short government debt with no coupon, bought below 100 and repaid at 100. US bills are quoted as a discount on a 360-day year: a 180-day bill at 4% costs 98.00.
Repo: A loan secured on a bond: sell it for cash and agree to buy it back later at a slightly higher price. Desks finance bonds this way, so a bond yielding 4.5% financed at 4.0% earns 0.5% a year.
General collateral and specials: Most government bonds repo at the rate, close to Bank Rate. A bond many traders want to borrow trades , at a lower rate, so its owner borrows cash cheaply.
Pricing the policy path: SONIA futures quote 100 minus the expected rate (96.20 implies 3.80%), and price the rate after each meeting. Bank Rate at 4.00% with 3.90% priced after the next meeting is 10 of a 25bp cut: about a 40% chance.

The chance of a move the market has priced

Chance of a 25bp move ≈ (rate priced − Bank Rate now) ÷ 0.25

rate priced: rate implied for the period after the meeting, %
0.25: the size of one move, in percentage points (25bp)

Remember: Chance of a cut ≈ bp priced ÷ 25bp. Money-market prices show what traders expect the central bank to do, and repo is the cost of holding every bond.

R5Supply, auctions and relative value

Auction formats: US Treasury auctions are single-price: bids fill from the lowest up, and all winners get the highest yield accepted, the stop-out. At conventional auctions, run by the , winners pay what they bid.
Bid-to-cover: Total bids divided by the amount sold: $105bn of bids for $42bn of bonds is about 2.5. It means little on its own, so traders compare it with recent auctions of the same bond.
Tail and stop-through: The is how far the stop-out lands above the yield at the bidding deadline: 4.238% against 4.215% is a weak 2.3. Landing below it is a stop-through, a sign of strong demand.
On-the-run bonds: The newest bond at each maturity is : the one everyone quotes, trades and borrows. Investors pay for that ease, so it yields a few basis points less than an older bond maturing at about the same time.
Supply and the curve: Heavier long-dated issuance means investors must be paid more to hold it, so long yields rise against short ones and the curve steepens, even if expected don’t move.

Judging a US auction

Tail (bp) = (stop-out yield − when-issued yield) × 100

stop-out: the highest yield accepted at the auction, %
when-issued: the new bond’s yield in the market at the bidding deadline, %

Remember: Tail = stop-out yield − when-issued yield. An auction that tails sold for less than the market price; one that stops through found more demand than was on offer.

R6Bond futures and hedging

Bond future: An exchange-traded agreement to buy or sell government bonds on a later date at a price fixed today. One ICE Long contract covers £100,000 of gilts. Contracts expire quarterly: H March, M June, U September, Z December.
Delivery basket and conversion factor: The seller can deliver any bond in a basket: for the Long Gilt, gilts with 8¾ to 13 years left. Each has a , about its price at a 4% ÷ 100, so coupons above 4% give factors above 1.
Cheapest to deliver: On delivery the seller is paid the futures price × the factor, plus accrued interest, and delivers whichever bond costs least against that. The future tracks this bond.
Hedge ratio: Hedge by DV01, not by the amount of bonds. A future’s DV01 is about the cheapest bond’s DV01 ÷ its factor: £82 ÷ 0.93 is about £88, so a position losing £33,000 per needs about 375 contracts sold.
What a hedge leaves: A DV01 hedge covers only moves where all yields shift together. 30-year gilts hedged with Long Gilt futures, which track a 10-year gilt, keep curve risk; the gap between your bonds and the cheapest-to-deliver is .

Futures needed to hedge

Contracts = position DV01 ÷ (CTD DV01 ÷ CF)

CTD DV01: DV01 of the cheapest-to-deliver bond, per contract
CF: that bond’s conversion factor

Remember: Contracts = position DV01 ÷ futures DV01. A futures hedge matches DV01, not the amount of bonds, and leaves you with the gap between your bonds and the cheapest-to-deliver.

R7Interest rate swaps

Interest rate swap: An exchange of fixed payments for floating ones, usually compounded or , on a notional that never changes hands. Each period only the difference is paid: 0.5% on £100m is £500,000.
Payer and receiver: The pays fixed and gains when rates rise; a company with a floating-rate loan pays fixed to lock in its cost. The gains when rates fall, as a bondholder does.
Swap rate: The fixed rate that makes both legs worth the same at the start, so a new swap costs nothing to enter and is worth zero to both sides.
Value after rates move: A swap gains or loses value as the market rate for its remaining term moves away from its fixed rate. Paying 3.80% for five years on £50m when the five-year rate is 4.10% is worth about £666,000 to the payer.
Swap spread: The swap rate minus the government bond of the same maturity. A negative spread usually means bonds are cheap after heavy supply, not that the government is riskier than a bank.

Value of a swap after rates move

Payer’s value ≈ (S − K) × A × N

S: today’s swap rate for the remaining years
K: the fixed rate in your swap
A: annuity factor: today’s value of 1 a year for the remaining years
N: notional. The receiver’s value is the same with the sign flipped

Remember: Payer’s value ≈ (S − K) × annuity × notional. A swap starts at zero; after that the payer gains when rates rise above the fixed rate and the receiver when they fall below it.

R8Curve trades, carry and roll-down

Curve trade: A bet on the gap between two , such as 2s10s, rather than their level. A buys the shorter bond and sells the longer one, gaining as the gap widens; a does the reverse.
Central-bank views: Two-year yields follow expected most closely. Expecting faster cuts than the market has priced suits a steepener; expecting more hikes than priced suits a flattener.
DV01 weighting: Size the legs to move the same per . £1m of 10-year moves about £820 per bp and £1m of 2-years about £190, so £50m of 2-years pairs with about £11.6m of 10-years.
Carry: What a bond position earns if nothing moves: its yield minus the rate that finances it. A bond yielding 4.5% financed at 4.0% 0.5% a year; financing above the yield is negative carry.
Roll-down: On an upward-sloping curve, a bond’s yield falls as it ages into a shorter maturity. A 5-year at 4.20% that becomes a 4-year at 4.05% gains about 0.15 × its , in per cent.

Sizing a curve trade

N_B = N_A × DV01_A ÷ DV01_B

N_A, N_B: notionals of the two legs
DV01_A, _B: DV01 of each bond per £1m of notional

Remember: Leg B = leg A × DV01 of A ÷ DV01 of B. Size curve trades by DV01 so you bet on the gap, and check carry and roll-down to see whether you are paid to wait.

R9Inflation-linked bonds and breakevens

Inflation-linked bond: A bond whose coupons and repayment are scaled by an : the price index now ÷ its level at issue. UK follow ; US follow CPI.
Real yield: A linker’s is a , earned on top of inflation: one yielding 1.2% real returns about 1.2% a year plus whatever inflation turns out to be.
Breakeven inflation: The inflation rate at which a linker and a conventional bond of the same maturity return the same: a 4.6% against a 1.2% real linker gives 3.4%. Inflation above it favours the linker; below it, the conventional bond.
Not a pure forecast: Breakevens also carry an , which lifts them, and a discount for linkers being harder to sell, which pulls them down when investors rush for cash.
5y5y: The inflation rate priced for years six to ten: about twice the 10-year breakeven minus the 5-year, so a 10-year at 2.8% and a 5-year at 3.0% give 2.6%. Central banks watch it to judge whether inflation will settle at target.
RPI and CPIH from 2030: RPI’s methods are due to be aligned with from February 2030. That is expected to lower RPI inflation, and with it what index-linked gilts pay after that date.

Breakeven inflation

Breakeven = conventional yield − real yield

conventional: yield on the ordinary fixed-payment bond, %
real yield: yield on the matching inflation-linked bond, %

Remember: Breakeven = conventional yield − real yield. If inflation averages more than the breakeven, the linker wins; if less, the conventional bond does.

R10What moves rates: event days

The front end: The 2-year sits close to the average expected over the next two years, so rate decisions, inflation figures, wage growth and jobs data move it first.
Surprises against pricing: Prices already hold what traders expect. A fully priced 25 cut barely moves 2-year yields; the gap between the outcome and what was priced does, along with hints about the next decision.
Hawkish and dovish surprises: A surprise lifts short yields most: a bear . A one lowers them most: a bull . Strong US payrolls usually bring a bear flattener in Treasuries.
The long end: Long yields also carry , so Budgets, borrowing plans and auctions move them. Heavier long borrowing usually brings a bear steepener; in September 2022 the 30-year yield rose 130bp in three days of trading.
Splitting a move: Read the 2-year move as the change in expected policy rates and the rest of the 10-year move as term premium and other factors: 2-year up 3bp and 10-year up 14bp leaves 11bp that isn’t about the central bank.

A rough split of a 10-year move

Δ10y = Δ2y + Δ2s10s

Δ10y: change in the 10-year yield, bp
Δ2y: change in the 2-year yield: the expected path of Bank Rate
Δ2s10s: change in the gap between them: term premium and other factors

Remember: Δ10y ≈ Δ2y (rate expectations) + Δ2s10s (the rest). Central-bank news moves the front end most; borrowing, supply and term premium move the long end.