AFIN8003 Week 13 - Review

Banking and Financial Intermediation

Dr. Mingze Gao

Department of Applied Finance

2025-06-05

Risk-based capital ratio

Under Basel III, depositary institutions (DIs) calculate and monitor four capital ratios:

Common equity Tier 1 (CET1) risk-based capital ratio $\text{CET1 capital ratio} = \frac{\text{CET1 capital}}{\text{Risk-weighted assets}} \qquad(1)$
Tier 1 risk-based capital ratio $\text{Tier 1 capital ratio} = \frac{\text{Tier 1 capital}}{\text{Risk-weighted assets}} \qquad(2)$
Total risk-based capital ratio $\text{Total capital ratio} = \frac{\text{Total capital}}{\text{Risk-weighted assets}} \qquad(3)$
Tier 1 leverage ratio $\text{Tier 1 leverage ratio} = \frac{\text{Tier 1 capital}}{\text{Total exposure}} \qquad(4)$

The calculation of these capital ratios is complex.
Use risk-weighted assets (RWA) to distinguish the different credit risks of different assets.
Measure a DI’s credit risk (on-and off-balance-sheet).

Important

Additional capital charges for market risk and operational risk.
- For now, we do not consider these risks and their impact on RWA.
- We briefly explain how they affect RWA and capital ratios at the end of this lecture.

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.

Duration model

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

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

$\frac{\Delta P}{P} = - D \times \frac{\Delta R}{1+R} = - MD \times \Delta R \qquad(6)$

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 interest rate risk management

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

$\Delta A = - D_A \times A \times \frac{\Delta R}{1+R} \qquad(7)$

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

$\Delta L = - D_L \times L \times \frac{\Delta R}{1+R} \qquad(8)$

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 7 and Equation 8, we have

$\Delta E = \left[- D_A \times A \times \frac{\Delta R}{1+R}\right] - \left[- D_L \times L \times \frac{\Delta R}{1+R}\right] \qquad(9)$

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

$\Delta E = - (D_A - D_L k) \times A \times \frac{\Delta R}{1+R} \qquad(10)$

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

Value at Risk (VaR)

Suppose we know the distribution of an asset’s returns over a specific period (in the future), then for a given confidence level $c\in[0,1]$ (e.g., $c=0.95$ ), we can partition the distribution into two parts: one (in red) that represents a proportion of $(1-c)$ of the distribution and the other (in blue) accounting the remaining $c$ proportion.

Figure 1: Distribution of expected returns and VaR

Therefore, the cutoff value of returns that separates the two parts defines:

a minimum return that we are confident 95% of the time, or
a maximum loss that we are confident 95% of the time.

	CBA	Westpac	NAB	ANZ	Macquarie
RWA for credit risk	362,869	339,758	355,554	349,041	97,485
RWA for market risk	61,968	51,676	38,274	41,967	11,663
RWA for operational risk	43,155	55,175	41,178	42,319	15,828
Other RWA	0	4,809	0	0	0
Total RWA	467,992	451,418	435,006	433,327	124,976

Contractually promised return on a loan

Direct and indirect fees and charges:

Loan origination fee ( $f$ ) for processing the application.
Compensating balance requirement ( $b$ ) to be held as (demand) deposits.
- A portion of the loan that the borrower must keep in a deposit account, typically earning little or no interest, which helps the bank secure funds and reduce the effective loan amount while still charging interest on the full loan value.
Reserve requirement ( $RR$ ) imposed by regulatory authority on the demand deposits.

The gross return on loan ( $k$ ) per dollar lent is given by $1+k = 1+\frac{f+(BR+\phi)}{1-[b(1-RR)]}$

The expected return on a loan

The promised return on a loan ( $k$ ) can differ from the expected return and the actual return on the loan because of default risk.

At the time the loan is made, the expected return $E(r)$ per dollar lent is related to the promised return $k$ as follows: $1+E(r) = p(1+k) + (1-p)\times 0$ where $p$ is the probability of complete repayment of the loan and $(1-p)$ is the probability of default, in which case the FI receives nothing.¹ The expected return is a weighted average of:

Full repayment: $(1+k)$ with probability of $p$
No repayment: 0 with probability of $(1-p)$

RAROC

Risk-adjusted return on capital (RAROC) was pioneered by Bankers Trust (acquired by Deutsche bank in 1998).

$RAROC = \frac{\text{One-year net income on a loan}}{\text{Loan risk}}$ where $\text{One-year net income on loan} = (\text{Spread} + \text{Fees}) \times \text{Dollar value of the loan outstanding}$ and Loan risk can be measure by, for example, duration. $\frac{\Delta LN}{LN} = - D_{LN} \frac{\Delta R}{1+R}$ so that $\underbrace{\Delta LN}_{\text{dollar risk exposure}} = - \underbrace{D_{LN}}_{\text{duration of loan}} \times \underbrace{LN}_{\text{loan amount}} \times \underbrace{\frac{\Delta R}{1+R}}_{\text{shock}}$

Loan approval if RAROC > benchmark return on capital.

Moody’s Analytics RiskFrontier Model (cont’d)

Required input variables:

$R_i$ : Expected return on a loan to a borrower $i$ is the loan’s all-in-drawn spread ( $AIS$ ) minus expected loss (expected default frequency times loss given default (LGD)).¹ $R_i = AIS_i - E(L_i) = AIS_i - (EDF_i\times LGD_i)$
- The Basel Committee assessed a fixed 45% LGD on secured loans if fully secured by physical, non–real estate collateral and 40% if fully secured by receivables.

$\sigma_i$ : Risk of a loan to borrower $i$ , “unexpected loss”, is the volatility of the loan’s default rate $\sigma_{D_i}$ times the amount lost given default (LGD).² $\sigma_i = UL_i = \sigma_{D_i} \times LGD_i = \sqrt{EDF_i(1-EDF_i)} \times LGD_i$
- Assume defaults are binomially distributed (default or not), the standard deviation of default rate ( $\sigma_{D_i}$ ) is $\sqrt{EDF_i(1-EDF_i)}$ .

$\rho_{ij}$ : Correlation of default risks between borrowers $i$ and $j$ is the correlation between the systematic return components of the asset returns of $i$ and $j$ .

To measure the unobservable default risk correlation between any two borrowers, the Moody’s Analytics Global Correlation Model (GCORR) uses the systematic asset return components of the two borrowers and calculates a correlation that is based on a factor model rather than direct historical observations.

Liquidity Coverage Ratio (LCR)

The Liquidity Coverage Ratio (LCR) is designed to ensure that a DI maintains sufficient high-quality liquid assets (HQLA).
These HQLA must be easily convertible to cash to meet liquidity needs over a 30-day period.
The LCR is based on an “acute liquidity stress scenario” defined by supervisors.
- The scenario includes both institution-specific and systemic shocks.
- It reflects actual conditions experienced during the global financial crisis.
The purpose of maintaining the LCR is to ensure that DIs can survive severe liquidity stress for at least 30 days.
The LCR is reported to DI supervisors on a monthly basis.

$\text{Minimum LCR} = \frac{\text{Stock of HQLAs}}{\text{Total net cash outflows over next 30 calendar days}} \ge 100\%$

Liquidity Coverage Ratio (LCR) (cont’d)

The total net cash outflows is defined as:

$\text{Total net cash outflows over next 30 calendar days} = Out - \min(In, 75\% \times Out)$

where

$Out$ is total expected cash outflows
$In$ is total expected cash inflows

Cash inflows and outflows are computed based on the type of assets/liabilities and associated draw-down factors.

Note

For a detailed calculation of cash outflows and inflows, refer to textbook Saunders, Cornett, and Erhemjamts (2023) or to the suggested readings.

Net Stable Funding Ratio (NSFR)

The Net Stable Funding Ratio (NSFR) focuses on long-term liquidity management on a DI’s balance sheet.
It evaluates liquidity across the entire balance sheet, encouraging the use of stable sources of financing.
The NSFR requires a minimum amount of stable funding over a one-year time horizon.
- It aims to limit the reliance on short-term wholesale funding, a significant issue during the financial crisis.
The NSFR has been reported to DI supervisors quarterly since 2018.

$\text{NSFR} = \frac{\text{Available amount of stable funding}}{\text{Required amount of stable funding}} \ge 100\%$

Foreign exchange risk

An FI’s overall FX exposure in any given currency can be measured by the net position exposure, which is measured in local currency as

$\begin{aligned} \text{Net exposure}_i &= (\text{FX assets}_i - \text{FX liabilities}_i) + (\text{FX bought}_i - \text{FX sold}_i) \\ &=\text{Net foreign assets}_i + \text{Net FX bought}_i \end{aligned}$

where $i$ represents the $i$ th currency.

FX rate volatility and FX exposure

We can measure the potential size of an FI’s FX exposure as:

$\begin{aligned} \text{Dollar loss/gains in currency } i &= \text{Net exposure in foreign currency } i \text{ measured in local currency} \\ &\times \text{Shock (volatility) to the exchange rate of local currency to foreign currency } i \end{aligned}$

Greater exposure to a foreign currency combined with greater volatility of the foreign currency implies greater daily earnings at risk (DEAR).
Reason for FX volatility: fluctuations in the demand for and supply of a country’s currency

Purchasing Power Parity

Nominal interest rate $R$ is basically the sum of inflation $i$ and real interest rate $r$ :

For two countries, e.g., Australia (AU) and United States (US), we have:

$\begin{aligned} R_{AU} &= i_{AU} + r_{AU} \\ R_{US} &= i_{US} + r_{US} \end{aligned}$

Assuming real interest rates are equal across countries: $r_{AU}=r_{US}$ , then

$R_{AU} - R_{US} = i_{AU} - i_{US}$

That is, the (nominal) interest rate spread between Australia and the United States represents the difference in inflation rates between the two countries.

Important

When inflation rates and/or interest rates change, foreign exchange rates (without government control) should adjust to account for relative differences in the price levels (inflation) between the two countries.

The Purchasing Power Parity (PPP) is one theory explaining how this adjustment takes place.

Purchasing Power Parity (cont’d)

Tip

The PPP says that the most important factor determining exchange rates is the price differences drive trade flows and thus demand for and supplies of currencies.

Specifically, the PPP theorem states that the change in the exchange rate between two countries’ currencies is proportional to the difference in the inflation rates in the two countries. That is:

$i_{domestic} - i_{foreign} = \frac{\Delta S_{domestic/foreign}}{S_{domestic/foreign}}$

where

$S_{domestic/foreign}$ is the spot exchange rate of the domestic currency for the foreign currency
$\Delta S_{domestic/foreign}$ is the change in the one-period spot exchange rate

Purchasing Power Parity (cont’d)

Suppose that the current spot exchange rate of Australian dollars for Chinese yuan, $S_{AUD/CNY}$ , is 0.17 (i.e. 0.17 dollars, or 17 cents is equal to 1 yuan). The price of Chinese-produced goods increases by 10 per cent (i.e. inflation in China $i_C$ , is 10 per cent), and the Australian price index increases by 4 per cent (i.e. inflation in Australia, $i_{AUS}$ , is 4 per cent). What will be the change of the exchange rate?

According to PPP:

$i_{AUS} - i_{C} = \frac{\Delta S_{AUD/CNY}}{S_{AUD/CNY}}$

So that

$0.04 - 0.1 = \frac{\Delta S_{AUD/CNY}}{0.17}$

Solving the equation, we get $\Delta S_{AUD/CNY} = −0.0102$ . Thus, it costs 1.02 cents less to receive a yuan. The Chinese yuan depreciates in value by 6 per cent against the Australian dollar as a result of its higher relative inflation rate. In other words, a 6 per cent fall in the yuan’s value translates into a new exchange rate of 0.1598 dollars per yuan.

Interest Rate Parity

To illustrate (covered) interest rate parity (IRP), let’s consider two investment strategies.

Strategy 1: Domestic Investment

You invest $1 domestically at an interest rate $r_d$ for one period.
After one period, your investment grows to $1 + r_d$ .

Strategy 2: Foreign Investment with Forward Contract

You convert $1 into foreign currency at the current spot exchange rate $S$ , which gives $1/S$ foreign currency.
You then invest the foreign currency amount in a foreign asset at the interest rate $r_f$ for one period.
After one period, this investment grows to $(1 + r_f)$ .
You lock in the forward exchange rate $F$ now to convert the foreign currency back to domestic currency after one period.

For these two strategies to have the same return (and eliminate arbitrage opportunities), the returns on both should be equivalent when the foreign investment is converted back using the forward rate. This gives us the CIRP condition:

$1 + r_d = \frac{(1 + r_f) \cdot F}{S}$

Rearranging, we can see the relation between interest rates and the forward rate:

$F = S \cdot \frac{1 + r_d}{1 + r_f}$

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AFIN8003 Week 13 - Review Banking and Financial Intermediation Dr. Mingze Gao Department of Applied Finance 2025-06-05