Properties of Interest Rates

Q: What distinguishes Treasury rates, SOFR, and Repo rates as benchmark interest rates?

Treasury rates are government yields with minimal default risk; SOFR is an overnight secured rate replacing LIBOR as the primary risk-free benchmark; and Repo rates represent the annualized cost of borrowing secured by underlying collateral.

Q: How is an interest rate converted between an $ m $-times-per-year compounding frequency and continuous compounding?

$ R_c = m \ln\left(1 + \frac{R_m}{m}\right) $ and $ R_m = m \left(e^{R_c / m} - 1\right) $.

Q: What is the fundamental difference between valuing a bond using spot rates versus using Yield to Maturity (YTM)?

Spot rate valuation discounts each individual cash flow using the zero rate corresponding to its exact maturity, whereas YTM uses a single average rate assuming a flat yield curve and constant reinvestment.

Q: What is the bootstrapping method, and how is it applied to determine zero rates?

Bootstrapping is an iterative procedure that calculates zero rates sequentially, starting from short-term zero-coupon rates and using them to discount intermediate coupons of longer bonds to isolate the final zero rate.

Q: What is the payoff formula for a Forward Rate Agreement (FRA) settled at the start of the forward period $ T_1 $?

$ \text{Cash Flow at } T_1 = \frac{L \times (R_M - R_K) \times \tau}{1 + R_M \times \tau} $

Q: How are Macaulay Duration and Modified Duration related under discrete compounding?

$ D_{\text{mod}} = \frac{D_{\text{mac}}}{1 + \frac{y}{m}} $

Q: Why is convexity included alongside duration when estimating bond price changes for large yield shifts?

Duration provides only a linear approximation; convexity accounts for the curvature of the price-yield relationship, yielding the second-order Taylor approximation: $ \frac{\Delta P}{P} \approx -D_{\text{mod}} \Delta y + \frac{1}{2} C (\Delta y)^2 $.

Q: How does Liquidity Preference Theory explain the predominantly upward slope of the yield curve?

Investors prefer short-term liquidity and demand a risk premium for holding longer-term bonds, causing forward rates to exceed expected future spot rates.

FRM Part I · FRM · 7 min read · Updated: 11 Nov 2026

Welcome to the World of Interest Rates!

Interest rates are often called the "heartbeat" of the financial world. Whether you are valuing a bond, pricing a swap, or managing balance sheet risk, understanding how interest rates work is essential. In this chapter, we explore the properties that govern how money grows over time, how curves are constructed, and how bond prices respond to shifts in rates.

1. The Different Types of Interest Rates

Depending on credit quality, collateral, and liquidity, different rates prevail across the financial system:

Treasury Rates

These are the yields paid by sovereign governments on debt obligations such as Treasury bills, notes, and bonds. Because governments can tax and print local currency, Treasury rates carry virtually zero default risk. However, government debt yields are influenced by regulatory demand and tax considerations, which can drive them slightly below other risk-free benchmarks.

SOFR and the Risk-Free Rate Benchmark

Historically, the London Interbank Offered Rate (LIBOR) served as the primary benchmark for floating-rate loans and derivatives. Because LIBOR reflected unsecured interbank borrowing, it embedded bank credit risk and suffered from declining transaction volume. Modern financial markets have transitioned to the Secured Overnight Financing Rate (SOFR) as the leading reference and risk-free benchmark. SOFR is an overnight rate fully collateralized by U.S. Treasury securities, making it essentially free of credit risk.

Repo Rates

In a Repurchase Agreement (Repo), an institution sells high-quality securities (such as Treasuries) to a counterparty and agrees to buy them back at a slightly higher price on a future date. The annualized difference between the sale price and the repurchase price is the Repo Rate. Because the transaction is secured by the underlying collateral, repo rates trade very close to risk-free rates.

2. Compounding Frequencies and Conversions

Compounding frequency determines how often earned interest generates future interest. For an annual nominal rate of interest $ R $ compounded $ m $ times per year, the terminal value of $ \$1 $ after $ n $ years is:

$ \text{Terminal Value} = \left(1 + \frac{R}{m}\right)^{m \times n} $

As $ m \to \infty $, compounding becomes continuous, yielding:

$ \text{Terminal Value} = e^{R_c n} $

Conversion Formulas

To convert an $ m $-times-per-year compounded rate $ R_m $ to an equivalent continuously compounded rate $ R_c $:
$ R_c = m \ln\left(1 + \frac{R_m}{m}\right) $

To convert from continuous compounding $ R_c $ back to $ m $-times compounding $ R_m $:
$ R_m = m \left(e^{R_c / m} - 1\right) $

3. Bond Valuation: Spot Rates vs. Yield to Maturity

A Zero Rate (or $ n $-year spot rate, denoted $ r_n $) is the yield earned on an investment that makes a single cash payment at maturity $ n $, with no intermediate coupons.

Valuing a Bond from Spot Rates

Because each coupon arrives at a different point in time, each cash flow should be discounted using the zero rate matching its specific maturity. For a bond paying cash flows $ C_i $ at times $ t_i $ and face value $ F $ at maturity $ T $ under continuous compounding:

$ P = \sum_{i=1}^N C_i e^{-r_i t_i} + F e^{-r_T T} $

Yield to Maturity (YTM)

The Yield to Maturity ($ y $) is the single discount rate that equates the present value of all future cash flows to the bond\'s observed market price $ P $:

$ P = \sum_{i=1}^N C_i e^{-y t_i} + F e^{-y T} $

While YTM offers a convenient summary metric, it assumes that the yield curve is flat and that all intermediate coupons can be reinvested at $ y $. In contrast, spot rates reflect true market discounting.

4. Bootstrapping Zero Rates

Zero rates for long maturities are rarely directly observable because most long-term bonds pay periodic coupons. The bootstrapping method iteratively determines the zero curve using money market rates and coupon-bearing bonds.

The Bootstrapping Procedure

Step 1: Obtain zero rates directly from short-term instruments (such as Treasury bills) that pay no coupons.
Step 2: Move to the shortest coupon bond (e.g., a 1-year bond paying semi-annual coupons). Discount its earlier coupons using the known shorter zero rates from Step 1.
Step 3: Solve for the unknown zero rate corresponding to the final maturity date.
Step 4: Repeat sequentially for longer-maturity bonds, using all previously solved zero rates to discount earlier cash flows.

5. Forward Rates

Forward rates are the interest rates implied by current spot rates for borrowing or lending during a future period.

Continuous Compounding Formulation

Let $ R_1 $ be the zero rate for maturity $ T_1 $, and $ R_2 $ be the zero rate for maturity $ T_2 $ (where $ T_2 > T_1 $). The implied forward rate $ R_F $ for the period between $ T_1 $ and $ T_2 $ is:

$ R_F = \frac{R_2 T_2 - R_1 T_1}{T_2 - T_1} $

If the zero curve is upward-sloping ($ R_2 > R_1 $), the forward rate sits above the spot rate ($ R_F > R_2 $). If the zero curve is inverted ($ R_2 < R_1 $), the forward rate sits below the spot rate.

6. Forward Rate Agreements (FRAs)

A Forward Rate Agreement (FRA) is an over-the-counter contract designed to fix the interest rate to be paid or received on a principal amount $ L $ over a forward interval from $ T_1 $ to $ T_2 $.

FRA Cash Flow Mechanics

Let $ R_K $ be the agreed fixed forward rate, and $ R_M $ be the reference market rate (such as SOFR compounded over the period) observed at time $ T_1 $. The maturity interval is $ \tau = T_2 - T_1 $.

The cash payoff calculated at maturity $ T_2 $ for the borrower (the party paying $ R_K $ and receiving floating $ R_M $) is:

$ \text{Payoff at } T_2 = L \times (R_M - R_K) \times \tau $

In standard market practice, FRAs are settled at the start of the loan period (time $ T_1 $). To reflect immediate payment, the payoff is discounted to $ T_1 $ at the prevailing floating rate $ R_M $:

$ \text{Cash Flow at } T_1 = \frac{L \times (R_M - R_K) \times \tau}{1 + R_M \times \tau} $

7. Duration and Convexity

When interest rates change, bond prices move in the opposite direction. Risk managers use duration and convexity to quantify this price sensitivity.

Macaulay Duration

Macaulay Duration ($ D_{\text{mac}} $) measures the weighted average time until a bond\'s cash flows are received, weighted by the present value of each cash flow:

$ D_{\text{mac}} = \sum_{i=1}^n t_i \left[\frac{C_i (1 + y/m)^{-m t_i}}{P}\right] $

Modified Duration and Dollar Duration

Modified Duration ($ D_{\text{mod}} $) directly links percentage price change to changes in yield:

$ D_{\text{mod}} = \frac{D_{\text{mac}}}{1 + \frac{y}{m}} $

Note: Under continuous compounding, $ D_{\text{mod}} = D_{\text{mac}} $.

Dollar Duration (also termed DV01 when scaled to a 1 basis point shift) measures the absolute dollar price change for a given change in yield:

$ \text{Dollar Duration} = D_{\text{mod}} \times P $

Convexity and the Combined Price Change Approximation

Duration provides a linear first-order approximation. Because the price-yield relationship of an option-free bond is convex (curved), duration underestimates price increases when yields fall and overestimates price decreases when yields rise.

Convexity ($ C $) captures the second derivative of price with respect to yield:

$ C = \frac{1}{P} \frac{d^2 P}{dy^2} $

Combining duration and convexity yields the second-order Taylor expansion for bond price changes:

$ \frac{\Delta P}{P} \approx -D_{\text{mod}} \Delta y + \frac{1}{2} C (\Delta y)^2 $

Or in total dollar change terms:
$ \Delta P \approx -D_{\text{mod}} P \Delta y + \frac{1}{2} C P (\Delta y)^2 $

8. Theories of the Term Structure

The term structure of interest rates (the yield curve) describes the relationship between zero rates and maturity. Three primary theories explain its shape:

1. Expectations Theory

Asserts that long-term yields equal the geometric average of expected future short-term rates. Under pure expectations theory, an upward-sloping curve implies that the market anticipates rising short-term interest rates in the future.

2. Market Segmentation Theory

Argues that financial institutions and borrowers have distinct maturity preferences based on liabilities (e.g., pension funds need long-term assets; commercial banks prefer short-term liquidity). Rates for each maturity are determined independently by supply and demand in that specific segment, with no arbitrage across maturities.

3. Liquidity Preference Theory

Recognizes that investors prefer short-term liquidity and demand a premium for tying up funds in longer-dated instruments subject to greater price risk. Consequently, forward rates overestimate expected future spot rates by the amount of the liquidity premium, explaining why yield curves are typically upward-sloping even when expectations of future rates are flat.

References

2026 FRM Exam Part I Learning Objectives — Global Association of Risk Professionals (GARP)
2026 FRM Candidate Guide — Global Association of Risk Professionals (GARP)

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What distinguishes Treasury rates, SOFR, and Repo rates as benchmark interest rates?

Treasury rates are government yields with minimal default risk; SOFR is an overnight secured rate replacing LIBOR as the primary risk-free benchmark; and Repo rates represent the annualized cost of borrowing secured by underlying collateral.

How is an interest rate converted between an $ m $-times-per-year compounding frequency and continuous compounding?

$ R_c = m \ln\left(1 + \frac{R_m}{m}\right) $ and $ R_m = m \left(e^{R_c / m} - 1\right) $.

What is the fundamental difference between valuing a bond using spot rates versus using Yield to Maturity (YTM)?

Spot rate valuation discounts each individual cash flow using the zero rate corresponding to its exact maturity, whereas YTM uses a single average rate assuming a flat yield curve and constant reinvestment.

What is the bootstrapping method, and how is it applied to determine zero rates?

Bootstrapping is an iterative procedure that calculates zero rates sequentially, starting from short-term zero-coupon rates and using them to discount intermediate coupons of longer bonds to isolate the final zero rate.

How is the continuously compounded forward rate calculated between two future dates from existing spot rates?

$ R_F = \frac{R_2 T_2 - R_1 T_1}{T_2 - T_1} $

What is the payoff formula for a Forward Rate Agreement (FRA) settled at the start of the forward period $ T_1 $?

$ \text{Cash Flow at } T_1 = \frac{L \times (R_M - R_K) \times \tau}{1 + R_M \times \tau} $

How are Macaulay Duration and Modified Duration related under discrete compounding?

$ D_{\text{mod}} = \frac{D_{\text{mac}}}{1 + \frac{y}{m}} $

Why is convexity included alongside duration when estimating bond price changes for large yield shifts?

Duration provides only a linear approximation; convexity accounts for the curvature of the price-yield relationship, yielding the second-order Taylor approximation: $ \frac{\Delta P}{P} \approx -D_{\text{mod}} \Delta y + \frac{1}{2} C (\Delta y)^2 $.

How does Liquidity Preference Theory explain the predominantly upward slope of the yield curve?

Investors prefer short-term liquidity and demand a risk premium for holding longer-term bonds, causing forward rates to exceed expected future spot rates.

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