FMP 16: Properties of Interest Rates
An interest rate is what a lender earns for parting with money for a while, and two things move it more than anything else. Credit risk comes first: the shakier the borrower looks, the more the lender demands. Liquidity comes second: how easily the instrument passes to another investor at a fair price. Gaps between rates are quoted in basis points, one basis point being 0.01%, so 2% amounts to 200 basis points.
Government borrowing rates
Each currency has an anchor rate, what its own government pays to borrow in it, and in the United States that anchor is the Treasury rate. Default by a developed country on debt in its own currency is thought very unlikely, since the government controls the supply of that currency and can create more to settle what it owes. Such borrowing counts as risk-free, and these rates sit under those paid by anyone else raising the same currency. Developing country governments have defaulted on domestic currency debt, though, and euro area members do not control the euro money supply.
Overnight interbank borrowing
Banks in most countries must hold cash, called reserves, at the central bank, in an amount that depends on their outstanding liabilities. By the close some banks are long and others short, and a very liquid overnight lending market grows out of that mismatch. The American rate for such lending is the federal funds rate, and the weighted average across the day’s trades is the effective federal funds rate, watched by the Federal Reserve and periodically pushed up or down by its own trading. Other currencies run the same arrangement. The United Kingdom has the sterling overnight index average, SONIA. The euro has ESTER, which took over from the older benchmark EONIA. Japan has the Tokyo overnight average, TONAR.
Repo rates
A repurchase agreement, always shortened to repo, dresses a secured loan as a pair of trades. Party A sells securities to Party B at a price of X today and undertakes to buy them back later at X plus e, so in substance Party B has lent X and collected e in interest. If Party A never repurchases, Party B keeps the securities outright, which is why the lender bears so little risk so long as they are worth roughly X and hold that value. Overnight repos dominate, though longer maturities trade. Two indices are built from these overnight secured transactions: SOFR, the Secured Overnight Financing Rate, in the American market, and SARON in the Swiss market.
Some contracts fix their rate in advance, as when a company borrows at 4% for five years. Many do not, and a floating contract must name a reference rate plus a rule turning it into the rate that applies. Libor filled that role for decades. A borrowing cost written as the three-month Libor fixing with 80 basis points added means that each future three-month period takes whatever three-month Libor is observed on its first day, plus 0.8%. A fixing of 3% on that day would set 3.8% for the quarter that follows.
How Libor was produced
LIBOR stands for London Interbank Offered Rate. From 2014 to 2021 the Intercontinental Exchange, ICE, put the numbers together daily by polling 16 global banks for estimates of what unsecured borrowing from another bank would cost them, with submissions due just before 11 a.m. United Kingdom time. Five currencies were covered, USD, GBP, EUR, CHF and JPY, across seven borrowing periods: overnight, one week, one month, two months, three months, six months and 12 months. Discarding the four highest and four lowest quotes and averaging the rest produced the day’s fixing. Submitting banks typically carried an AA credit rating, so the number read as the unsecured funding cost of an AA-rated bank over one horizon, and contracts worth hundreds of trillions of dollars rested on those fixings.
The failure and the replacements
Trouble surfaced in 2011. Genuine interbank borrowings had become too scarce to point at, so submissions rested on judgement about what borrowing would have cost rather than on trades that had happened. An international investigation in 2012 then found that several large banks had manipulated their submissions and colluded with one another. Two motives were at work. A bank due to pay or receive Libor plus a spread over the coming six months gains by nudging the relevant fixing, and a bank wanting to look creditworthy gains by quoting low. Trimming the extreme quotes limited the damage, but collusion got round that safeguard. Fines ran to billions of dollars, lawsuits followed, traders lost their jobs and some went to prison. Regulators then retired Libor in favour of rates built from actual transactions.
| Currency | Secured or unsecured | Replacement rate | Built from |
|---|---|---|---|
| US dollar (USD) | Secured | SOFR | Overnight repo transactions |
| Swiss franc (CHF) | Secured | SARON | Overnight repo transactions |
| British pound (GBP) | Unsecured | SONIA | Overnight borrowing between banks |
| Euro (EUR) | Unsecured | ESTER | Overnight borrowing between banks |
| Japanese yen (JPY) | Unsecured | TONAR | Overnight borrowing between banks |
Source: the replacement rates named in the chapter, grouped here by secured against unsecured rather than in the chapter’s order.
All five carry virtually no risk, and they are known collectively as risk-free rates, or RFRs.
Replacing Libor with an overnight rate shifts the timing of information, which matters to anyone drafting a contract. A three-month Libor fixing was published on the first day of the period it covered, so both sides knew their rate before any interest accrued. A rate assembled from overnight observations cannot be complete until the last business day of the period has passed. Libor was a forward-looking rate. Rates derived from overnight benchmarks are backward-looking rates.
Compounding the daily rates
Turning a run of overnight observations into a rate for a three-month period means compounding them. Suppose the period holds n business days, that the overnight rate on the ith of them is , and that this rate covers calendar days.
Usually equals 1. Weekends and public holidays break that pattern, because the rate seen on the last business day before a break has to cover the whole break, and a Friday observation normally carries a of 3.
Credit sensitive rates
Libor moved with bank credit quality and the new benchmarks do not. In calm conditions three-month USD Libor stood about ten basis points above a three-month rate derived from overnight rates, but stress widened the gap sharply, and it peaked at 364 basis points (3.64%) in October 2008. Losing that credit signal left a hole, since the RFRs are neither credit sensitive nor forward looking, and the market has been hunting for a forward-looking credit sensitive rate to sit beside SOFR. The Bloomberg Short-Term Bank Yield Index, BSBY, is one proposal: a proprietary index published each day at 8:00 a.m. (ET), which watches real transactions to gauge what large global banks pay for unsecured wholesale funding, then fits a curve to give maturities from overnight out to 12 months, with one-month, three-month and six-month points between. AMERIBOR, from the American Financial Exchange, is another, reflecting the funding costs of thousands of community and regional banks across the United States.
Derivatives on the new rates
An active derivatives market referencing the overnight rates matters more than it might seem, because it is what allows a full term structure based on RFRs to be pinned down, and forward-looking term rates follow from that structure. On exchange, the most successful contract used to be Eurodollar futures, whose payoff came from fixings of three-month USD Libor. A comparable SOFR contract is now listed by the CME group and has proved popular.
A quoted rate says nothing on its own. It has to be read with a compounding frequency, which fixes how often interest begins earning interest of its own. Put USD 100 at 5% for five years with annual compounding: the balance is 105 after a year, since 100 x 1.05 = 105, then 110.25 after two years, and roughly 127.63 by the fifth. Read the same 5% semi-annually and half of it, 2.5%, is credited every six months, so 100 becomes 102.5, then 105.12, then 107.75, and about 128.01 after five years. Quarterly compounding credits 1.25%, which is 0.05 divided by 4, and the same 100 reaches roughly 128.20.
An investor prefers 5% semi-annually to 5% annually, since 128.01 beats 127.63, and a borrower prefers the opposite. Grams and pounds both measure weight; annual and semi-annual compounding both measure interest, and one outcome can be quoted either way.
Converting from one frequency to another
Let carry compounding times a year and be the equivalent rate carrying compounding times a year. Equivalence means an amount A reaches the same future value under either quotation, so the annual growth factors must match, and rearranging that equality delivers the conversion.
Usual conventions
Convention ties compounding frequency to payment frequency. Instruments maturing within a year of issue are money market instruments and settle principal and interest together at maturity, so a three-month one is normally quoted quarterly and a one-year one annually. That can mislead. Take a one-year money market rate of 10% alongside a one-month rate of 9.8%. The annual figure looks larger, but monthly compounding sits behind the 9.8% and annual compounding behind the 10%, and restating the shorter rate annually turns it into 10.25%, the higher of the two. Bonds issued for more than a year normally pay every six months, so their yields carry semi-annual compounding. The link is not universal. Canadian mortgage rates are quoted semi-annually although payments fall monthly or fortnightly, while the same mortgage would be quoted monthly in the United States and annually in the United Kingdom. These are regulatory choices, meant to let borrowers within one country compare offers with as little confusion as possible.
Nothing stops the frequency from climbing further. Compound hourly, then by the minute, then by the second, and the growth factor keeps inching up while each increment shrinks. The limit is continuous compounding, where Equation (16.1) collapses into an exponential.
Invest USD 100 for five years at 5% compounded continuously and the balance reaches 128.40. Daily compounding lands on the identical figure once you round to two decimal places. This convention is standard when options and other complex derivatives are valued, and it suits work on yields and futures prices, largely because exponential formulas are tidier than their periodic counterparts.
Moving between continuous and periodic rates
Let be quoted with compounding m times a year and be the same rate quoted continuously. Equating growth over one year gives , and logarithms turn that into a matched pair of conversions.
Three conversions show both directions at work. To restate 6% with semi-annual compounding as a continuously compounded rate, Equation (16.4) with m = 2 asks for twice the natural logarithm of 1.03, which comes to 0.0591, or 5.91%. That sits below 6% because continuous compounding works harder, so a smaller quoted rate produces the same growth. Going the other way, 9% continuously compounded becomes four times the quantity e raised to 0.0225, less 1, which is 0.0910, or 9.10% quarterly compounded, and the rate rises because coarser compounding needs more of it. Where neither quotation is continuous the work stays inside Equation (16.2): 10% compounded monthly, restated semi-annually, means raising 1 + 0.10/12 to the power 6, subtracting 1 and doubling, giving 0.1021, or 10.21%.
Compounding runs forward, from money now to money later. Discounting runs backward, asking what a future sum is worth today, and mechanically it is the same growth factor inverted.
Discount USD 500 arriving in three years at 4% compounded semi-annually and it is worth 443.99 today. The multiplier doing the work is the discount factor, and here it equals 0.8880. Every line of a bond valuation table is one cash flow multiplied by the discount factor for its own maturity.
Zero rates
Zero-coupon interest rate, zero rate and spot rate are three names for one idea: the rate applying when an investor collects everything, interest and principal alike, at the end of T years and nothing before. No interim payment means no reinvestment at an unknown future rate, which is what makes the number unambiguous.
Instruments maturing inside a year settle in that single lump, so they hand over their zero rates directly. Anything issued for longer usually pays coupons along the way, and a coupon bond will not reveal a zero rate on inspection. STRIPS are the exception, having had their coupons removed. For the rest, zero rates must be extracted from market prices by the bootstrap procedure covered later.
| Maturity (years) | Discount factor | Zero-coupon rate |
|---|---|---|
| 0.5 | 0.9852 | 3.0% |
| 1.0 | 0.9631 | 3.8% |
| 1.5 | 0.9368 | 4.4% |
| 2.0 | 0.9095 | 4.8% |
| 2.5 | 0.8817 | 5.1% |
| 3.0 | 0.8548 | 5.3% |
Source: the chapter’s zero-coupon rate table. The discount factor column is derived here using Equation (16.6).
Reading down the discount factors shows the term structure at work: each falls below the last because maturity and rate are both rising. These six rates carry the valuation, yield, duration and forward rate work that follows.
Valuing a bond takes two steps. List the cash flows with their dates, then discount each at the zero rate belonging to its own date and add up.
A bond maturing in three years carries a principal of USD 100, also called the par value or face value, which is the amount repaid at maturity. Semi-annual coupons run at 6% a year, so USD 3 arrives every six months, and the closing payment of 103 combines the last coupon with the principal. Discount at the six semi-annually compounded zero rates of 3.0%, 3.8%, 4.4%, 4.8%, 5.1% and 5.3%.
Yield duration gauges how sensitive a bond price is to a move in its own yield. Write B for the price, D for duration, for the yield change and for the resulting price change.
Which D belongs there depends on how the yield has been quoted. Macaulay duration is correct for a continuously compounded yield, modified duration for any other frequency. Taking the continuous case first, Macaulay duration is the average time a bondholder waits for the present value of the money, each date weighted by its share of the total, with the yield as the discount rate.
A miniature case makes the weighting concrete. A bond worth USD 106 delivers a present value of USD 6 in one year and USD 100 in two years, so its Macaulay duration is 1.9434, being 1 year weighted by 6/106 plus 2 years weighted by 100/106. A 5-basis point rise in the continuously compounded yield, or 0.0005, then shifts the price by -0.103, from multiplying 0.0005 by 1.9434 and by 106, so the price slips to USD 105.897.
Duration of the three-year bond
The bond valued in the previous section costs USD 102.0695 and yields 5.2455% semi-annually compounded. Restating that yield continuously through Equation (16.4) means taking the natural logarithm of 1 + 0.052455/2 and doubling it, which gives 0.051779, or 5.1779%. Discounting the six cash flows at that rate gives the present values below. The earliest is 2.9233, and dividing it by the price gives 0.02864, so 2.864% is the weight on T = 0.5. The next is 2.8486, weighted 2.791%. Summing time multiplied by weight produces a Macaulay duration of 2.79238.
| Time (years) | Present value | Time x weight | Square of time x weight | Weight |
|---|---|---|---|---|
| 3.0 | 88.1811 | 2.59180 | 7.77539 | 0.86393 |
| 2.5 | 2.6357 | 0.06456 | 0.16139 | 0.02582 |
| 2.0 | 2.7049 | 0.05300 | 0.10600 | 0.02650 |
| 1.5 | 2.7758 | 0.04079 | 0.06119 | 0.02720 |
| 1.0 | 2.8486 | 0.02791 | 0.02791 | 0.02791 |
| 0.5 | 2.9233 | 0.01432 | 0.00716 | 0.02864 |
| Total | 102.0695 | 2.79238 | 8.13904 | 1.00000 |
Source: the chapter’s duration and convexity tables, merged, ordered from the longest maturity down and with the weight column moved to the end.
The fourth column belongs to the next section but costs nothing to build here. The weights are strikingly lopsided: principal repayment at three years carries 86.393% of the present value by itself, which is why a duration of 2.79238 sits so near the three-year maturity. The compounding convention leaves these present values alone, provided each is computed consistently with the yield, but it does alter the sensitivity of price to yield.
Once a yield carries compounding m times a year instead of continuous compounding, Equation (16.8) needs one correction: divide the duration computed as a weighted average of times by 1 + y/m. The result is modified duration, and it is what belongs in Equation (16.8) whenever the yield is periodically compounded.
For the three-year bond the semi-annually compounded yield is 5.2455% and the tabulated duration is 2.79238, so dividing by 1 + 0.052455/2 leaves a modified duration of 2.7210. Feed the unmodified figure into a semi-annually compounded yield change instead and the predicted price move is overstated every time.
Dollar duration
Modified duration answers in percentages. Multiply it by the price and you get dollar duration, which answers in currency.
So D gives the sensitivity of proportional price changes to a change in yield, while dollar duration gives the sensitivity of the actual change. A manager adding exposures across bonds at different prices can sum the currency version, and cannot sum the percentage one.
The three-year bond is priced at USD 102.0695, yielding 5.1779% continuously compounded and 5.2455% semi-annually compounded, with a Macaulay duration of 2.79238 and a modified duration of 2.7210. Two tests follow, one in each convention, both for a rise of 10 basis points.
Duration performs well inside a narrow set of conditions and degrades outside them. Two assumptions carry the weight. One is that the shift in rates is parallel, every point on the term structure moving by the same amount, so any given bond’s yield moves by very nearly that amount too: a uniform 10 basis point rise would shift the yield by close to 10 basis points and Equation (16.8) would forecast the price fall well. The other is that the change is small. Where the shift is uneven across maturities, or simply large, the equation cannot be trusted.
The reason a big move defeats it is geometric: Equation (16.8) is the tangent to the price and yield curve at the current yield, and that curve bends away from its tangent in both directions.
Adding the convexity term
Convexity measures that curvature, and a second term recovers most of what duration misses. Writing C for convexity refines the relationship into something that survives reasonably large parallel shifts.
Computing convexity repeats the duration calculation with one change to the final column: each weight is multiplied by the square of the time. That column already appeared above, totalling 8.13904. Pair modified convexity with modified duration, and the unadjusted figures with a continuously compounded yield.
The bond costs 102.0695 and yields 5.2455% semi-annually, with a modified duration of 2.7210 and a convexity of 8.13904. Rates now shift by a full 2%, twenty times the earlier move, and the exact revalued price is 96.6951.
Forward rates are the future interest rates that today’s zero-coupon rates already imply. Given a half-year rate of 3.0% and a one-year rate of 3.8%, both semi-annually compounded, the rate covering the six months beginning six months from now is not open to opinion: it is whichever six-month rate, stacked on top of today’s 3.0%, reproduces the one-year figure of 3.8%. Anything else would let an investor lock in a profit by preferring one path to the other.
The general procedure
Three steps handle the interval between and , where and , what one dollar becomes by each date. Divide the second by the first, giving what a dollar held at is worth at . Then find the rate over the interval that matches the ratio.
Apply that to the half-year to one-year interval using the zero rates tabulated earlier. A dollar reaches 1.015 by six months and 1.038361 by one year, and dividing gives 1.023016. Quoting F semi-annually means 1 + F/2 matches that ratio, so F = 0.04603, a forward rate of 4.603%. From one year to 1.5 years the growth factors are 1.038361 and 1.067463, their ratio is 1.028027, and F = 0.05605, or 5.605%.
| Period | Growth ratio V2 divided by V1 | Forward rate |
|---|---|---|
| 0.5 year to 1.0 year | 1.023016 | 4.603% |
| 1.0 year to 1.5 years | 1.028027 | 5.605% |
| 1.5 years to 2.0 years | 1.030024 | 6.005% |
| 2.0 years to 2.5 years | 1.031522 | 6.304% |
| 2.5 years to 3.0 years | 1.031515 | 6.303% |
Source: forward rates from the chapter’s zero-coupon rate table. The growth ratio column is derived here to expose the intermediate step.
The continuous compounding version
Continuous compounding replaces the ratio with a subtraction. With and , the ratio becomes , and equating that to gives the forward rate in one line.
Suppose the three-year zero rate is 5% and the four-year zero rate is 6%, both continuously compounded. Subtracting 3 x 0.05 from 4 x 0.06 and dividing by the one-year gap gives 0.09, a forward rate of 9% for the fourth year. Requote both annually and ratios return: one dollar grows to 1.05 cubed and to 1.06 to the fourth power, the ratio is 1.0906, and the forward rate becomes 9.06%.
A forward rate agreement locks an interest rate today for a period beginning later. Formally it swaps a pre-specified fixed rate against a floating rate, both applied to an agreed principal over an agreed future period. The floating leg once referenced Libor and now uses the replacement benchmarks. An interest rate swap is simply a portfolio of these agreements.
Settlement
Interest usually falls due at the close of the period it covers, whereas an FRA settles at the start. The convention is to pay the present value of the interest difference, discounted back over the period at the realised floating rate.
An FRA pays a fixed rate of 4% and receives the three-month floating rate, both applied to a principal of USD 1 million across the three-month period starting one year from now. Rates carry quarterly compounding and day count issues are set aside. That floating rate turns out to be 4.8%.
The discount rate there is normally drawn from overnight index rates and the swaps written on them, the market’s working definition of a risk-free rate for valuing derivatives.
Everything above assumes a set of zero-coupon rates, and markets rarely hand them over. Money market instruments under a year supply theirs directly, returning everything at maturity. Longer instruments pay coupons along the way, so their prices blend many maturities and the zero rates inside have to be dug out. The standard method works outward from the short end, solving for one new rate at each step while holding the rates already determined fixed. It is called bootstrapping, since each answer feeds the next.
Semi-annually compounded zero rates are already fixed at 2.0% for half a year, 2.3% for one year and 2.5% for eighteen months. A two-year bond of par value USD 100 trades at USD 102.7 and pays semi-annual coupons at 4% a year, so USD 2 every six months and USD 102 at maturity.
The shape of the resulting curve
A zero curve relates zero-coupon rates to their maturities, and two conventions fill the ends: rates are held constant out to the first maturity with data, and constant again past the longest. Between observed points the usual choice is linear interpolation, giving a piecewise linear curve.
When the maturities do not line up
That arithmetic was unusually convenient, because the bond matured on a date the curve already covered. Real data is rarely so tidy. Suppose the next available price belonged to a bond running 2.4 years. Setting its zero rate to R does not close the problem, since the bond pays a coupon at 1.9 years and no rate for that date exists. Interpolating between the known 1.5-year figure of 0.025 and the unknown R gives a 1.9-year rate of 0.4R plus half of 0.025, all divided by 0.9, which still depends on R. Coupon dates at 1.4, 0.9 and 0.4 years interpolate from rates already settled, and an iterative search finds the R that reproduces the price.
The term structure describes how rates vary with maturity. Upward sloping means long-term rates sit above short-term rates, downward sloping the reverse. Three theories compete to explain the shape.
Market segmentation theory
The first says short, medium and long maturities draw entirely separate clienteles, so each stretch of the curve is priced in isolation by its own traders. Short rates would answer only to participants who care about short investments. This is now generally seen as unrealistic, because investors do not confine themselves to one segment. Pension plans are natural buyers of long-maturity bonds, yet they shift into medium or short maturities when the yields there look better.
Expectations theory
The second says the curve is a picture of where the market thinks rates are heading. Expected increases give an upward-sloping structure, expected declines a downward-sloping one. Put precisely, expectations theory holds that forward rates equal expected future spot rates.
Observation is awkward for it. Curves slope upward far more often than downward, yet the market presumably expects declines about as often as increases, all the more so because short-term rates are mean reverting, being tugged back toward a long-run average. Across a long history they spend roughly half the time below that average and expected to rise, and half above it and expected to fall. If expectations were the whole story, downward-sloping curves would be as common as upward-sloping ones.
Liquidity preference theory
The third theory accounts for the imbalance. Imagine rates did nothing but reflect expectations. Most investors would then pick the short-term investment, because a short maturity brings the money back sooner and keeps it available for whatever need appears. Borrowers lean the opposite way, since short-term borrowing has to be rolled over, and should the market’s view of the borrower’s financial health sour, rightly or wrongly, that rollover may not be available at a competitive rate.
So lenders want short maturities, borrowers want long ones, and the two do not meet. Financial intermediaries such as banks close the gap by lifting long-term rates above the level expectations alone would justify. Long-term borrowing then loses appeal to borrowers, and short-term lending loses appeal to lenders. The conclusion is that forward rates sit persistently above the spot rates the market genuinely expects.
A stripped-down case shows the adjustment happening. Say only two maturities are quoted, three months and five years, both at 2.5% a year, and that this genuinely reflects expectations. With the curve flat at 2.5%, liquidity considerations send lenders to the three-month maturity and borrowers to the five-year maturity, so the two sides fail to match. Market forces then push the five-year rate above 2.5%. It might emerge that a five-year rate of 3.5%, with the three-month rate held at 2.5%, tempts enough borrowers into three-month borrowing and enough lenders into five-year lending to balance both points.