Question

You are the bank manager of a bank with 10m. capital and 90m. deposits.

Deposits cost you 1% Choice of two assets: Safe asset earning 5% risk free Risky asset earning 25% and loses 20% with equal probabilities

Calculate the expected ROA and ROE for both choices (Hint: Calculate the profits net of deposit interests for the good / bad scenario first. Remember: For owners the loss is limited toequity).

If the bank only holds the risk free asset, its ROA is ____% and the ROE is ____%. If the bank only holds the risky asset, its ROA is ____% and the ROE is ____%. To maximize share holder return the bank manager will chose the (risk free or risky?) asset. Would the depositors be happy with 1% interest if they knew what the banker is doing? The interest rate on deposits would (increase / decrease) if the depositors had perfect information about the bank's risk profile. Repeat your calculations for the case that the bank uses 30m capital and only 70m deposits. Investing in the risky asset would now have an expected ROA of________ % and ROE of ___________%. Sit back and think of the importance of bank equity for the choices that bank managers will make. Which is better for society? Can you see why we need bank regulation?

Answer 1

Case 1: Capital - 10 Mn Deposits - 90 Mn

Case A: If we go for risk free asset, Return = 5% * Total Assets = 5% * (Capital + Deposits) = 5% * 100 = 5 Mn

Depositors Interest Payout = 1% * 90 Mn = 0.9 Mn

Net Profit to bank/equity shareholders = 5 - 0.9 = 4.1 Mn

Hence, Return on Asset (RoA) = Net Profit to Bank / Total Asset = 4.1 / 100 = 4.1%

Return on Equity(RoE) = Net Profit to Equity Shareholders / Capital = 4.1/10 = 41%

Case B: If we go for Risky Asset, we have 50% chance of gaining 25% & 50% chance of losing 20%. Thus, our net return is computed by weighted average of two return = 50* 25% - 50%* 20% = 2.5%

Thus Return = 2.5% * Total Assets = 2.5% * (Capital + Deposits) = 2.5% * 100 = 2.5 Mn

Net Profit to bank = 2.5 - 0.9 = 1.6 Mn

Hence, Return on Asset = Net Profit to Bank / Total Asset = 1.6 / 100 = 1.6%

In this case as there is a case of 20% loss, then at 20% loss complete capital is eroded away & hence, equity investors suffer loss of 10 Mn only. But the excess loss will not pass through to the equity shareholders. In the case of 10 Mn loss no depositor interest would need to be given by bank as it will declare bankruptcy but in 25 Mn profit case it will pay back 0.9 Mn. Hence, their gain will range from -10 Mn to 24.1 Mn with 50% probability to both.

Net profit to equity shareholders = (24.1 - 10 ) *50% = 7.05

Return on Equity = Net Profit to Equity Shareholders / Capital = 7.05/10 = 70.5%

Hence, to maximize the shareholder return banker will choose the risky asset even though the RoA overall is less than RoE in this case.

Here, we see that in both cases RoA is positive for the bank with alot of value being generate for the bank. Hence, if depositors knew about the value creation disbalance, they would have been unhappy with getting only 1% interest rate & would have asked for higher returns.

Case 1: Capital - 30 Mn Deposits - 70 Mn

Case A: If we go for risk free asset, Return = 5% * Total Assets = 5% * (Capital + Deposits) = 5% * 100 = 5 Mn

Depositors Interest Payout = 1% * 70 Mn = 0.7 Mn

Net Profit to bank/equity shareholders = 5 - 0.7 = 4.3 Mn

Hence, Return on Asset (RoA) = Net Profit to Bank / Total Asset = 4.3 / 100 = 4.3%

Return on Equity(RoE) = Net Profit to Equity Shareholders / Capital = 4.3/30 = 14.3%

Case B: If we go for Risky Asset, we have 50% chance of gaining 25% & 50% chance of losing 20%. Thus, our net return is computed by weighted average of two return = 50* 25% - 50%* 20% = 2.5%

Thus Return = 2.5% * Total Assets = 2.5% * (Capital + Deposits) = 2.5% * 100 = 2.5 Mn

Net Profit to bank = 2.5 - 0.7 = 1.8 Mn

Hence, Return on Asset = Net Profit to Bank / Total Asset = 1.8 / 100 = 1.8%

In this case as capital is higher not depositors do not suffer & hence don't feel the loss of principal or the interest payments. Hence, bank's gain will range from -18 Mn to 25 Mn with 50% probability to both. In both cases it will pay out 0.7 Mn as depositor's interest payment.

Net profit to equity shareholders = (25 - 18 ) *50% - 0.7 = 2.8

Return on Equity = Net Profit to Equity Shareholders / Capital = 2.8/30 = 9.33%

Thus, we see that increasing the equity rapidly reduces the RoE measure for the bank but improves the RoA. This is so because of the risk diversion & more risk being taken on now by the bank in the operations. Hence, from a society point of view risk sharing is more beneficial due to the increase of overall return measure (RoA).

As banks would prefer to maximize shareholder returns, hence they'll try to get away with minimum capital involvement. Hence, there is a need for banking regulations to keep a certain check in terms of minimum required capital. It is done by the use of ratios like capital adequacy ratio.