Chapter 4 - Deriving the Zero Coupon Bond Yield

What is the general bond pricing formula?

P₀ = Σ Fₜ × B(0,t), where each cash flow is discounted using its maturity-specific discount factor.

What does B(0,t) represent?

The discount factor; the present value of $1 received at time t.

What does R(0,t) represent?

The spot rate; the annualized interest rate for maturity t on a zero-coupon bond.

What is the relationship between discount factors and spot rates?

B(0,t) = 1 / (1 + R(0,t))^t.

Why are discount factors often preferred over spot rates?

They do not require assumptions about compounding frequency.

What is the term structure of interest rates?

The collection of spot rates for all maturities.

Do different maturities usually have the same interest rate?

No, interest rates generally differ by maturity.

What is a zero-coupon (pure discount) bond?

A bond that pays no coupons and pays a single amount at maturity.

How do we obtain spot rates in practice?

By inferring them from coupon bond prices using bootstrapping.

Why is there a “chicken-and-egg” problem in bond pricing?

Bond prices depend on discount rates, but discount rates are implied from bond prices.

What is relative pricing?

Using market prices of basic securities to infer discount rates and price other securities.

Why does arbitrage matter in bond pricing?

Any deviation from pricing rules creates arbitrage opportunities, which should not persist.

How do you calculate a spot rate from a zero-coupon bond price?

R(0,t) = (FV / Price)^(1/t) − 1.

What is bootstrapping?

A method of deriving spot rates sequentially from coupon bond prices.

What is the first step in bootstrapping?

Use short-term zero-coupon bonds to obtain initial spot rates.

How are longer-term spot rates obtained in bootstrapping?

By subtracting the present value of earlier coupons using known spot rates.

What is the direct method for recovering the term structure?

Solving discount factors directly from bond prices with matching coupon dates.

How can a zero-coupon bond be replicated?

By combining long and short positions in coupon bonds to eliminate intermediate cash flows.

Why does replication help determine spot rates?

It enforces no-arbitrage pricing.

What does B(0,1) represent numerically?

The present value today of $1 received in one year.

What happens if two bonds with same cash flows have different prices?

An arbitrage opportunity exists.

Why do we need interpolation in yield curves?

Market rates are only available for certain maturities.

What is linear interpolation?

Estimating a rate between two known maturities by assuming a straight-line relationship.

Write the linear interpolation formula for rates.

R(t) = R(t₁) + [(t − t₁)/(t₂ − t₁)] × [R(t₂) − R(t₁)].

What is the goal of deriving the zero-coupon yield curve?

To obtain discount rates for all maturities for accurate pricing.

Why is the zero-coupon yield curve important?

It allows correct valuation of bonds and other fixed-income securities.