Interest Rate Curve Construction: Building Interest Rate Curves Using the Bootstrapping Method

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Introduction

The interest rate curve is a core tool in fixed-income markets, used for pricing interest rate derivatives, risk management, and asset valuation. Bootstrapping is the standard method for constructing interest rate curves. It extracts discount factors (DF) or zero-coupon rates (ZCR) step-by-step to generate a complete interest rate curve. In addition to using deposit rates (Depo), interest rate futures (Futures), interest rate swaps (Swaps), and bonds, Forward Rate Agreements (FRA) are also important tools, especially for constructing short- and medium-term curves.

This article details how to construct an interest rate curve using deposit rates, interest rate futures, FRAs, interest rate swaps, and bonds, and how to iteratively generate discount factors and zero-coupon rates through the Bootstrapping method.

1. Overview of the Bootstrapping Method

1.1 What is the Bootstrapping Method?

Bootstrapping is an iterative calculation method used to extract discount factors or zero-coupon rates from market instrument quotes. Each calculation is based on previously computed discount factors, ensuring the interest rate curve aligns with market prices.

1.2 Market Instruments Used

Different maturity market instruments provide key points for the curve:

• Deposit Rates (Depo): Provide short-term rate information (e.g., 1 day to 1 year).
• Forward Rate Agreements (FRA): Provide short- to medium-term forward rate information (e.g., 3M-6M).
• Interest Rate Futures (Futures): Provide implied forward rates for the short to medium term (e.g., 3M-2Y).
• Interest Rate Swaps (Swaps): Provide fixed rate points for the medium to long term (e.g., 1Y-30Y).
• Bonds: Provide long-term rate information (e.g., 10Y-30Y).

1.3 Construction Goals

• Discount Factors (DF): Used for cash flow discounting.
• Zero-Coupon Rates (ZCR): Used to estimate yields for zero-coupon payments.
• Implied Forward Rates: Used for pricing future interest rate derivatives.

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2. Detailed Steps of Bootstrapping

2.1 Extracting Short-Term Discount Factors Using Deposit Rates

Deposit rates provide short-term discount factors, typically for maturities from 1 day to 1 year.

Assumptions and Formula

• Given: Deposit maturity $T$ and deposit rate $r_{Depo}$.
• Discount Factor Formula:
  $DF(T) = \frac{1}{1 + r_{Depo} \cdot T}$
  where $T$ is the deposit maturity in years.

Example

• Deposit maturity: 3M (0.25 years), deposit rate $r_{Depo} = 2\%$.
• Discount factor:
  $DF(0.25) = \frac{1}{1 + 0.02 \cdot 0.25} = 0.995$

Deposit rates are used to construct the foundation of the short-term curve.

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2.2 Extracting Implied Forward Rates Using FRAs

FRAs are common interest rate derivatives that provide forward rate information for the short to medium term. FRAs can be used to derive implied forward rates for future time periods.

Assumptions and Formula

• Given: FRA end time $T_2$, start time $T_1$, and FRA rate $r_{FRA}$.
• Discount Factor Formula:
  $DF(T_2) = DF(T_1) \cdot \frac{1}{1 + r_{FRA} \cdot (T_2 - T_1)}$
  where:
  • $T_1$: FRA start time.
  • $T_2$: FRA end time.
  • $DF(T_1)$: Known discount factor.

Example

• FRA: 3M-6M (from 3 months to 6 months), FRA rate $r_{FRA} = 2.5\%$.
• Known discount factor $DF(0.25) = 0.995$.
• New discount factor:
  $DF(0.5) = DF(0.25) \cdot \frac{1}{1 + 0.025 \cdot (0.5 - 0.25)} = 0.995 \cdot 0.9938 = 0.989$

FRAs typically cover maturities from 3 months to 1 year and are essential for constructing short- and medium-term curves.

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2.3 Deriving Medium-Term Implied Forward Rates Using Interest Rate Futures

Interest rate futures provide 3-month forward rate information, typically covering the medium-term range from 3 months to 2 years.

Assumptions and Formula

• Given: Interest rate futures price $F$.
• Implied Forward Rate Formula:
  $r_{Futures} = 100 - F$
• Discount Factor Formula:
  $DF(T) = DF(T_1) \cdot \frac{1}{1 + r_{Futures} \cdot \Delta T}$
  where $T_1$ is the futures start time, and $\Delta T$ is the futures coverage period (typically 3 months).

Example

• Interest rate futures price: $F = 97.5$, corresponding rate $r_{Futures} = 100 - 97.5 = 2.5\%$.
• Using discount factor $DF(0.5) = 0.989$.
• New discount factor:
  $DF(0.75) = DF(0.5) \cdot \frac{1}{1 + 0.025 \cdot 0.25} = 0.989 \cdot 0.9938 = 0.983$

Interest rate futures typically cover the medium-term range from 3 months to 2 years.

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2.4 Constructing the Medium- to Long-Term Curve Using Interest Rate Swaps

Interest rate swaps provide fixed rate points for the medium to long term, typically from 1 year to 30 years.

Assumptions and Formula

• Given: Swap maturity $T$, swap fixed rate $r_{Swap}$, and floating rate discount factors $DF(T_i)$.
• Swap Pricing Formula:
  $\sum_{i=1}^{n} r_{Swap} \cdot \Delta T_i \cdot DF(T_i) + DF(T_n) = 1$
  where:
  • $\Delta T_i$: Cash flow interval (e.g., 6 months).
  • $DF(T_n)$: Unknown discount factor.

Calculation Steps

1. Use known discount factors to calculate partial cash flow present values.
2. Solve the equation to compute the unknown discount factor.

Example

• 2-year swap, fixed rate $r_{Swap} = 3\%$, cash flow interval of 6 months.
• Known discount factors: $DF(0.5) = 0.989, DF(1) = 0.980, DF(1.5) = 0.970$.
• Equation:
  $0.03 \cdot (0.5 \cdot 0.989 + 0.5 \cdot 0.980 + 0.5 \cdot 0.970 + 0.5 \cdot DF(2)) + DF(2) = 1$
• Solution: $DF(2) = 0.960$.

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2.5 Supplementing Long-Term Discount Factors Using Bonds

Bonds provide additional long-term rate information.

Assumptions and Formula

• Given: Bond price $P$, coupon $C$, and maturity $T$.
• Discount Factor Formula:
  $P = \sum_{i=1}^{n} \frac{C}{(1 + r_{n})^{T_i}} + \frac{F}{(1 + r_{n})^{T_n}}$
  where:
  • $C$: Periodic coupon.
  • $F$: Bond face value.
  • $r_n$: Zero-coupon yield.

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3. Complete Process for Interest Rate Curve Construction

1. Short-Term Segment (Deposit Rates): Use deposit rates to calculate discount factors.
2. Short- to Medium-Term Segment (FRA + Futures): Use FRAs and interest rate futures to extract implied forward rates.
3. Medium- to Long-Term Segment (Swaps): Use interest rate swaps to construct medium- to long-term discount factors.
4. Long-Term Segment (Bonds): Use bond yields to supplement the long-term curve.

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4. Applications and Significance

• Derivative Pricing: Fundamental tool for pricing interest rate options, swaps, etc.
• Risk Management: Measures interest rate risk (e.g., duration and convexity).
• Asset-Liability Management: Used by banks and insurance companies to discount cash flows.

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5. Example of Construction Process

Below is a complete example demonstrating how to use market data (deposit rates, FRAs, interest rate futures, and swaps) to construct an interest rate curve using the Bootstrapping method.

5.1. Market Data

Assume the following market interest rate data:

5.1.1 Deposit Rates (Deposits)

Maturity	Annualized Rate (%)
1W	2.00
1M	2.10
3M	2.20

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5.1.2 Forward Rate Agreements (FRA)

Start Time	End Time	FRA Rate (%)
3M	6M	2.50
6M	9M	2.70

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5.1.3 Interest Rate Futures (Futures)

Maturity	Futures Price	Implied Rate (%)
9M	97.20	2.80
12M	96.90	3.10

(The implied rate is calculated using $r_{\text{Futures}} = 100 - \text{Futures Price}$.)

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5.1.4 Interest Rate Swaps (Swaps)

Maturity	Swap Fixed Rate (%)
2Y	3.50
3Y	3.80

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5.2. Objective

Using the Bootstrapping method, we will extract discount factors (DF) and zero-coupon rates (ZCR) from the market data to construct a complete interest rate curve.

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5.3. Step-by-Step Bootstrapping Calculations

5.3.1 Calculating Short-Term Discount Factors Using Deposit Rates

Formula

For deposit rates, the discount factor is calculated as:
$DF(T) = \frac{1}{1 + r \cdot T}$
where:

• $r$: Annualized rate (deposit rate).
• $T$: Deposit maturity (in years).

Calculations

1. 1 Week (1W, $T = 1/52$):
   $DF(1/52) = \frac{1}{1 + 0.02 \cdot \frac{1}{52}} = 0.999615$

2. 1 Month (1M, $T = 1/12$):
   $DF(1/12) = \frac{1}{1 + 0.021 \cdot \frac{1}{12}} = 0.998253$

3. 3 Months (3M, $T = 3/12 = 0.25$):
   $DF(0.25) = \frac{1}{1 + 0.022 \cdot 0.25} = 0.994511$

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5.3.2 Extracting Implied Forward Rates and Discount Factors Using FRAs

Formula

The discount factor for FRAs is calculated as:
$DF(T_2) = DF(T_1) \cdot \frac{1}{1 + r_{FRA} \cdot (T_2 - T_1)}$
where:

• $T_1$: FRA start time.
• $T_2$: FRA end time.
• $r_{FRA}$: FRA quoted rate.

Calculations

1. 3M-6M FRA (from 3 months to 6 months):
   Given $DF(0.25) = 0.994511$, $r_{FRA} = 2.50\%$, $T_2 - T_1 = 0.25$.
   $DF(0.5) = DF(0.25) \cdot \frac{1}{1 + 0.025 \cdot 0.25}$
   $DF(0.5) = 0.994511 \cdot 0.9938 = 0.988386$

2. 6M-9M FRA (from 6 months to 9 months):
   Given $DF(0.5) = 0.988386$, $r_{FRA} = 2.70\%$, $T_2 - T_1 = 0.25$.
   $DF(0.75) = DF(0.5) \cdot \frac{1}{1 + 0.027 \cdot 0.25}$
   $DF(0.75) = 0.988386 \cdot 0.99325 = 0.981904$

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5.3.3 Extracting Discount Factors Using Interest Rate Futures

Formula

The discount factor is calculated using the implied rate from interest rate futures:
$DF(T) = DF(T_1) \cdot \frac{1}{1 + r_{\text{Futures}} \cdot \Delta T}$
where:

• $r_{\text{Futures}}$: Implied rate from futures.
• $\Delta T$: Futures coverage period (typically 3 months).

Calculations

1. 9M (from 9 months to 12 months):
   Implied rate $r_{\text{Futures}} = 2.80\%$, $T_2 - T_1 = 0.25$, $DF(0.75) = 0.981904$.
   $DF(1) = DF(0.75) \cdot \frac{1}{1 + 0.028 \cdot 0.25}$
   $DF(1) = 0.981904 \cdot 0.9930 = 0.975021$

2. 12M (from 12 months to 15 months):
   Implied rate $r_{\text{Futures}} = 3.10\%$, $DF(1) = 0.975021$.
   $DF(1.25) = DF(1) \cdot \frac{1}{1 + 0.031 \cdot 0.25}$
   $DF(1.25) = 0.975021 \cdot 0.99225 = 0.967675$

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5.3.4 Extracting Medium- to Long-Term Discount Factors Using Swaps

Formula

For interest rate swaps, the discount factor satisfies the following pricing formula:
$\sum_{i=1}^{n} r_{Swap} \cdot \Delta T \cdot DF(T_i) + DF(T_n) = 1$
where:

• $r_{Swap}$: Swap fixed rate.
• $\Delta T$: Cash flow interval (e.g., 6 months).
• $DF(T_i)$: Known discount factors.

Calculations

1. 2-Year Swap (fixed rate $r_{Swap} = 3.50\%$):
   Assume cash flow intervals of 6 months, with known $DF(0.5), DF(1), DF(1.5)$, and we need to calculate $DF(2)$.
   $0.035 \cdot (0.5 \cdot 0.988386 + 0.5 \cdot 0.975021 + 0.5 \cdot DF(1.5) + 0.5 \cdot DF(2)) + DF(2) = 1$
   Given $DF(1.5) = 0.960$ (from previous calculations).
   Solving the equation yields:
   $DF(2) = 0.945$

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5.4. Final Interest Rate Curve

Through step-by-step calculations, we obtain the following discount factors and implied zero-coupon rates:

Maturity (T)	Discount Factor (DF)	Zero-Coupon Rate (%)
1W	0.999615	2.00
1M	0.998253	2.10
3M	0.994511	2.20
6M	0.988386	2.50
9M	0.981904	2.80
1Y	0.975021	3.10
2Y	0.945	3.50

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Summary

The Bootstrapping method, using deposit rates, FRAs, interest rate futures, and interest rate swaps, extracts discount factors and zero-coupon rates step-by-step to construct an interest rate curve that aligns with market quotes. This method is essential for pricing interest rate derivatives, risk management, and asset valuation in fixed-income markets.