AFIN8003 Week 4 - Interest Rate Risk

Banking and Financial Intermediation

Dr. Mingze Gao

Department of Applied Finance

2024-08-14

Interest rate risk

What is interest rate risk?

Interest Rate Risk

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:
1. refinancing and reinvestment risks
2. financial instruments have different levels of value sensitivity to interest rate changes
Interest rate is volatile and can be unexpected.

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;
  function pv(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"}),
  ]
})

maturity4 = 15

couponRate4 = 0.05

d = Object {YTM: Array(20), Price: Array(20)}

data4 = Array(20) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object]

Assume $100 bond, annual coupons paid in arrears and effective annual discount rate.

Measuring interest rate risk

	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

		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

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.

$\Delta NII_i = (GAP_i) \times \Delta R_i = (RSA_i - RSL_i) \times \Delta R_i$

where:

$\Delta NII_i$ 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:

$\Delta NII_i = GAP_i \times \Delta R_i = -10 \text{ million} \times 0.01 = -100,000$

So, the repricing model suggests a loss of $100,000 in net interest income for the FI.

One-year repricing gap

It is common to also estimate the one-year (cumulative) repricing gap, which is -$15 million from Table 3.

$\Delta NII_i = CGAP_i \times \Delta R_i = -15 \text{ million} \times 0.01 = -150,000$

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?

$\Delta NII_i = RSA_i \times \Delta R^A_i - RSL_i \times \Delta R^L_i = (\$155M \times 0.012) - (\$155M \times 0.01) = \$310,000$

Suppose the RSA is $155 million and the RSL is $140 million. Interest rates rise by 1.2% on RSAs and 1% on RSLs. What is the change in NII?

$\Delta NII_i = RSA_i \times \Delta R_i^A - RSL_i \times \Delta R_i^L = (\$155M \times 0.012) - (\$140M \times 0.01) = \$460,000$

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

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 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.

Duration and price changes: example

Consider the previous example of a 5-year annual-coupon bond in Table 5 with a 4% coupon rate and 6% yield. It has a duration of 4.61 years.

What is the price change if the interest rate (yield) decreases by 1 percent ( $\Delta R=-0.01$ )?

Note

Recall Equation 2 that $\frac{\Delta P}{P} = - D \times \frac{\Delta R}{1+R} = - MD \times \Delta R$ .

$\text{Modified duration} = MD = \frac{D}{1+R} = \frac{4.61}{1+0.06} = 4.349$

So, we have

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

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.

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 $\Delta E = \Delta A - \Delta L$ .

Making use of the previous results Equation 3 and Equation 4, we have

$\Delta E = \left[- D_A \times A \times \frac{\Delta R}{1+R}\right] - \left[- D_L \times L \times \frac{\Delta R}{1+R}\right] \qquad(5)$

If the level of interest and expected shock to interest rates are the same for both assets and liabilities, then:

$\Delta E = - (D_A - D_L k) \times A \times \frac{\Delta R}{1+R} \qquad(6)$

where $k=\frac{L}{A}$ measures the FI’s leverage.

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:

$\Delta E = - (D_A - D_L k) \times A \times \frac{\Delta R}{1+R} = -(5 - 3\times 0.9) \times \$100 \times \frac{0.01}{1+0.1} = -\$2.09$

This means that the FI could lose $2.09 million in net worth if interest rates rose by 1 per cent.

How can the FI manage the interest rate exposure?

Reduce $D_A$ , and/or
Increase $D_L$ , and/or
Increase leverage $k$ , and/or
…

Immunization and regulatory considerations

Regulators, like APRA in Australia, require banks to hold a minimum amount of capital against their (risk-weighted) assets.

We have discussed capital management and adequacy last week.
The simplest ratio is: $\frac{E}{A}$ .

Thus, in order to comply with regulations, the aim of risk management should not be $\Delta E=0$ but $\Delta (\frac{E}{A}) = 0$ .

Instead of setting $D_A-D_L k=0$ , the bank now needs to target $D_A-D_L=0$ .

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AFIN8003 Week 4 - Interest Rate Risk Banking and Financial Intermediation Dr. Mingze Gao Department of Applied Finance 2024-08-14

AFIN8003 Week 4 - Interest Rate Risk
Interest rate risk
What is interest rate risk?
Maturity mismatch: refinancing risk
Maturity mismatch: reinvestment risk
Financial instruments’ value sensitivity to interest rate
The level and movement of interest rates
The level and movement of interest rates
The level and movement of interest rates
The market is bad at guessing interest rates
Interest rate risk and Basel framework
Interest rate risk and Basel framework (cont’d)
Measuring interest rate risk
An overview
Interest rate risk models
Repricing model
The repricing model
Repricing gap, RSA and RSL
Maturity (Tenor) buckets
Maturity buckets and RSA/RSL
Repricing gaps
The repricing model
The repricing model: examples
The repricing model: examples
Real-world application of repricing model
The repricing model: problems
Duration model
Motivation
Duration
Duration: example
Duration: another example
The economic meaning of duration
The economic meaning of duration
Duration and price changes: example
Features of duration
Duration and interest rate risk management
Duration and interest rate risk management: single security
Duration and interest rate risk management: balance sheet
The duration model
The duration model
The duration model: example
Immunization and regulatory considerations
Difficulties in applying duration model
Finally…
Suggested readings
References