## **Learning Outcomes**

After reading this article, you will be able to identify the main forms of interest rate risk and explain the difference between fixed and floating rate exposure. You will learn how to calculate and interpret Macaulay and modified duration, assess repricing gaps, and use these tools to measure and manage interest rate risk in financial decisions.

## **ACCA Advanced Financial Management (AFM) Syllabus**

For ACCA Advanced Financial Management (AFM), you are required to understand how interest rate risk arises and how to measure and manage it effectively. This article covers the following syllabus points:

- Identify different types of interest rate risk and their impact on organisations
- Calculate and interpret Macaulay duration and modified duration for debt securities and portfolios
- Explain the significance of duration and its limitations, including the effects of convexity
- Illustrate the use of repricing gap analysis for managing asset and liability mismatch risk
- Apply gap and duration methods in practical interest rate risk management

## **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._

1. What does a large positive repricing gap in the 0–3 month band indicate for a bank’s interest income if rates rise?
2. Which is the primary quantitative measure for price sensitivity of a fixed-rate bond to yield changes?
   a) Time to maturity  
   b) Coupon rate  
   c) Duration  
   d) Convexity
3. True or false? Modified duration shows the approximate percentage change in bond price for a 1% change in yield to maturity.
4. Why is duration a better measure of interest rate sensitivity than simple maturity?
5. Briefly describe how a financial institution can use gap analysis to reduce exposure to adverse rate movements.

## **Introduction**

Interest rate risk is an important concern for organisations involved in borrowing or lending, especially financial institutions. Unmanaged, shifts in interest rates can significantly affect cash flows, asset values, and reported earnings. Identifying and measuring this risk is a prerequisite for risk control. This article explains the types of interest rate risk, introduces duration and repricing gap analysis, and shows how to interpret these measures for practical risk management.

> **Key Term: interest rate risk**
> The risk that fluctuations in market interest rates will adversely affect an organisation's cash flows, asset values, or net interest income.

## **Forms of Interest Rate Risk**

Interest rate risk can arise in several ways for banks, corporations, and investors.

### Transaction and Economic Interest Rate Risk

**Transaction risk** occurs whenever the value of future cash flows from an asset or liability depends on prevailing interest rates. For instance, a company with floating-rate borrowings exposes itself to payment uncertainty as rates change.

**Economic risk** reflects the risk that changes in long-term interest rates reduce the present value of future cash flows, affecting valuations of both fixed and variable rate instruments.

> **Key Term: repricing risk**
> The risk that assets and liabilities will reprice at different times or intervals, causing mismatches that affect net interest income or value.

## **Measuring Interest Rate Risk: Duration**

Duration is the principal measurement of interest rate sensitivity for fixed-interest instruments and portfolios.

### Macaulay Duration

Macaulay duration summarises the weighted average time to receive a bond's cash flows, with each period's weight proportional to the present value of that cash flow. A higher duration indicates greater price sensitivity to interest rate changes.

> **Key Term: Macaulay duration**
> The weighted average time, measured in years, for a bondholder to be repaid by the present value of all future cash flows.

### Modified Duration

For ease of application, modified duration adjusts Macaulay duration to estimate the bond’s percentage price change for a given change in yield.

> **Key Term: modified duration**
> Macaulay duration divided by (1 + yield per period); it estimates the percentage change in price for a 1% change in yield.

### **Worked Example 1.1**

A corporation holds a four-year bond with a \$1,000 face value and 6% annual coupons. The bond's required yield is 5%.

**Calculate Macaulay duration and modified duration.**

> **Answer:**
>
> 1. Find present values: Year 1–3: \$60 per year, Year 4: \$1,060
>
> Year 1: \$60 / 1.05 = \$57.14  
> Year 2: \$60 / 1.05² = \$54.42  
> Year 3: \$60 / 1.05³ = \$51.83  
> Year 4: \$1,060 / 1.05⁴ = \$871.75  
> Total PV = \$57.14 + \$54.42 + \$51.83 + \$871.75 = \$1,035.14
>
> Time-weighted PV sum:  
> (1 × \$57.14) + (2 × \$54.42) + (3 × \$51.83) + (4 × \$871.75) = \$57.14 + \$108.84 + \$155.49 + \$3,487.00 = \$3,808.47  
> Macaulay duration = \$3,808.47 / \$1,035.14 = 3.68 years  
> Modified duration = 3.68 / 1.05 = 3.50  
> Interpretation: For a 1% change in yield, bond price will change by about 3.5%.

### How Does Duration Measure Risk?

- Longer duration indicates higher price sensitivity to interest rate movements.
- Duration incorporates all cash flows’ timing and size, providing a superior risk metric compared to simple maturity.

> **Key Term: convexity**
> The extent to which the price–yield relationship of a bond bends or curves, indicating how duration changes as yields change.
>
> **Key Term: repricing gap**
> The difference between the amount of assets and liabilities that reprice or mature within a given time window.

## **Limitations of Duration Analysis**

- Duration best approximates price sensitivity for small, parallel shifts in yields.
- It assumes the cash flows themselves do not change (no optionality).
- For instruments with embedded options or variable coupons, more advanced modelling is needed.
- Duration does not fully capture risk for large yield movements—convexity becomes important.

### **Exam Warning**

> Duration is a linear approximation. For substantial interest rate changes, the actual bond price change may differ due to convexity. Always mention convexity in exam answers when discussing the limits of duration.

## **Repricing Gap Analysis**

Gap analysis quantifies the mismatch between the timing of asset and liability repricing, often within specified ‘time buckets’ (e.g., 0–3 months, 3–12 months).

- For each time band, calculate:
  Repricing gap = Rate-sensitive assets – Rate-sensitive liabilities

### Interpreting Gaps

- **Positive gap:** Assets reprice before liabilities. If rates rise, net interest income tends to increase.
- **Negative gap:** Liabilities reprice before assets. If rates rise, net interest income tends to fall.

### **Worked Example 1.2**

A bank’s interest rate sensitivity table for the next 6 months is as follows:

| Time Band  | Rate-Sensitive Assets | Rate-Sensitive Liabilities |
| ---------- | --------------------- | -------------------------- |
| 0–3 months | \$40 million          | \$30 million               |
| 3–6 months | \$25 million          | \$32 million               |

**Calculate the repricing gap for each band and comment.**

> **Answer:**
>
> 0–3 months: Gap = \$40m – \$30m = +\$10m. There is more exposure to rising rates, since assets will reprice faster.
>
> 3–6 months: Gap = \$25m – \$32m = –\$7m. If rates rise, more liabilities than assets reprice, decreasing income.

### Managing Interest Rate Risk Using Gaps

A bank or company can structure its balance sheet to limit repricing gaps that would expose it unduly to rate changes. However, perfect matching is rare—acceptable levels depend on risk appetite and profitability goals.

### **Revision Tip**

> In calculations, always specify the time bands and be clear whether assets or liabilities are repricing faster—direction is essential for interpreting risk.

## **Summary**

Measuring interest rate risk is central to controlling its impact. Duration provides a standardised gauge for how fixed-rate instruments respond to rate changes. Repricing gap analysis complements this by focusing on the timing mismatch of rate resets across assets and liabilities. Both methods inform risk management strategies used by financial managers and treasurers to protect income and asset values.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Identify transaction, economic, and repricing forms of interest rate risk
- Calculate Macaulay and modified duration and explain their uses
- Distinguish between duration, maturity, and convexity for measuring price sensitivity
- Recognise limitations of duration and the impact of convexity
- Calculate and interpret repricing gaps by time band for asset–liability management
- Apply gap and duration analysis in formulating interest rate risk management strategies

## **Key Terms and Concepts**

- interest rate risk
- repricing risk
- Macaulay duration
- modified duration
- convexity
- repricing gap

Risk identification and measurement - Interest rate risk, duration, and repricing gaps