The yield curve: spot and forward rates · RiskAtoms Academy

Know these first

L1.3.03 Present value and discounting
L1.3.07 Real vs nominal rates

Math Atoms

∑ Exponents

By the end you will be able to

1Read a yield curve and derive forward rates from spot rates.
2Describe normal, flat, inverted and humped curves, and what each may signal.
3Turn spot rates into discount factors and value cash flows off a curve.
4Explain a forward rate as the break-even rate that rules out arbitrage.

Why it matters

In July 2022 the two-year US Treasury yield rose above the ten-year yield, and stayed there for about two years. Markets call that an inverted yield curve, and in the US it has come before most recessions of the past half-century. One chart, read by every central bank and trading desk in the world.

Core explanation

One rate for every date

So far we have used a single rate to discount every cash flow. In reality, the rate depends on how long you lend for. Lending to the US government for one year might pay 3%, for five years 4%. Plot those rates against maturity and you have a yield curve.

The cleanest version uses spot rates: the rate for money lent today and paid back in one lump at a single date, with nothing in between. A spot rate for each maturity gives a discount factor for each date, so every cash flow can be discounted at the rate for its own date. That is what L1.4.03 meant by “each cash flow could use its own discount rate”.

Upward sloping (normal)

Long rates above short rates. The usual shape: lenders want more for tying money up longer.

Flat

Similar rates at every maturity. Often seen when the market expects rates to stay put.

Inverted

Short rates above long rates. Often when the central bank has raised rates and markets expect cuts later.

Humped

Rates rise, then fall. Less common; often seen as a curve moves between shapes.

Forward rates. The spot curve also tells you something about the future. Suppose the one-year rate is 3.0% and the two-year rate is 3.5%. You could lend for two years straight, or for one year and then roll over for a second. If you could fix today the rate for that second year, what would it have to be for both routes to pay the same? That rate is the forward rate. Here it is 4.00%.

It must be that rate, because otherwise there would be a free lunch: lend on the better route, borrow on the worse one, and pocket the difference with no risk. That is an arbitrage, and markets close it quickly. The forward is not a promise of where rates will be. It is the rate that makes today's spot rates consistent with each other.

Read the curve

Decide, then see why

The spot curve rises from 3% at one year to 4% at five years. Are the one-year forward rates above or below the spot rates?
The spot curve is flat at 4% for every maturity. What are the forward rates?

The formal version

New notation: $s_t$ is the spot rate for $t$ years, annually compounded, and $f_{1,2}$ is the forward rate from year 1 to year 2. Everything else is from L1.3.03.

Any forward period. The same argument works between any two dates $t_1 < t_2$ . Growing to $t_2$ directly must equal growing to $t_1$ and then at the forward rate:

(1 + s_{t_2})^{t_2} = (1 + s_{t_1})^{t_1}\,(1 + f_{t_1,t_2})^{t_2 - t_1}

f_{t_1,t_2} = \left(\frac{(1 + s_{t_2})^{t_2}}{(1 + s_{t_1})^{t_1}}\right)^{\frac{1}{t_2 - t_1}} - 1

The formulas, for reference

\mathrm{DF}_t = (1 + s_t)^{-t} \qquad P = \sum_{t} \mathrm{CF}_t\,\mathrm{DF}_t \qquad f_{t-1,t} = \frac{\mathrm{DF}_{t-1}}{\mathrm{DF}_t} - 1

Compounding all the one-year forwards gives back the spot rate: $(1+s_T)^T = (1+f_{0,1})(1+f_{1,2})\cdots(1+f_{T-1,T})$ , with $f_{0,1} = s_1$ .

Try it

Drag the spot rates and watch the forwards (the steps) move. Make the curve inverted and the forwards drop below the spots. Then lift only the five-year dot: just the last forward jumps, because it has to carry the whole change on its own.

Try itDrag the yield curve

Open the zero curve builder →

Start from:

This curve is upward sloping (normal) spot rate (drag the dots) one-year forward rate

Drag a dot, or focus it and use the up and down arrow keys.

t	Spot	DF	Forward
1	3.00%	0.9709	3.00%
2	3.50%	0.9335	4.00%
3	3.80%	0.8941	4.40%
4	4.00%	0.8548	4.60%
5	4.10%	0.8180	4.50%

Where the spot curve rises, forwards sit above it; where it falls, below.

Worked example

From spot rates to forwards and a bond price

A treasury analyst at a bank in Hong Kong has these annual spot rates for US Treasuries: 1-year 3.0%, 2-year 3.5%, 3-year 3.8%. Annual compounding throughout. Try each step before revealing it.

Forwards, discount factors and a price

Step 1 of 7

1
Growth factors
$1.035^2 = 1.071225$ and $1.038^3 = 1.118387$ .

Regulatory lens

Banks must measure how their value and earnings change when the yield curve moves. The Basel Committee's standards on interest rate risk in the banking book (IRRBB, 2016) prescribe six shock scenarios, including parallel moves, a steepening and a flattening of the curve. European insurers discount their liabilities with risk-free curves that EIOPA publishes every month under Solvency II, extended beyond the market's longest maturities towards an “ultimate forward rate”.

Looking ahead: both return in Level 4 (IRRBB, and Solvency II in L4.6).

Sparks

SparkA PhD that became a recession signal

Campbell Harvey's 1986 doctoral thesis at the University of Chicago showed that an inverted yield curve had come before US recessions. The idea has been watched ever since, though an inversion is a signal, not a timetable: the gap between inversion and recession has varied a lot, and there have been false alarms.

SparkForward rates are not forecasts

It is tempting to read the forward rate as the market's forecast of future rates. Partly it is, but lenders may also want extra pay for tying money up longer, a term premium. That premium pushes forwards above expected future rates, so a gently rising curve can be consistent with rates that are expected to stay flat.

Key takeaways

A yield curve plots rates against maturity; spot rates s_t are for single payments at date t.
Each cash flow is discounted at the spot rate for its own date: DF_t = (1 + s_t)^−t.
Forward rates come from no-arbitrage: (1 + s₂)² = (1 + s₁)(1 + f₁,₂).
A rising spot curve has forwards above it; an inverted curve has forwards below it.
A bond's single yield is a blend of the spot rates for its cash flow dates.

Reaction

Check what stuck

Question 1 · objective 1

The 1-year spot rate is 2% and the 2-year spot rate is 3%. What is the one-year forward rate starting in one year, in %, to two decimals?

Question 2 · objective 3

The 2-year spot rate is 3%. What is the 2-year discount factor, to four decimals?

Question 3 · objective 2

What does an inverted yield curve show?

ALong-term rates are above short-term ratesBShort-term rates are above long-term ratesCAll rates are negativeDRates are the same at every maturity

Question 4 · objective 3

The 2-year spot rate is 4%. What is a payment of 100 in two years worth today?

Question 5 · objective 4

Why must (1 + s₂)² equal (1 + s₁)(1 + f₁,₂)?

ABecause regulators set forward rates that wayBOtherwise investors could lock in a riskless profit by lending one way and borrowing the otherCBecause the forward rate is the market's exact forecast of next year's rateDBecause spot rates never change

Glossary

Yield curve: A chart of interest rates against maturity, for borrowers of the same credit quality, usually a government.
Spot (zero) rate: The annual rate for money lent today and repaid in one payment at a given date. Also called a zero rate. Written s_t.
Forward rate: A rate agreed today for lending over a future period, implied by the spot curve. Written f with the start and end dates.
Inverted curve: A yield curve on which short-term rates are above long-term rates.
Arbitrage: A riskless profit from buying and selling the same thing at two prices. Prices adjust until it disappears.
Term premium: The extra return investors may demand for lending long rather than rolling short loans.

Deep Dives & Origins

Deep Dives

Tuckman and Serrat, Fixed Income Securities: Tools for Today's Markets, 3rd edition (Wiley, 2011), chapters 1 and 2, on discount factors, spot and forward rates. Intermediate.
Hull, Options, Futures, and Other Derivatives (Pearson), chapter 4, on zero rates and forward rates. Intermediate.

Origins

Tuckman and Serrat, Fixed Income Securities, 3rd edition (Wiley, 2011).
Basel Committee on Banking Supervision, Interest rate risk in the banking book (April 2016).
EIOPA, risk-free interest rate term structures (monthly publication and technical documentation).
Campbell R. Harvey, PhD dissertation (University of Chicago, 1986); and “The real term structure and consumption growth”, Journal of Financial Economics 22 (1988).
US Department of the Treasury, daily Treasury par yield curve rates (2022 to 2024).

Module complete

That is the last lesson of Module 1.3. Put it all together on applied problems: Try the Lab Test →

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Real vs nominal rates

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What is a bond?