Spot rates and pricing a bond with the spot curve

Why do bond yields differ with maturity?

Two bonds can yield different amounts for several reasons. Credit risk, currency, liquidity, tax treatment and the periodicity of the quote all matter. One more is time to maturity: the maturity structure of interest rates , also called the term structure of interest rates , describes how yields change as the time to maturity changes.

Difference in yield-to-maturityseveral causes can be mixed together

• Credit risk the issuer may fail to pay
• Currency different base interest rates
• Liquidity how easily the bond trades
• Tax status for example, discount bonds taxed differently
• Periodicity annual or semiannual quoting
• Time to maturity Memorize the only one the term structure isolates

To isolate maturity, every other cause must be held equal. The ideal data are yields on default-free zero-coupon bonds of every maturity, in one currency, with the same liquidity and tax status. Zero-coupon bonds also remove coupon reinvestment risk, because there are no coupons to reinvest.

What is a spot rate?

Memorize

spot rates are the yields to maturity on default-free zero-coupon bonds, one for each maturity. The set of them, plotted against maturity, is the spot curve . It is also called the zero curve , or the strip curve , because the coupons of a government bond can be “stripped” off and sold separately as zeros.

$$z_N = \left(\frac{FV}{PV}\right)^{1/N} - 1$$

The shape of the spot curve carries information. It usually slopes up, because investors want more yield for tying money up longer. Sometimes it is flat. Sometimes short maturities yield more than long ones, an inverted yield curve .

Spot curvespot rate plotted against maturity

• Upward sloping Memorize longer maturities yield more; the normal case
• Flat about the same yield at every maturity
• Downward sloping Memorize an inverted yield curve: short maturities yield more

How is a real spot curve built?

True zeros exist for few maturities, so analysts estimate the curve from the government bonds that trade. Only the most recently issued, actively traded bonds are used: they have similar liquidity and trade close to par, so tax effects are small. Gaps between maturities are filled by interpolation. Short bills are quoted as discount rates and are converted to bond equivalent yields (add-on rates on a 365-day year), and every point is put on the same periodicity before the curve is compared.

Go deeper Bootstrapping the spot curve from par bonds

Another route is to work forward from the shortest maturity. A one-year bond that pays par gives the 1-year spot rate. A 2-year bond priced at par gives one equation with one unknown, the 2-year spot rate, once the 1-year rate is known. Repeating this builds the whole curve. It is the reverse of the par-rate calculation in the next lesson, and the full method is a Level II topic.

Go deeper Why recently issued bonds are used

An older bond is held by buy-and-hold investors and trades less often, so its price carries a liquidity premium. It also sits at a discount or premium, and some tax systems treat the price gain differently from coupon income. Both effects would bend the curve for reasons unrelated to maturity. Recent issues avoid most of that.

How do you price a bond with spot rates?

Give each cash flow the rate for its own date. A single yield is replaced by a sequence of spot rates.

$$PV = \frac{PMT}{(1 + z_1)^{1}} + \frac{PMT}{(1 + z_2)^{2}} + \cdots + \frac{PMT + FV}{(1 + z_N)^{N}}$$

The result is the no-arbitrage price . If a bond trades above it, an arbitrageur sells the bond and buys its cash flows as separate zeros for less. If it trades below, the trade runs the other way. Transaction costs aside, the gap closes. For a corporate bond, a spread for credit risk is added to each spot rate before discounting.

What is the YTM of that bond, and why is it not the 4-year spot rate?

The yield to maturity is the single rate that gives the same price. Solve the usual bond equation for the price 102.313, entering the 100 repaid at maturity as the future value (FV).

Discounting at 4.357% for every cash flow gives the same total, 102.313, but different individual present values: 4.791 for the first coupon (against 4.845) and 88.532 for the last payment (against 88.387). The pieces differ, the sum does not.

Exam tip On an upward-sloping curve, the YTM of a coupon bond is a little below its maturity’s spot rate. On an inverted curve, a little above.

Go deeper Negative spot rates

Spot rates have been negative for some government bonds. The mathematics does not change: a spot rate of −0.30% means a growth factor of 0.997 per year, and the same price formula applies. Discount factors are then above 1.