Fixed income I: bond markets, yields and the term structure

Bond markets are larger than share markets, and their prices, interest rates, affect every other asset. FIN-201 lesson 5 covered the basics from the issuer's point of view: pricing a bond from its yield, finding the yield to maturity, clean and dirty prices, a first look at duration, credit ratings and spot rates. This lesson takes the investor's point of view. It surveys the kinds of fixed-income securities investors hold, explains why the yield to maturity is not the return an investor will actually earn, shows how to extract spot and forward rates from bond prices, and examines what the shape of the yield curve says about expected interest rates, risk premiums and the economy. Lesson 10 then deals with managing interest rate and credit risk.

The fixed-income universe

Fixed-income securities differ by issuer, structure and protection. Government bonds of countries that borrow in their own currency, such as Canada and the United States, carry essentially no default risk in nominal terms and form the benchmark for every other rate. Sub-national bonds, issued by provinces or states and cities, and agency bonds come next. Corporate bonds range from highly rated issues of banks and utilities to high-yield bonds of riskier companies. Securitized products pool mortgages or other loans and pass their cash flows to investors in tranches with different priorities. Structures vary too. Zero-coupon bonds, including government strip bonds created by separating coupons from principal, pay only at maturity. Floating-rate notes reset their coupon to a reference rate plus a spread, so their prices barely move with rates. Inflation-indexed bonds adjust their principal for inflation, so they promise a real rather than a nominal return.

The Government of Canada sells treasury bills and bonds through regular auctions run by the Bank of Canada, to a group of dealers who distribute them. Canadian real return bonds, indexed to the consumer price index, were issued from 1991 until the government stopped new issues in 2022; existing ones still trade. The benchmark reference rate for floating-rate debt and derivatives is now CORRA, the Canadian Overnight Repo Rate Average, which replaced CDOR after June 2024. Most individual Canadians hold bonds through funds, GICs (guaranteed investment certificates issued by banks and protected by deposit insurance within limits) or exchange-traded funds rather than directly.

Yield measures and realized returns

The yield to maturity is a promised yield. It equals the return an investor actually earns only if the bond is held to maturity, every payment is made on time, and every coupon is reinvested at the same yield. If rates fall, coupons are reinvested at lower rates and the realized return falls short; if rates rise, it exceeds the yield. This reinvestment risk is the mirror image of price risk: rising rates hurt a bond's price but help the reinvestment of its coupons. The realized compound yield is the rate that grows the purchase price into the actual ending wealth, including reinvested coupons. Horizon analysis extends the idea to a holding period shorter than maturity: the investor forecasts the bond's price at the horizon, from a forecast of the yield then, plus the value of reinvested coupons.

For callable bonds, investors compute the yield to call as well as the yield to maturity and focus on the lower of the two, the yield to worst. For bonds with a chance of default, the promised yield overstates the expected return, as FIN-201 lesson 5 noted; the gap is the expected loss from default, which lesson 10 quantifies.

Spot rates and bootstrapping

FIN-201 introduced spot rates: the yields on zero-coupon bonds maturing at each date. They are the fundamental building blocks for pricing any bond, because a coupon bond is simply a package of zero-coupon payments. When strip bonds are not available for every maturity, spot rates can be bootstrapped from coupon bonds. Start with the shortest bond, whose yield is the first spot rate. The next bond's first coupon is discounted at that known spot rate, which leaves one unknown, the second spot rate. Repeat for each longer maturity.

# Bootstrapping spot rates from annual-coupon bonds, then implied forward rates
bonds = [(1, 3.0, 99.50), (2, 4.0, 100.20), (3, 5.0, 101.00), (4, 5.0, 100.40), (5, 5.5, 101.30)]
spots = []
for n, cpn, price in bonds:
    pv_known = sum(cpn / (1 + spots[t]) ** (t + 1) for t in range(n - 1))
    s = ((100 + cpn) / (price - pv_known)) ** (1 / n) - 1
    spots.append(s)
for n, s in enumerate(spots, start=1):
    print(f"spot rate, {n} year(s): {s:.3%}")

# One-year forward rate starting in year n-1:  (1+s_n)^n / (1+s_{n-1})^(n-1) - 1
fwd = [spots[0]] + [(1 + spots[n]) ** (n + 1) / (1 + spots[n - 1]) ** n - 1 for n in range(1, len(spots))]
for n, f in enumerate(fwd, start=1):
    print(f"forward rate for year {n}: {f:.3%}")

# Price a new 5-year 4 % bond off the spot curve
price = sum(4 / (1 + spots[t]) ** (t + 1) for t in range(4)) + 104 / (1 + spots[4]) ** 5
print(f"5-year 4 % bond priced off spot curve: {price:.2f}")

spot rate, 1 year(s): 3.518%
spot rate, 2 year(s): 3.902%
spot rate, 3 year(s): 4.680%
spot rate, 4 year(s): 4.941%
spot rate, 5 year(s): 5.282%
forward rate for year 1: 3.518%
forward rate for year 2: 4.287%
forward rate for year 3: 6.253%
forward rate for year 4: 5.728%
forward rate for year 5: 6.657%
5-year 4 % bond priced off spot curve: 94.76

Forward rates

Spot rates imply forward rates: the interest rate for a future period that can be locked in today. If the two-year spot rate is 3.9 % and the one-year spot rate is 3.5 %, an investor can earn 3.9 % a year for two years, or 3.5 % for one year and then reinvest. The second-year rate that makes the two strategies equal, about 4.3 %, is the one-year forward rate starting one year from now. Investors can actually lock it in by buying the two-year zero and selling the one-year zero, so forward rates are real market prices, not just arithmetic. They are used to price forward rate agreements and interest rate swaps, and to compare the market's view of future rates with one's own.

Theories of the term structure

Why does the yield curve usually slope upward, and what makes it change shape? The expectations hypothesis says forward rates equal expected future short-term rates, so an upward-sloping curve means the market expects rates to rise and a downward-sloping one that it expects them to fall. Under pure expectations, every strategy over a given horizon has the same expected return. The liquidity preference theory adds that investors demand a premium for holding longer bonds, whose prices are more volatile; forward rates then exceed expected future rates by that term premium, which explains why the curve slopes up most of the time even when rates are not expected to rise. Market segmentation and preferred habitat theories stress the demand of particular investors, such as pension funds and insurers that need long bonds to match long liabilities, and central banks, whose large bond purchases during quantitative easing pushed long yields down. Each theory contains some truth; estimates suggest the term premium varies over time and was close to zero, or even negative, in the late 2010s before rising again.

Reading the yield curve

The yield curve is watched as an economic signal. Short rates are anchored by the central bank's policy rate; long rates reflect expected future short rates, expected inflation and the term premium. An inverted curve, with short rates above long rates, has preceded most US recessions since the 1960s, because it typically means the central bank has tightened policy and markets expect it to cut later as growth slows. The signal is not perfect: the curve was deeply inverted in 2022–2024 in both countries, yet neither economy fell into an official recession in that period. The spread between nominal and inflation-indexed bond yields of the same maturity, the breakeven inflation rate, measures the inflation investors expect plus an inflation risk premium.

In late October 2026, approximate Government of Canada yields were about 2.9 % for two years and 3.8 to 3.9 % for ten years, with the Bank of Canada's policy rate at 2.25 %, so the curve sloped upward. These figures are rounded and illustrative; current yields are published daily by the Bank of Canada.