Interest rate risk is the possibility of a financial loss due to changes in interest rates.

Recall that in Week 1, we explained one of the key functions performed by financial institutions (especially depositary institutions) is maturity transformation. For DIs, they take short-term deposits from depositors and make long-term loans to borrowers.

Table 1: Typical balance sheet of a DI

Assets

Liabilities and Equity

Loans (relatively long-term)

Deposits (relatively short-term)

Other assets

Other liabilities

Equity

As a result,

This maturity transformation means that FIs have mismatches of maturities of their liabilities (e.g., deposits) and assets (e.g., loans).

Maturity mismatch exposes FIs to (unexpected) interest rate changes:

refinancing and reinvestment risks

financial instruments have different levels of value sensitivity to interest rate changes

Interest rate is volatile and can be unexpected.

Maturity mismatch: refinancing risk

Consider an example bank that has $2 million of 10-year fixed-rate loans and $1 million of 1-year fixed-rate term deposits now. The bank expects to have the same balance sheet in the future.

What would happen in 1 year from now?

Term deposits mature. Some deposits may be withdrawn.

In this case, the bank needs to take additional deposits to maintain the size of balance sheet.

However, the interest rate (deposit rate) could be higher in the future, which means higher costs for the bank.

This is a typical refinancing risk - the costs of rolling over funds or re-borrowing funds will rise above the returns generated on investments.

Maturity mismatch: reinvestment risk

Consider another example bank that has $2 million of 1-year fixed-rate loans and $1 million of 2-year fixed-rate term deposits now. The bank expects to have the same balance sheet in the future.

What would happen in 1 year from now?

Loans mature.

In this case, the bank needs to reinvest into some other assets (e.g., make loans).

However, the interest rate (on the loan) could be lower in the future, which means reduced returns for the bank.

This is a typical reinvestment risk - the returns on funds to be reinvested will fall below the cost of funds.

Financial instruments’ value sensitivity to interest rate

The value of longer-maturity instruments typically is more sensitive to interest rate.

For example, a bond’s price is given by

\[
P = \sum_{t=1}^T\frac{C}{(1+r)^t} + \frac{F}{(1+r)^T}
\qquad(1)\]

where \(C\) is coupon payment, \(F\) face value, \(T\) maturity, and \(r\) interest rate.

If a FI is financed by issuing long-term bonds and invests in short-term loans, given a change in interest rate, its liabilities’ value would change by more than the change in its assets’ value, thereby causing an impact on its net worth.

Code

viewof maturity4 = Inputs.range( [1,30], {value:15,step:1,label:"Maturity (years):"})viewof couponRate4 = Inputs.range( [0.01,0.2], {value:0.05,step:0.01,label:"Coupon rate:"})d = {const f =100;let c, m; c = couponRate4; m = maturity4;functionpv(c, f, t, r) {return c * (1- (1+r)**(-t)) / r + f / (1+r)**(t) }const prices = {"YTM": [],"Price": []};let coupon = f * c;for (let ytm =0.01; ytm <20; ytm++) {let price =pv(coupon, f, m, ytm/100); prices["YTM"].push(ytm); prices["Price"].push(price); }return prices;}data4 =transpose(d)Plot.plot({caption:"Assume $100 bond, annual coupons paid in arrears and effective annual discount rate.",x: {padding:0.4,label:"YTM (%)"},grid:true,marks: [ Plot.ruleY([0,100]), Plot.ruleX([0]), Plot.lineY(data4, {x:"YTM",y:"Price",stroke:"blue"}), ]})

The level and movement of interest rates

The relative degree of interest rate volatility is directly linked to the monetary policy of the Reserve Bank of Australia (RBA).

For example, if RBA smooths or targets the level of interest rates, unexpected interest rate shocks and interest rate volatility tend to be small.

A change in approach from targeting interest rate (cash rate) goals towards CPI growth rate increased the interest rate volatility.

No matter what, interest rate volatility exists and thus, the appropriate measurement of management of interest rate risk is important to every FI.

From here, we’re to study specific models and technique for measuring FI’s interest rate risk. However, it is helpful to first explain the general rule so to better understand the big picture.

A risk causes an unwanted negative impact on a FI. To measure such risk, we need to specify a few things:

Which FI’s characteristic are we concerned about? Income, net worth, or something else?

How sensitive is that FI’s characteristic to this risk?

What is the potential size of this risk?

Is it based on book value or market value?

What is the time frame we are looking at?

…

Interest rate risk models

The rest of this lecture discusses two models for measuring interest rate risk with different focuses.

Table 2: A simple comparison of two interest rate risk models

Repricing model

Duration model

Which characteristic?

Net interest income (NII)

Net worth

Sensitivity measurement?

By assumption

Duration

Size of risk?

Flexible

Flexible

Book value or market value?

Book value

Market value

Time frame?

Flexible

Flexible

Repricing model

The repricing model

Repricing model is also known as the funding gap model.

A book value accounting cash flow analysis of the repricing gap between the interest income earned on an FI’s assets and the interest paid on its liabilities over some period.

Focus on the changes of net interest income (NII).

A simple model used by small FIs.

APRA requires smaller ADIs to use the repricing method to estimate interest rate exposures in their banking book for capital adequacy purposes.

A revision has just been completed in early July, 2024, which will be effective in 2025.^{1}

Repricing gap, RSA and RSL

Repricing gap is the difference between rate-sensitive assets (RSA) and rate-sensitive liabilities (RSL).

RSA: assets whose interest rates will be repriced or changed over some future period (rate-sensitive assets)

RSL: liabilities whose interest rates will be repriced or changed over some future period (rate-sensitive liabilities)

An asset (or liability) is “rate-sensitive” if it is repriced at or near current market interest rates within a certain time horizon (or maturity bucket). For example,

a variable-rate mortgage’s interest rate may be reset every quarter based on the prevailing interest rate.

interest rate of term deposits are also adjusted frequently.^{1}

Maturity (Tenor) buckets

Under repricing model, banks report their repricing gaps for various maturity buckets. For example,

One day (overnight).

More than 1 day to 3 months.

More than 3 months to 6 months.

More than 6 months to 12 months.

More than 1 year to 5 years.

More than 5 years.

How to determine RSA/RSL for each bucket?

Ask a simple question: Will or can this asset or liability have its interest rate changed within the maturity bucket? If the answer is yes, it is a rate-sensitive asset or liability. If the answer is no, it is not rate sensitive.

Maturity buckets and RSA/RSL

Let’s practice. Try to identify the one-year RSA and RSL given the following assets and liabilities of a bank.

Assets:

Liabilities:

Repricing gaps

Table 3: Repricing gaps ($ millions)

Assets

Liabilities

Gaps

Cumulative gap

1

One day

20

30

-10

-10

2

One day to three months

30

40

-10

-20

3

Three months to six months

70

85

-15

-35

4

Six months to 12 months

90

70

+20

-15

5

One year to five years

40

30

+10

-5

6

Over five years

10

5

+5

0

Total

$260

$260

For example, a negative $10 million difference between its RSA and RSL being repriced in one day (one-day bucket). A rise in the overnight rate would lower the FI’s net interest income since the FI has more rate-sensitive liabilities than assets in this bucket.

A refinancing risk given RSA < RSL. Increase in interest rate means more interest paid on liabilities than earned from assets.

The repricing model

The repricing model can be used to estimate the change in the FI’s net interest income in a particular repricing bucket if interest rates change.

\(\Delta NII_i\) is the change in net interest income in the \(i\)th bucket.

\(GAP_i\) is the dollar size of the gap between the book value of RSA and RSL in maturity bucket \(i\).

\(\Delta R_i\) is the change in the level of interest rates impacting assets and liabilities in the \(i\)th bucket.

The repricing model: examples

One-day repricing gap

In the first bucket of Table 3, if the gap is negative $10 million and overnight interest rate rises by 1 percent, the annualized change in the FI’s future NII is:

If the average interest rises by 1 percent, the model suggests a loss of $150,000 in net interest income.

The repricing model: examples

Assume both RSAs and RSLs equal $155 million. Suppose that there is a general interest rate rise and that the rates on RSAs rise by 1.2 percent and rates on RSLs rise by only 1 percent. What is the resulting change in NII?

Let see a real-world application of repricing model by Texas Capital Bank, a (relatively small) commercial bank headquartered in Dallas, Texas, United States.

We can find its annual reports from SEC’s EDGAR system, which allows us to search for any corporate filings.

Check Item 7A. Quantitative and qualitative disclosure about market risk.

The repricing model: problems

The repricing model is very simple and intuitive, but is NOT an accurate measure of interest rate risk.

Major shortcomings:

It is over-aggregative.

It ignores the market value effects of interest rate changes: it focuses on the income and ignores the capital gains/losses.

It uses the banking book and hence book values of assets and liabilities.

It fails to deal with the problem of rate-insensitive asset and liability runoffs and prepayments.

It ignores cash flows from off-balance-sheet activities.

Duration model

Motivation

In the early 2000s, the BIS issued a consultative document suggesting a standardized model be used by regulators in evaluating a bank’s interest rate risk exposure. The approach suggested is firmly based on market value accounting and the duration gap model.

The duration gap (model):

considers market values and the maturity distributions of an FI’s assets and liabilities.

considers the degree of leverage on an FI’s balance sheet.

considers the timing of the payment or arrival of cash flows on assets and liabilities.

is a more comprehensive measure of an FI’s interest rate risk.

Bigger banks have adopted duration model as their primary measure of interest rate risk.^{1}

Duration

The essence of duration gap model is the concept of duration, which is covered in introductory finance courses.

Simply put, duration is the weighted-average time to maturity on the loan using the relative present values of the cash flows as weights.

\[
D = \sum_{t=1}^{N} w_t \times t
\]

where \(D\) is the duration, and

\[
w_t = \frac{\text{PV of the cash flow at time } t}{\text{Total PV of all cash flows}}
\]

Duration: example

Consider the duration of a 3-year coupon bond with a face value of $1,000, a coupon rate of 5%, and a yield to maturity of 4%. The bond pays annual coupons.

Code

import pandas as pd# Parametersface_value, coupon_rate, yield_to_maturity, years =1000, 0.05, 0.04, 3annual_coupon_payment = face_value * coupon_rate# Initialize and calculate valuesdata = [ (t, annual_coupon_payment if t < years else annual_coupon_payment + face_value)for t inrange(1, years +1)]total_pv =sum(payment / (1+ yield_to_maturity) ** t for t, payment in data)table = [ ( t, payment, pv := payment / (1+ yield_to_maturity) ** t, weight := pv / total_pv, t * weight, )for t, payment in data]# Create DataFramedf = pd.DataFrame( table, columns=["Time (Years)","Payment ($)","PV of Payment ($)","Weight (PV/Total PV)","Weighted Time", ],)# Add totals rowtotals = df.sum(numeric_only=True)totals = pd.DataFrame(totals).Ttotals["Time (Years)"] ="Total"df = pd.concat([df, totals], ignore_index=True)df.style.highlight_max(["Weighted Time"]).format(precision=2).hide()

Table 4: Calculating the duration of a 3-year bond

Time (Years)

Payment ($)

PV of Payment ($)

Weight (PV/Total PV)

Weighted Time

1

50.00

48.08

0.05

0.05

2

50.00

46.23

0.04

0.09

3

1050.00

933.45

0.91

2.72

Total

1150.00

1027.75

1.00

2.86

The bond’s duration is 2.86 years.

Duration: another example

Consider the duration of a 5-year coupon bond with a face value of $1,000, a coupon rate of 4%, and a yield to maturity of 6%. The bond pays annual coupons.

Code

import pandas as pd# Parametersface_value, coupon_rate, yield_to_maturity, years =1000, 0.04, 0.06, 5annual_coupon_payment = face_value * coupon_rate# Initialize and calculate valuesdata = [ (t, annual_coupon_payment if t < years else annual_coupon_payment + face_value)for t inrange(1, years +1)]total_pv =sum(payment / (1+ yield_to_maturity) ** t for t, payment in data)table = [ ( t, payment, pv := payment / (1+ yield_to_maturity) ** t, weight := pv / total_pv, t * weight, )for t, payment in data]# Create DataFramedf = pd.DataFrame( table, columns=["Time (Years)","Payment ($)","PV of Payment ($)","Weight (PV/Total PV)","Weighted Time", ],)# Add totals rowtotals = df.sum(numeric_only=True)totals = pd.DataFrame(totals).Ttotals["Time (Years)"] ="Total"df = pd.concat([df, totals], ignore_index=True)df.style.highlight_max(["Weighted Time"]).format(precision=2).hide()

Table 5: Calculating the duration of a 5-year bond

Time (Years)

Payment ($)

PV of Payment ($)

Weight (PV/Total PV)

Weighted Time

1

40.00

37.74

0.04

0.04

2

40.00

35.60

0.04

0.08

3

40.00

33.58

0.04

0.11

4

40.00

31.68

0.03

0.14

5

1040.00

777.15

0.85

4.24

Total

1200.00

915.75

1.00

4.61

The bond’s duration is 4.61 years.

The economic meaning of duration

We are interested in the sensitivity of bond price to interest rate.

That is, how much the price of a bond changes when the interest changes.

Duration directly measures the interest rate sensitivity of an asset or liability:

\[
\frac{\Delta P}{P} = - D \times \frac{\Delta R}{1+R} = - MD \times \Delta R
\qquad(2)\]

where

\(\Delta P\) is the price change of assets or liabilities

\(\Delta R\) is the change of interest rate

\(MD\) is the modified duration that equals to \(\frac{D}{1+R}\)

The larger the numerical value of \(D\), the more sensitive is the price of that asset or liability to changes or shocks in interest rates.

Important

The relationship is only true for small changes in the yield.

The economic meaning of duration

Why does it work?

\[
\frac{\Delta P}{P} = - D \times \frac{\Delta R}{1+R} = - MD \times \Delta R
\]

We can write out the price formula of a bond and take its partial derivative with respect to yield.

\[
\Delta P = - MD \times P \times \Delta R = - 4.349 \times \$1000 \times (-0.01) = \$43.49
\]

That is, a one percentage point (100 basis points) decrease in yield would increase bond price by $43.49.

Note that \(MD\times P\) is also named “dollar duration”, i.e., modified duration times the bond price.

Features of duration

Duration increases with the maturity of a fixed-income security, but at a decreasing rate.

Duration decreases as the yield on a security increases.

Duration decreases as the coupon or interest payment increases.

Duration and interest rate risk management

Now that we have refreshed our knowledge of duration, how is duration relevant in FI’s interest rate risk management?

Let’s consider two cases.

A single security.

A portfolio of securities (e.g., the balance sheet).

Duration and interest rate risk management: single security

Superannuation funds or insurers often have to make a specific amount of payment to their policyholders at a given future date. How to guarantee it? Investments may decrease in value if interest rate changes over the period.

There are two options:

Buy a zero-coupon bond which matures exactly when the payment is made in the future.

Buy bond(s) with a duration that matches the payment date.^{1}

However,

Option 1 is very difficult because there simply may not be any such zero-coupon bonds with desirable maturities.

Option 2 is manageable and practical!

Duration and interest rate risk management: balance sheet

Duration can be used to measure a financial institution’s duration gap to evaluate the FI’s overall interest rate exposure. It is possible to calculate the duration of the asset portfolio and of the liability portfolio.

Duration of a portfolio is the weighted average duration of its components.

Duration of assets portfolio:

\[
D_A = \sum_{i=1}^{N_A} w_{iA} \times D^A_i
\]

where \(N_A\) is the total number of assets, \(w_{iA}\) is the market value weight of asset \(i\), \(D^A_i\) is the duration of asset \(i\).

So, change of assets value for a given change in interest rate is

\[
\Delta A = - D_A \times A \times \frac{\Delta R}{1+R}
\qquad(3)\]

Duration of liabilities portfolio:

\[
D_L = \sum_{i=1}^{N_L} w_{iL} \times D^L_i
\]

where \(N_L\) is the total number of liabilities, \(w_{iL}\) is the market value weight of liability \(i\), \(D^L_i\) is the duration of liability \(i\).

So, change of liabilities for a given change in interest rate is

\[
\Delta L = - D_L \times L \times \frac{\Delta R}{1+R}
\qquad(4)\]

The duration model

We know that total assets (\(A\)) is the sum of liabilities (\(L\)) and equity (\(E\)): \(A=E+L\).

Therefore, \(E = A - L\), and \(\Delta E = \Delta A - \Delta L\).

\[
\Delta E = - (D_A - D_L k) \times A \times \frac{\Delta R}{1+R}
\]

The effect of interest rate changes on the market value of an FI’s net worth breaks down into three effects:

The leverage-adjusted duration gap: \((D_A-D_L k)\)

The size of the FI: \(A\)

The size of the interest rate shock: \(\frac{\Delta R}{1+R}\)

The duration model: example

Suppose a FI has total assets of $100 million and total liabilities of $90 million.

Assume that the average duration of assets is 5 years, while the average duration of liabilities is 3 years. The current interest rate is 10%, but is expected to increase to 11% in the future.

We can calculate the expected change in the FI’s net worth as follows: