PastPaperHero | Valuation and term structure - Spot rates forward rates and no arbitrage

Learning Outcomes

This article explains how to use spot and forward interest rates within an arbitrage‑free term structure framework for fixed‑income valuation. It clarifies the economic meaning of spot rates and forward rates, contrasts their roles in discounting and forecasting, and links both to the shape and interpretation of the yield curve. It demonstrates how to compute present values of single cash flows and coupon bonds by applying the correct spot rate to each maturity, and how to extract implied forward rates from a given set of spot rates or market bond prices using the appropriate compounding relationships. The article also explains how the no‑arbitrage principle constrains the term structure, ensuring that bonds with identical cash flows must trade at the same price. Worked examples illustrate how to test whether a quoted bond price is consistent with spot rates, construct an arbitrage‑free price, and identify mispricing. Finally, the article highlights typical CFA Level 1 exam pitfalls, including averaging yields instead of compounding and misinterpreting forward rates as guaranteed future spot rates.

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are expected to understand the core principles of term structure valuation, with a focus on the following syllabus points:

Defining and comparing spot rates and forward rates
Calculating the present value of cash flows using spot rates
Deriving forward rates implied by spot rates and market securities
Applying the no-arbitrage principle in pricing fixed-income instruments
Identifying arbitrage opportunities when the term structure is inconsistent

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

What is a spot rate, and how is it used in pricing a fixed-income security?
How can you derive a 2-year forward rate one year from now using spot rates?
Describe the no-arbitrage principle in the context of term structure valuation.
If a 2-year bond is priced above its no-arbitrage value (given spot rates), what might an investor do to exploit this?

Introduction

The valuation of fixed-income securities relies on understanding the time value of money across different maturities. Spot rates and forward rates are central to this process. Term structure valuation ensures that securities are consistently priced without arbitrage. For CFA Level 1, knowing how to use spot and forward rates for discounting and identifying pricing inconsistencies is a required skill.

Key Term: spot rate
The yield on a zero-coupon bond for a particular maturity; used to discount a single cash flow occurring at that time.

Key Term: forward rate
The implied interest rate for a period in the future, agreed today, derived from the current term structure of spot rates.

Key Term: no-arbitrage principle
The concept that available assets or portfolios with identical future cash flows must have the same present value and price; otherwise, riskless profit opportunities would exist.

THE TERM STRUCTURE OF INTEREST RATES

The term structure of interest rates, also called the yield curve, shows how spot rates vary with different maturities for otherwise similar default-free securities.

Each spot rate corresponds to the annualized yield of a zero-coupon bond maturing at a given time. Discounting future cash flows using spot rates gives the true present value of those payments.

Valuing Cash Flows Using Spot Rates

To value a fixed-income security, discount each future cash flow by the corresponding spot rate:

\text{PV} = \sum_{t=1}^{N} \frac{\text{CF}_t}{(1 + s_t)^t}

where $s_t$ is the spot rate for maturity $t$ , and $\text{CF}\_t$ is the cash flow at time $t$ .

Worked Example 1.1

A 3-year bond pays $50 coupons annually and $1,000 principal at maturity. Spot rates for 1, 2, and 3 years are 2%, 2.5%, and 3%.

Question: What is the present value of this bond? Answer:
Compute each discounted cash flow:

Year 1: $50 / (1.02) = $49.02

Year 2: $50 / (1.025)^2 = $47.62

Year 3: $1,050 / (1.03)^3 = $961.13 Total present value: $49.02 + $47.62 + $961.13 = $1,057.77

Spot Rate Curve and Par Yield Curve

The spot rate curve is essential for accurate bond valuation and comparison across securities. Par yields represent coupon rates at which a bond’s price equals its par value, and a flat yield curve indicates identical spot and par yields across maturities.

FORWARD RATES AND THEIR RELATIONSHIP TO SPOT RATES

Forward rates represent expectations (under no-arbitrage) for future interest rates derived from current spot rates.

Forward Rate Formula

For a one-period simple forward rate, starting at time $t$ for investment between years $t$ and $t+n$ :

(1 + s_{t+n})^{t+n} = (1 + s_t)^t \cdot (1 + f_{t, n})^n

Solve for the forward rate $f\_{t, n}$ :

1 + f_{t, n} = \frac{(1 + s_{t+n})^{t+n}}{(1 + s_t)^t} ^ {1/n}

Worked Example 1.2

Suppose 1-year and 2-year spot rates are 3% and 4%, respectively. What is the 1-year forward rate one year from now?

Answer:
$(1 + 0.04)^2 = (1 + 0.03)^1 \times (1 + f_{1,1})$ Solving for $f_{1,1}$ : $(1 + f_{1,1}) = (1.04^2) / 1.03 = 1.0816 / 1.03 = 1.05$ So, $f_{1,1} = 5\%$ .

Exam Warning

The most common CFA exam error in term structure valuation is forgetting that spot and forward rates must compound over their respective periods; do not simply average spot rates to find a forward rate.

NO-ARBITRAGE VALUATION

The no-arbitrage principle ensures that assets with the same cash flows have the same price. In fixed-income valuation, this means we must use the spot rate for each cash flow. If market prices differ, arbitrage strategies exist.

Worked Example 1.3

A 2-year bond with annual $100 coupon and $1,000 face value is priced at $1,050. Spot rates are 3% (1 year) and 5% (2 years).

Question: Does an arbitrage opportunity exist? Answer:
Compute the no-arbitrage value:

Year 1: $100/(1.03) = $97.09

Year 2: $1,100/(1.05)^2 = $997.73 Total: $97.09 + $997.73 = $1,094.82 The bond trades at $1,050, below its no-arbitrage value. An investor could buy the bond, receive its cash flows, and lock in a profit against securities priced using the spot curve.

Arbitrage Strategies in the Term Structure

If a bond trades above its valuation based on spot rates, an investor can sell/short the bond, buy corresponding zero-coupon bonds for each cash flow, and lock in riskless profit.

Key Term: arbitrage portfolio
A portfolio constructed to exploit mispricing by taking offsetting positions in overvalued and undervalued assets to guarantee a risk-free profit.

Summary

The term structure of interest rates allows for precise valuation of future cash flows. Spot rates discount single cash flows; forward rates are derived from spot rates to reflect no-arbitrage pricing. Securities must be priced using spot rates for each cash flow; otherwise, arbitrage exists. Command of spot and forward rates and the no-arbitrage principle is essential for CFA Level 1 success.

Key Point Checklist

This article has covered the following key knowledge points:

Distinguish spot rates from forward rates, and understand their use in time value valuation
Calculate bond prices by discounting each cash flow using its corresponding spot rate
Calculate forward rates implied by spot rates using the correct compounding formula
Apply the no-arbitrage principle to ensure bonds are priced consistently across the term structure
Identify arbitrage opportunities arising from mispriced bonds relative to spot rates

Key Terms and Concepts

spot rate
forward rate
no-arbitrage principle
arbitrage portfolio

Valuation and term structure - Spot rates forward rates and ...