VRM 10: Interest Rates
Discount factors alone are enough to value any Treasury instrument, and observed Treasury prices can be used to imply the discount factors behind them. Markets nevertheless quote the time value of money as interest rates, and a rate is half a number until a compounding frequency is attached. Frequency sets the unit of measurement, and changing it changes what the same quoted number is worth.
Take 8% per annum with annual compounding. USD 100 becomes 108 after a year, and in the second year the whole 108 earns 8%, so the balance reaches 116.64, which is 100 multiplied by 1.08 squared, and after n years 100 multiplied by 1.08 to the power n. Measure the same 8% with semi-annual compounding and 4% arrives every six months: a year gives 108.16, two years 116.99, and n years 100 multiplied by 1.04 to the power 2n. Quarterly compounding credits 2% a quarter, giving 100 multiplied by 1.02 to the power 4n.
A lender wants the rate compounded as often as possible and a borrower wants the opposite. The gap is real without being dramatic: 8.16% with annual compounding buys exactly the growth that 8% with semi-annual compounding buys. Present values move on the same lever.
| Compounding frequency | Times compounded per year | Value of USD 100 after one year | Present value of USD 100 due in five years | Extra growth over annual compounding |
|---|---|---|---|---|
| Annual | 1 | 108.00 | 68.06 | 0.00 |
| Semi-annual | 2 | 108.16 | 67.56 | 0.16 |
| Quarterly | 4 | 108.24 | 67.30 | 0.24 |
| Monthly | 12 | 108.30 | 67.12 | 0.30 |
| Weekly | 52 | 108.32 | 67.05 | 0.32 |
| Daily | 365 | 108.33 | 67.03 | 0.33 |
| Continuous | no limit | 108.33 | 67.03 | 0.33 |
Source: GARP, Valuation and Risk Models, Chapter 10. The final column is derived.
The quarterly present value follows straight from the mechanics: discounting at 2% per three-month period over the 20 periods in five years, 100 divided by 1.02 to the power 20 is 67.30.
Compounding frequency usually matches payment frequency but does not have to, generally because a government has legislated a quoting convention. A Canadian fixed-interest mortgage rate is expressed with semi-annual compounding even where payments fall monthly or every two weeks, while the same rate would use monthly compounding in the United States and annual compounding in the United Kingdom.
Once frequency is understood as a unit, moving between units is mechanical. Let R1 be a rate compounded m1 times a year and R2 the equivalent rate compounded m2 times a year. Equivalent means an amount A reaches the same balance after a year either way.
Continuous compounding
Daily compounding is not the end of the road. Nothing stops the frequency rising to every hour, then every minute, then every second. The limit is continuous compounding: at a continuously compounded rate R an amount A grows to A multiplied by e to the power RT by time T, where e is approximately 2.71828.
Extending the earlier table confirms where the sequence lands. At 8% continuously compounded, USD 100 becomes 108.33 after a year and USD 100 received in five years is worth 67.03 today, both matching the daily figures to two decimal places. Derivatives markets normally quote rates this way, partly because the formulas come out simpler.
One rate is quoted at 5% with semi-annual compounding. A second is quoted at 6% with continuous compounding.
Cash that comes back at a single future date, with nothing paid along the way, earns what the market calls a spot rate. The same number has two other names, the zero-coupon interest rate and, casually, the zero.
Suppose USD 100 goes out today and USD 120 comes back in three years with nothing in between. The three-year spot rate is whatever reconciles those two amounts. With annual compounding it solves 100 multiplied by the quantity one plus R, cubed, equal to 120, giving 6.27%. Build the frequency in from the start instead, using six half-year periods, and the answer is 6.17%.
Spot rates and discount factors carry identical information in different clothing. Write d(t) for the discount factor at t years and r(t) for the t-year spot rate quoted with semi-annual compounding, the usual Treasury convention. An investment of 100 grows by time t to 100 multiplied by the quantity one plus r(t) over two, raised to the power 2t, and the discount factor brings that back to exactly 100.
Treasury bonds pay coupons twice a year at a stated annual rate, so a coupon rate of 7% delivers 3.5% of face value every six months. Hold the maturity T fixed and turn the coupon up from zero. At zero the instrument is the equivalent of a stripped bond and, as long as rates are not negative, is worth less than face value. Raising the coupon raises the price, and at one coupon rate the price lands exactly on face value: the par rate.
Payments fall at 0.5, 1, 1.5 and so on up to T years. Writing p for the par rate, a bond paying p divided by two on each date and 100 at maturity is worth exactly 100.
It tidies up once the coupon strip is treated as one instrument. Let A(T) stand for an annuity that pays USD 1 on each payment date out to T, so A(T) is just the sum of the discount factors. The condition then collapses to something you can apply directly.
The annuity factor earns its keep again when the coupon is not the par rate. A bond of maturity T with coupon c and face value 100 is worth c over two multiplied by A(T), plus 100 multiplied by d(T). Substituting the par condition turns that into a statement about the coupon gap.
Four discount factors are known: 0.98038 at half a year, then 0.95181, then 0.92184 at 1.5 years, and 0.88849 at two years. Read as semi-annually compounded spot rates they run 4%, then 5%, then 5.5%, then 6%.
A forward rate is the future spot rate that today’s spot rates already imply. Say a bank offers 3% for one year and 4% for two, with annual compounding on both. Rough averaging puts the second year at 5%, since 3% and 5% average to 4%.
Doing it properly shifts the answer. The forward rate F must make one year at 3% followed by one year at F match two years at 4%, so 100 multiplied by 1.03 multiplied by the quantity one plus F equals 100 multiplied by 1.04 squared. F comes out at 5.01%, near the approximation but not equal to it, because annual compounding carries non-linearities.
Fixed-income markets normally work in semi-annual compounding, and extending the argument covers a six-month window opening at time T. Write R1 for the spot rate at maturity T and R2 for the spot rate at T plus 0.5. The annualized forward rate is twice the expression below.
Continuous compounding is where the arithmetic turns clean, because the non-linearities vanish and the averaging argument becomes exact. Change the example so both rates are continuously compounded, 3% for one year and 4% for two. The requirement is that e to the power 0.03, multiplied by e to the power F, equals e to the power 0.08, and F is exactly 5%. In general the forward rate between T1 and T2 is a difference of rate-weighted times divided by the length of the period.
Running the process in reverse rebuilds the spot rates, since compounding successive forward rates across n six-month periods reproduces the n-period spot rate.
Forward rates are not only arithmetic. A large financial institution able to borrow and lend at spot rates can transact today at the forward rate, which is why it is the benchmark any rate view has to beat.
Two continuously compounded spot rates are quoted, 5% out to 5 years and 5.2% out to 5.5 years. A financial institution wants to fix the return on funds invested between year five and year 5.5.
An instrument guaranteeing a particular rate over a future period is a forward rate agreement. Struck at the prevailing forward rate it is worth zero at inception, for the reason the trade above demonstrates. Struck instead at some rate R while the forward rate is F, it is worth the present value of R less F applied to the principal, which is positive when R sits above F.
Trading a view against the forward rate
Suppose an investor expects the rate that actually turns up between T1 and T2 to come in below the forward rate, and assume the investor borrows and lends at the same rate, roughly the position of a large financial institution. The trade is to borrow for the shorter period T1 and invest for the longer period T2. If the rate paid to roll the borrowing lands below the forward rate, financing costs less than the return earned.
Put numbers on it. Continuously compounded rates of 4% out to two years and 5% out to three years place the third-year forward rate at 7%. An investor confident that the third-year rate will print below 7% borrows over two years at 4% and invests over three years at 5%, and if the view proves right the overall borrowing cost lands below 5%. The opposite view calls for the opposite trade, borrowing over three years and investing over two.
A bond has T years to run and pays a coupon of c. Six months pass and the term structure has not moved. Has the price changed?
It has, and the reason shows up by comparing the cash flows the bond offers today with those it will offer six months from now, measured from that later starting point. They agree everywhere except at the end.
| Time (years) | Cash flows starting today | Cash flows starting in six months | Difference |
|---|---|---|---|
| 0.5 | c/2 | c/2 | 0 |
| 1.0 | c/2 | c/2 | 0 |
| 1.5 | c/2 | c/2 | 0 |
| T-1.0 | c/2 | c/2 | 0 |
| T-0.5 | c/2 | 100 + c/2 | -100 |
| T | 100 + c/2 | none | 100 + c/2 |
Source: GARP, Valuation and Risk Models, Chapter 10, transposed so that time runs down the rows.
Read the difference column on its own and a familiar instrument appears. Paying out 100 at time T minus 0.5 and receiving 100 plus c over two at time T is lending 100 for the final six months at an annual rate of c: a forward rate agreement struck at the coupon rate.
Its value follows the rule from the previous section. When the coupon exceeds the forward rate for that final period the agreement is worth something positive, so the later stream is worth less than today’s and the bond price falls. When the coupon falls short, the agreement is worth less than nothing and the price rises.
An upward-sloping term structure tends to put the forward rate for the last period above the coupon, so bond prices there drift upward as time to maturity shortens.
Three rate curves have now been defined off one set of discount factors, and their relative positions are fixed by the slope of the term structure and nothing else. A flat term structure collapses the distinction, since every par rate and every forward rate then equals the common spot rate.
Once the curve tilts, the ordering separates. With an upward-sloping term structure the par rate for a maturity sits below the spot rate for that maturity, and the forward rate for a period starting at T sits above the spot rate for maturity T. A downward-sloping structure reverses both. The forward curve is the outermost in the direction the curve leans, the par curve the innermost.
The table sets an upward-sloping structure beside a downward-sloping one, all rates semi-annually compounded. Each forward column is the rate for a six-month period beginning on the maturity shown, which is why it stops one row early.
| Maturity (yrs) | Spot, upward | Spot, downward | Par, upward | Par, downward | 6-mnth fwd, upward | 6-mnth fwd, downward |
|---|---|---|---|---|---|---|
| 0.5 | 2.01 | 5.06 | 2.01 | 5.06 | 4.04 | 4.24 |
| 1 | 3.02 | 4.65 | 3.01 | 4.66 | 4.86 | 3.73 |
| 1.5 | 3.63 | 4.35 | 3.62 | 4.36 | 5.27 | 3.73 |
| 2 | 4.04 | 4.19 | 4.01 | 4.21 | 5.83 | 3.17 |
| 2.5 | 4.40 | 3.99 | 4.36 | 4.01 | 5.94 | 3.07 |
| 3 | 4.65 | 3.84 | 4.60 | 3.86 | 6.24 | 3.12 |
| 3.5 | 4.88 | 3.73 | 4.82 | 3.76 | 6.36 | 3.08 |
| 4 | 5.06 | 3.65 | 4.99 | 3.68 | 6.54 | 3.10 |
| 4.5 | 5.23 | 3.59 | 5.14 | 3.62 | 6.67 | 3.08 |
| 5 | 5.37 | 3.54 | 5.28 | 3.57 | none | none |
Source: GARP, Valuation and Risk Models, Chapter 10, the two tables placed side by side.
The columns behave as the rules predict at every maturity. At five years the upward-sloping par rate of 5.28 sits below its spot rate of 5.37, while the downward-sloping par rate of 3.57 sits above its spot rate of 3.54. The forward column also swings much wider than the spot column, because spot rates are averages of forward rates.
Two labels describe how a term structure moves, and both are defined on the gap between long-maturity and short-maturity rates rather than on either rate alone.
A flattening term structure arises in two ways. Long-maturity and short-maturity rates can both move down with the long end falling further, which the market calls a bull flattener, or both can move up with the short end rising further, a bear flattener. A steepening is the mirror of each: rates move down together with the short end falling further, or up together with the long end rising further.
The vocabulary is about relative movement and makes no claim about the shape the curve ends up in. A curve that already slopes upward becomes more upward-sloping on a steepening and less so on a flattening. A downward-sloping curve does the reverse, so a term structure described as flattening does not always finish flatter than it started.
A view on that gap becomes a position through two bonds of different maturities. Take a trader who believes an upward-sloping term structure will steepen, with 20-year rates increasing faster than ten-year rates. The position is a short in the 20-year bonds against a long in the ten-year bonds, and it pays off because the 20-year bonds lose value against the shorter ones. Expecting a flattening instead, the trader reverses both legs.
Everything so far has been built on Treasury rates. Libor was for many years the reference rate behind trillions of dollars of transactions, and the scandals surrounding it ended that role. Replacing it are rates calculated by compounding overnight reference rates.
The United States settled on the Secured Overnight Financing Rate, or SOFR, computed as an average across overnight repo transactions. Elsewhere the equivalent comes from unsecured interbank borrowing at each day’s close, where a bank with spare funds lends to one that is short and a broker usually matches them. Britain’s version is the sterling overnight index average, or SONIA, and the euro area runs the euro overnight index average, or EONIA.
What a swap transaction looks like
A swap exchanges one interest stream for another on an agreed principal. Party A might undertake to pay Party B 4% per annum, fixed, on USD 100 million of notional principal, while Party B pays across three-month SOFR, quarter by quarter, for five years.
Take the swap above, with three-month SOFR setting at 3% for the first three months and 3.2% for the second.
| Payment date (years) | Party A payment, fixed | SOFR rate for previous 3 months | Party B payment, floating | Fixed rate less SOFR | Net payment Party B to Party A |
|---|---|---|---|---|---|
| 5.00 | 1.0 | 5.6% | 1.40 | -1.6% | 0.40 |
| 4.75 | 1.0 | 5.4% | 1.35 | -1.4% | 0.35 |
| 0.75 | 1.0 | 3.4% | 0.85 | 0.6% | -0.15 |
| 0.50 | 1.0 | 3.2% | 0.80 | 0.8% | -0.20 |
| 0.25 | 1.0 | 3% | 0.75 | 1.0% | -0.25 |
Source: GARP, Valuation and Risk Models, Chapter 10. Payments in USD million; the rate gap column is derived and the latest dates are listed first.
Why the swap market defines par rates
The USD 100 million carries the label notional principal because it never changes hands. Exchanging it after five years would change nothing, since each party would hand the other the same amount. Adding that neutral exchange turns the swap into a fixed-rate bond traded against a floating-rate bond.
A floating-rate bond is worth par. Since the swap has no value when it is initiated, the fixed-rate bond on the other side must be worth par too. So the swap rate is by construction the coupon rate that prices a bond at par, which is the definition of a par rate. Swap rates across maturities are a set of par bonds, and those convert back into discount factors and spot rates.