28.04.2023 Banks II#

Rates#

banks set lending rate \(r > r^p\) policy rate and based on:

  • Risk of default

  • risk tolerance

  • amount of bank equity

  • degree of competition in banking sector

the higher \(r \uparrow \implies L^S \downarrow\) Liquidity Supply lower

Setting the Rate / Liquidity#

based on the interest rate

Formula for Rate $\( r = (1+ \mu^B) \ r^P ,\text{ with mark-up} = \mu^B \)$

Profit of a commercial Bank: $\( V_B = \frac{rL^S - r^p (L^S-D-e)}{e} -\frac{var(v)}{2 \tau}* \frac{L^S}{e}^2 \)$

  • r = lending rate

  • \(r^p\) = policy rate

  • \(L^S\) = credit supply

  • \(D\) = customer deposits

  • \(e\) = equity

  • \(\tau\) = risk tolerance

  • \(v\) = uncertain part of returns on loands

  • \(var(v)\) = riskiness of v

Terms explained:

  • \(rL^S\) = interest return on Loans

  • \(rL^S - r^p (L^S-D-e)\) = financing costs for credit not backed

  • \((\frac{L^S}{e})^2\) = Bank Leverage

  • \(var(v) * \frac{L^S}{e}^2\) = total risk of bank

Refinancing Options of Banks:

  • Interbanking Market

  • loans from CB

  • new deposits

Derivation

What is the optimal credit supply? $\( \frac{\delta V_B}{\delta L^S} = \frac{r-r^p}{e} -\frac{var(v)L^S}{\tau e^2} = 0 \)\( rewrite to determine the mark-up \)\( r - r^p = \frac{var(v)L^S}{\tau e} \\ \implies L^S = \frac{\tau e}{var (v)} * (r-r^p) \)$ logic: when to supply more?

  • higher risk tolerance \(\tau\)

  • higher equity \(e\)

  • larger proft margin (markup) = higher incentive to supply

  • higher risk = lower supply

Demand Liquidity#

credit demand of private sector

\[ I = L^D = \bar{I} - a * r \]

  • I = autonomous / independent demand

  • a = interest rate elasticity of investment demand

    • how flexible are firms to „switch banks“

Equilibrium#

\[ \overbrace{\frac{\tau e}{var (v)} * (r-r^p)}^{Supply} = \overbrace{\bar{I}-a*r}^{Demand} \]

solve for r

\[\begin{split} r [\frac{\tau e}{var (v)}+a] = \frac{r^p \tau e}{var(v)}+ \bar{I} \\ \end{split}\]

leads to:

\[ r = \frac {\frac{r^p \tau e}{var(v)}+ \bar{I}} {\frac{\tau e}{var (v)}+a} = \frac{r^p + \frac{\bar{I}* var(v)}{\tau e}} {1+ \frac{a * var(v)}{\tau e}} \]

Conclusions: interest rate increases:

  1. with higher autonomous demand \(\bar{I}\) :

    • more competition for loans (constant)

  2. with higher policy rate \(r^p\)

    • because of passtrough

  3. lower interest rate elasticity a

    • lower a = firms demand less flexible = less punishable = easier to hike markup

  4. riskiness of loans

  5. lower risk tolerance / lower equity

claims 3+4+5 further proofs needed

Information asymmetry#

  • Moral Hazard (ex post)

  • adverse selection (ex ante)

    • low risks dont want average rate, but high risks do

    • banks react and limit credit

=> banks require equity as incentive

=> in exam: do math and get points

obwohl diese Mathematik extrem kompliziert ist…