Spot, Par and Forward Curve Relationships

Ask three traders for "the two-year rate" and you can get three different-looking numbers that are all correct. One quotes a zero-coupon rate, one quotes the coupon a new two-year bond would need to price at par, and one quotes the rate for money borrowed a year from now for one year. None of them disagrees about the market. They are just reading the same curve through three different windows.

The one thing underneath: discount factors

A spot rate $z_T$ is the rate that discounts a single cash flow at time $T$ back to today, so a dollar at $T$ is worth $1/(1+z_T)^T$ today. Spot rates are what you get by stripping coupons off — they are literally the zero-coupon curve.

A par yield is the coupon rate $c$ that makes a bond maturing at $T$, paying that coupon annually, price exactly at 100. It blends every spot rate out to $T$ into one number, weighted by the coupon and principal cash flows, so a par yield is always a mixture of shorter spot rates plus the final one.

A forward rate $f_{t_1,t_2}$ is the rate for a loan that starts at $t_1$ and ends at $t_2$, both in the future, implied by today's spot curve. It answers "what one-year rate is the market pricing in for next year," using no information beyond today's prices.

Worked example: building all three from two spot rates

Suppose the one-year spot rate is 4.00% and the two-year spot rate is 4.50%.

Discount factors.

$$d_1 = \frac{1}{1.04} = 0.9615, \qquad d_2 = \frac{1}{1.045^2} = 0.9158$$

Par yield for a two-year bond. A par bond with annual coupon $c$ must satisfy $c \cdot d_1 + (100+c)\cdot d_2 = 100$. Solving:

$$c = \frac{100 - 100\, d_2}{d_1 + d_2} = \frac{100 - 91.58}{0.9615+0.9158} = 4.49$$

So a new two-year note would need to carry roughly a 4.49% coupon to trade at par — a hair below the 4.50% spot rate, because the coupon paid at year one is discounted less harshly than the principal at year two, pulling the blended yield down slightly.

Forward rate, one year forward for one year. The no-arbitrage condition is that rolling one-year money twice must earn the same as locking in two years:

$$(1+z_1)(1+f_{1,2}) = (1+z_2)^2 \;\Rightarrow\; f_{1,2} = \frac{1.045^2}{1.04} - 1 = 5.00\%$$

The curve is telling you that, absent any change in expectations, one-year money one year from now should cost about 5.00% — steeper than either spot rate, because it has to make up for the lower rate locked in during year one.

forward par spot

Why the ordering always holds

On an upward-sloping curve the ranking is always forward $>$ spot $>$ par at a given maturity. A forward rate has to compensate for locking in later than a spot rate, so it sits above it. A par yield is a weighted average of spot rates along the way to maturity, including earlier, lower ones, so it sits below the matching spot rate. Flip the curve upside down — inverted, short rates above long — and every one of these inequalities flips too.

Where this matters in practice

Curve traders quote and risk-manage in par-yield space because that is how bonds are issued and displayed on a screen, but they price everything — including that very bond — off the stripped spot curve, because only spot rates are additive across cash flows. Forward rates are the input to relative-value trades like riding the curve, and to testing the expectations hypothesis: if realized future spot rates consistently fall short of today's forwards, that gap is the term premium, not a forecasting error.

Key terms

• Spot rate — the discount rate for a single cash flow at one future date; the zero-coupon curve.
• Par yield — the coupon rate that prices a bond at exactly 100; a blend of spot rates.
• Forward rate — the rate for a future period implied by today's spot curve, with no forecasting assumption attached.
• Discount factor — $1/(1+z_T)^T$; the common currency all three curve types are built from.