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6.2 Relationships among Spot Rates, Forward Rates, Yield to Maturity, Expected and Realized Returns on Bonds, and the Shape of the Yield Curve.

At any point in time, the price of a risk-free single-unit payment (e.g., $1) at time T is called the "discount factor"r with maturity T, denoted by P(T).

Spot rate: The yield to maturity of the payment, denoted by r(T)

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  • The spot curve represents the term structure of interest rates at any point in time.
  • The spot curve shows, for various maturities, the annualized return on an option-free and default-risk-free zero-coupon bond with a single payment of principal at maturity.
  • The discount factor, P(T), and the spot rate, r(T), for a range of maturities in years T> 0 are called the "discount function" and the "spot yield curve" (or, more simply, spot curve), respectively.
  • Note that the discount function completely identifies the spot curve, and vice versa. The discount function and the spot curve contain the same set of information about the time value of money.

Forward Rate: It is an interest rate that is determined today for a loan that will be initiated in a future time period.The term structure of forward rates for a loan made on a specific initiation date is called the "forward curve." The Relationship Forward rates and forward curves can be mathematically derived from the current spot curve.

  • When the spot curve is upward sloping, the forward curve will lie above the spot curve.
  • Conversely, when the spot curve is downward sloping, the forward curve will be below the spot curve. The relationship between spot and forward curves is a reflection of the basic mathematical truth that when the average is rising (falling), the marginal data point must be above (below) the average. In this case, the spot curve represents an average over a whole-time period and the forward rates represent the marginal changes between future time periods.

Lesson Wrap-Up

This lesson should leave you able to explain the relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve. in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve. in your own words.
  2. State one exam-style risk, valuation, or market implication of the relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve..