The yield curve, term-structure theories and forward rates

◈ 8 cards

Read a GoC curve as normal, flat or inverted, explain its slope with expectations, liquidity preference or segmentation, predict what a tightening does to it, and bootstrap the implied one-year forward from two spot rates.

One issuer, many maturities

The yield curve plots the yields of one issuer's bonds — for Canada, Government of Canada bonds, so credit risk is held constant — against their maturities. Each point is a spot rate: today's yield on a zero-coupon claim to that date. Suppose the GoC curve today reads 1-year 3.5 %, 2-year 4.0 %, 10-year 4.4 %. Rising with maturity, it is a normal (upward-sloping) curve. If the long end sat below the short end it would be inverted; if level, flat. An inverted curve has preceded most Canadian and US recessions — but it is a signal with false alarms, not a guarantee.

Three explanations of the slope

Expectations theory. Long rates are the average of the short rates the market expects. The curve slopes up only because short rates are expected to rise; there is no built-in bias toward an upward slope, so a flat curve means rates are expected to stay put.
Liquidity preference. Lenders prefer short claims and demand a premium to lock money away, so the curve has an upward bias even when no rise is expected. An inverted curve therefore signals expected cuts big enough to overwhelm the premium.
Market segmentation. Short and long bonds trade in separate markets — banks at the short end, pension funds and insurers at the long end — and each segment's supply and demand sets its own rate. Preferred habitat is the softened version: investors will leave their segment for a large enough premium.

The CSC register is these three. Bigel lists four and calls the implied future rate a "spot rate"; this course uses spot for today's zero yield and forward for the implied future rate — see the callout.

Worked example — the rate the curve implies

Under expectations theory, two years at the 2-year spot must equal one year at the 1-year spot followed by one year at the implied forward rate $f$ :

$(1 + r_{2})^{2} = (1 + r_{1}) (1 + f) ⟹ f = \frac{( 1 + r _{2} ) ^{2}}{1 + r _{1}} - 1 = \frac{1.0 4 ^{2}}{1.035} - 1 = 4.5024 %$

=(1+0.04)^2/(1+0.035)-1. The market's 2-year rate of 4.0 % is consistent with 3.5 % now and 4.5 % next year — a rise is priced in.

Run it forward: if the forward rate for year 3 is 4.5 %, the 3-year spot the curve should show is the geometric average of all three years,

$r_{3} = [(1 + r_{2})^{2} (1 + f_{3})]^{1/3} - 1 = [1.0 4^{2} \times 1.045]^{1/3} - 1 = 4.1664 %$

=((1+0.04)^2*(1+0.045))^(1/3)-1. Spot to forward and back — the two directions of the same identity.

What a tightening does to the curve

A hike moves the short end up at once — the 1-year is anchored to the overnight rate. The long end moves less, or falls, because a credible hike lowers expected future inflation. The curve flattens; a large enough tightening inverts it. That is the mechanism behind the recession signal: an inverted curve is the market pricing the cuts that follow a slowdown.

Recompute

Spots 1-year 3.0 %, 2-year 3.6 %: =(1.036)^2/1.03-1 = 4.2035 %. A 3.6 % 2-year with a 4.2 % year-3 forward: =((1.036)^2*1.042)^(1/3)-1 = 3.7996 %.

Same maturities, three shapes. Normal (rising) is the usual case; flat says rates are expected to hold; inverted has preceded most recessions without guaranteeing one. Higher on the figure is a higher yield.