The Term Structure of Interest Rates Spot, Par, and Forward Curves (2024 CFA® Level I Exam – FI 9)

Summary

The video script provides an in-depth exploration of fixed income securities, focusing on the complexities of bond pricing in relation to the term structure of interest rates. It begins by revisiting the basic model of bond pricing, which assumes a constant yield to maturity, and then transitions into a more sophisticated model that accounts for fluctuating interest rates. This involves understanding and calculating spot rates, par rates, and forward rates, which are essential for constructing an arbitrage-free bond price. The presenter uses various examples to illustrate how these rates are derived and how they interact with each other. The importance of these concepts is emphasized through their impact on the shape of the yield curve, which can be upward sloping, downward sloping, or flat. The script concludes with practical advice for students preparing for the CFA exam, suggesting that while calculating par rates might not be a common exam question, understanding the relationship between spot rates, par rates, and forward rates is crucial. The presenter encourages students to practice these calculations to ensure they can apply them effectively in different scenarios.

Takeaways

• 📈 **Spot Rates**: The yield on zero-coupon bonds, used to price bonds by discounting each cash flow at the appropriate rate for its term.
• 🔄 **Par Rates**: The yield to maturity that makes a bond sell at its par value; slightly different from spot rates due to mathematical adjustments.
• 🔗 **Forward Rates**: The expected interest rate for borrowing or lending at a future time period, derived from spot rates to prevent arbitrage.
• ↗️ **Upward Sloping Curve**: When short-term rates are lower than long-term rates, spot rates and par rates align closely with forward rates being higher.
• ↘️ **Downward Sloping Curve**: When short-term rates are higher than long-term rates, the par curve is slightly above the spot curve, and forward rates are lower.
• 🔲 **Flat Curve**: When all rates are equal across different maturities, spot, par, and forward rates converge.
• 💡 **Arbitrage-Free Pricing**: Using spot, par, and forward rates ensures that bond pricing is accurate and prevents arbitrage opportunities.
• 📚 **CFA Level Model**: A more complex model than the basic 'kindergarten' model, allowing interest rates to change over the life of a bond.
• 🧮 **Calculation Complexity**: Computing bond prices using spot and forward rates is more complex and requires algebraic manipulation.
• 📊 **Yield Curve Shapes**: Understanding the shape of the yield curve is crucial for interpreting spot, par, and forward rates.
• 📈 **Increasing Short-Term Yields**: An increase in short-term yields with a smaller increase in long-term yields can lead to a flattening of the yield curve.
• ⏱️ **Time Value of Money**: The concept of time value of money is fundamental in calculating present values of future cash flows at different rates.

Q & A

What is the basic assumption made in the kindergarten model of bond pricing?

The basic assumption made in the kindergarten model of bond pricing is that the yield to maturity on the bond is fixed over the life of the bond, meaning the term structure of interest rates is unchanging.

How does the CFA level model differ from the kindergarten model in terms of interest rates?

The CFA level model allows for interest rates to change over the life of the bond, as opposed to the kindergarten model which assumes a fixed yield to maturity.

What is a spot rate?

A spot rate, also known as a zero rate, is the yield on a zero-coupon bond that provides a specific cash flow for a single period. It is used to discount a cash flow back to its present value at a specific point in time.

What is the significance of forming a spot curve, par curve, and forward curve?

Forming these curves allows for a more accurate and comprehensive understanding of bond pricing. The spot curve represents the yields of zero-coupon bonds, the par curve shows the yields of bonds priced at par, and the forward curve indicates the expected interest rates for future periods.

How does arbitrage play a role in the pricing of bonds?

Arbitrage opportunities arise when there are inconsistencies in bond pricing. An arbitrage-free bond price ensures that a bond cannot be bought and sold in a way that generates a risk-free profit, thus maintaining market efficiency.

What is the relationship between spot rates and par rates?

Par rates are the yields to maturity that make a bond sell at its par value. They are derived from spot rates and are generally slightly lower than the corresponding spot rates due to the mathematics involved in their calculation.

How are forward rates determined?

Forward rates are determined by ensuring that there is no arbitrage opportunity between a single longer-term bond and a series of shorter-term bonds that match the time horizon of the longer-term bond. They represent the expected interest rate for a future period.

What is the typical maximum number of years for which the CFA exam might ask to compute bond prices using spot rates?

The typical maximum number of years for which the CFA exam might ask to compute bond prices using spot rates is four years, as computations for longer periods become excessively complex.

What is the implication of an upward sloping yield curve?

An upward sloping yield curve implies that long-term interest rates are higher than short-term rates, which can indicate a healthy economy with expected growth and inflation.

What does a downward sloping yield curve suggest about future interest rates?

A downward sloping yield curve, also known as an inverted yield curve, suggests that investors expect future interest rates to decrease, which can be a signal of a potential economic downturn.

How are forward rates used in calculating the price of a bond?

Forward rates are used to discount the expected cash flows of a bond at each future period. This allows for the calculation of a bond's price based on the expected interest rates for each period, rather than a single yield to maturity.

What is the relationship between the spot curve and the forward curve on a graph?

On a graph, the spot curve represents the yields of zero-coupon bonds, while the forward curve represents the expected interest rates for future periods. The forward rates are typically higher than both the spot and par rates in an upward sloping yield curve scenario, but can be lower in a downward sloping yield curve scenario.