6A.10 Sector valuation I: banks and NBFCs · M06A Events, sectors, factors

Motivation

A bank's assets are loans and securities, carried near their recoverable value; its raw material is capital, and regulation ties how fast it can grow to how much equity it has. Book value is therefore meaningful in a way it is not for a software company, and banks are valued on P/B. M06 derived the justified P/B from the Gordon model (6.18). Here it comes from residual income, which handles changing ROE and makes the role of book explicit.

6A.10.1Residual income and P/B#

Setup

$B_t$ book equity at the end of year $t$ ; $NI_t$ net income; $D_t$ dividends net of new equity; $\text{ROE}_t = NI_t/B_{t-1}$ ; cost of equity $k$ . Clean surplus (M06 (6.2)): $B_t = B_{t-1} + NI_t - D_t$ .

Proposition 6A.14residual income valuation

If $V_0 = \sum_{t \ge 1} D_t/(1+k)^t$ and $B_T/(1+k)^T \to 0$ , then

V_0 = B_0 + \sum_{t=1}^{\infty} \frac{(\text{ROE}_t - k)\,B_{t-1}}{(1+k)^t}. \tag{6A.16}

Proof

From clean surplus, $D_t = NI_t - (B_t - B_{t-1}) = (\text{ROE}_t - k)B_{t-1} + (1+k)B_{t-1} - B_t$ . Divide by $(1+k)^t$ and sum from 1 to $T$ :

\sum_{t=1}^T \frac{D_t}{(1+k)^t} = \sum_{t=1}^T \frac{(\text{ROE}_t - k)B_{t-1}}{(1+k)^t} + \sum_{t=1}^T\left[\frac{B_{t-1}}{(1+k)^{t-1}} - \frac{B_t}{(1+k)^t}\right].

The last sum telescopes to $B_0 - B_T/(1+k)^T$ . Let $T \to \infty$ .∎

Value is book plus the present value of future residual income, the profit above the charge $k B_{t-1}$ for the equity used. A bank that will earn exactly its cost of equity is worth exactly its book, whatever its growth.

Proposition 6A.15steady-state and two-stage P/B

With constant ROE and book growth $g < k$ ,

\frac{V_0}{B_0} = 1 + \frac{\text{ROE} - k}{k - g} = \frac{\text{ROE} - g}{k - g}. \tag{6A.17}

If instead ROE and growth are $\text{ROE}_1, g_1$ for $T$ years and $\text{ROE}_2, g_2 < k$ afterwards,

\frac{V_0}{B_0} = 1 + (\text{ROE}_1 - k)\sum_{t=1}^{T}\frac{(1+g_1)^{t-1}}{(1+k)^t} + \frac{(\text{ROE}_2 - k)(1+g_1)^T}{(k - g_2)(1+k)^T}. \tag{6A.18}

Proof

With constant inputs $B_{t-1} = B_0(1+g)^{t-1}$ , so (6A.16) is $B_0$ plus a growing perpetuity with first term $(\text{ROE} - k)B_0/(1+k)$ and growth $g$ : by M06 (6.8) its value is $(\text{ROE}-k)B_0/(k-g)$ . Combine the fractions for the second form. For the two-stage case, the first $T$ residual incomes use $B_{t-1} = B_0(1+g_1)^{t-1}$ ; from $T+1$ on, residual income is $(\text{ROE}_2 - k)B_T(1+g_2)^{s-1}$ in year $T+s$ , a growing perpetuity worth $(\text{ROE}_2 - k)B_T/(k - g_2)$ at $T$ , discounted by $(1+k)^T$ , with $B_T = B_0(1+g_1)^T$ .∎

Growth multiplies the residual-income spread: $\partial(V/B)/\partial\text{ROE} = 1/(k - g)$ . With $k = 13.5\%$ and $g = 10\%$ that is 28.6, so one point of sustainable ROE is worth 0.29× book.

6A.10.2Drivers of bank ROE#

The ROE tree As a fraction of average total assets $A$ :

\text{ROA} = (\text{NIM} + \text{fees} - \text{opex} - \text{credit cost})(1 - \tau), \qquad \text{ROE} = \text{ROA} \times \frac{A}{B}. \tag{6A.19}

NIM (net interest margin): interest income minus interest expense, over average assets (banks often quote it over interest-earning assets; be consistent).
CASA ratio: current and savings account deposits over total deposits. These pay little or no interest, so a high CASA ratio means cheap funding and a higher NIM. It is the main franchise advantage of large private banks; NBFCs have no CASA and fund themselves from banks and bond markets, which is why their NIMs must come from higher lending yields and why their asset-liability mismatch matters (IL&FS and DHFL, M06 · 6.10).
Credit cost: provisions for loan losses over average assets (or loans). It is the most cyclical line.
Leverage $A/B$ : regulators cap it indirectly through capital adequacy (CRAR, capital to risk-weighted assets).

6A.10.3Asset quality and coverage#

Asset quality For gross loans $L$ : gross NPA ratio $\text{GNPA} = $ non-performing loans$/L$; provisions held against them $= \text{PCR} \times$ GNPA amount, where PCR is the provision coverage ratio; net NPAs are gross NPAs minus provisions.

Proposition 6A.16net NPA ratio

Measuring net NPAs against net loans,

\text{NNPA} = \frac{\text{GNPA}(1 - \text{PCR})}{1 - \text{GNPA}\cdot\text{PCR}}. \tag{6A.20}

Proof

Net NPAs are $gL - p\,gL = gL(1-p)$ with $g$ the GNPA ratio and $p$ the PCR. Net loans are $L - pgL$ . Divide.∎

Raising coverage from $p_0$ to $p_1$ costs $(p_1 - p_0)gL(1 - \tau)$ of equity, i.e. a fraction $(p_1 - p_0)g(L/B)(1-\tau)$ of book. A bank whose PCR is low relative to peers has book value that the market will discount (Yes Bank, M06 · 6.12.3).

6A.10.4Two banks and the limits of book#

Worked example

Two stylised banks (illustrative, not specific companies), $k = 13.5\%$ for both, tax 25.17%:

	NIM	Fees	Opex	Credit cost	$A/B$	ROA	ROE
Private bank	4.0%	1.3%	2.2%	0.6%	8.5	1.87%	15.9%
PSU bank	2.9%	0.9%	1.9%	0.7%	15.0	0.90%	13.5%

The PSU bank reaches a similar ROE only through almost twice the leverage, which is less capital to absorb a credit-cost shock. With ROE 16% and growth 11% a bank is worth $(0.16 - 0.11)/(0.135 - 0.11) = 2.0\times$ book; with ROE 11% and growth 8% it is worth $0.03/0.055 = 0.55\times$ . A market P/B of 3.0 with $g = 11\%$ implies a sustainable ROE of $0.11 + 3.0 \times 0.025 = 18.5\%$ . On asset quality, GNPA 5% with PCR 70% gives NNPA 1.55%; GNPA 2% with PCR 80% gives 0.41%.

Why banks are valued on book, and when not to Book is meaningful when assets are near recoverable value (adequate provisions, no evergreening), when growth is constrained by capital, and when ROE is stable enough to project. It fails when book is wrong (divergence, under-provisioning), when much of the value is fee franchises with little capital (asset managers, brokers), or for insurers whose book reflects accounting conventions; there use embedded value or earnings multiples.

Practitioner notes

For NBFCs add: cost of funds and its spread over the repo rate, ALM buckets (assets and liabilities by maturity; the RBI's structural liquidity statement), share of wholesale funding, and segment credit costs (vehicle, gold, microfinance, unsecured personal loans behave differently through a cycle). For both, read the RBI divergence note and the restructured-book disclosures.