Spot Rates, Spot Curve, and Bond Pricing

Spot Rates

Spot rates are the market discount rates for default-risk-free zero-coupon bonds. Unlike typical bonds that offer periodic interest payments, these bonds are sold at a discount and repaid at face value upon maturity. Sometimes referred to as “zero rates,” using a sequence of spot rates ensures a bond price that prevents arbitrage opportunities. In finance, this no-arbitrage condition ensures consistent asset pricing across markets, eliminating the chance for investors to gain risk-free profit from price differentials.

Spot Curve

The spot curve visually charts the yield-to-maturity of default-risk-free zero-coupon bonds against their maturities. Often termed the “zero” or “strip” curve, the “strip” terminology originates from the stripping of periodic coupon payments, converting bonds to zero-coupon status. An example of this is the spot curve of Canadian Government bonds shown below:

Types of Spot Curves

• Upward sloping spot curve: this is observed when longer-term government bonds yield higher than shorter-term bonds. It is a typical pattern under normal market conditions.
• Downward sloping (inverted) yield curve: this rarer configuration, where shorter-term yields are higher than longer-term yields, can signal impending economic downturns. It suggests that investors anticipate lower future rates, often due to expected economic slowdowns, and are thus more inclined to accept lower yields for longer-term bonds.

The spot curve is pivotal for maturity structure analysis, especially with government bonds that standardize elements like currency, credit risk, liquidity, and tax status. Notably, the absence of coupon reinvestment risk in zero-coupon bonds simplifies their evaluation.

Calculating the Price of a Bond Using Spot Rates

To determine bond prices using the spot curve, each cash flow date corresponds to a specific discount rate. The goal is to achieve “no-arbitrage” prices. The bond’s price is determined by discounting its cash flows with the corresponding spot rates. For bonds with periodic payments and a final principal repayment, the price is:

\[PV = \frac{PMT}{\left( 1 + Z_{1} \right)^{1}} + \frac{PMT}{\left( 1 + Z_{2} \right)^{2}} + \ldots + \frac{PMT + FV}{\left( 1 + Z_{N} \right)^{N}}\]

Where:

• \(PV\) is the present value or price of the bond.
• \(PMT\) is the periodic payment or coupon.
• \(FV\) is the bond’s face value.
• \(Z_{1},Z_{2},\ldots Z_{N}\) are the spot rates for periods \(1,2,\ldots N\) respectively.

This approach ensures that the bond price remains consistent, whether discounted using spot rates or yield-to-maturity. 

Example: Calculating the Price of a Bond Using Spot Rates

Given the term structure of government bonds:

$$\begin{array}{c|c} \hline \textbf{Maturity} & \textbf{Yield-to-maturity} \\ \hline 1-Year & 1.5000\% \\ 2-Year & 1.2500\% \\ 3-Year & 1.0000\% \\ 4-Year & 0.7500\% \\ 5-Year & 0.5000\% \\ \hline \end{array}$$

Calculate the price of a 1.00% coupon, four-year government bond.

Formula:

\[PV = \frac{PMT}{\left( 1 + Z_{1} \right)^{1}} + \frac{PMT}{\left( 1 + Z_{2} \right)^{2}} + \ldots + \frac{PMT + FV}{\left( 1 + Z_{N} \right)^{N}}\]

\[PMT\ = \ 1\% \times 100\ = \ 1\]

\[PV = \frac{1}{(1 + 0.015)^{1}} + \frac{1}{(1 + 0.0125)^{2}} + \frac{1}{(1 + 0.01)^{3}} + \frac{1 + 100}{(1 + 0.0075)^{4}} = 100.957\]

Question

Which of the following best describes a spot rate?

1. The yield-to-maturity of a coupon-bearing bond.
2. The market discount rate is applied to default-risk-free zero-coupon bonds.
3. The annual interest rate of a bond with periodic payments.

Solution

The correct answer is B.

Spot rates are market discount rates applied to default-risk-free zero-coupon bonds.

A is incorrect: The yield-to-maturity usually applies to coupon-bearing bonds, not specifically zero-coupon bonds.

C is incorrect: Spot rates are particularly associated with zero-coupon bonds and not bonds with periodic payments.