✏️ Balance Sheet Reserve Ratio Questions

Note: rather than being core skills that you must be on top of, the examples on this page are relatively advanced. If you have limited time to study, you may want to skip this page.

Occasionally, Bruce will add questions on an exam or a problem set that require involve both a reserve ratio, such as R or E, and a balance sheet. There are generally two types of problems like this:

• In one type, you are given R, E, or R+E, and you need to draw conclusions about a balance sheet.
• In another type, you are given a balance sheet - likely with entries missing - and you need to draw conclusions about R or E.

This type of question can be very challenging, so this page gives you many examples, starting from the least challenging and building up toward more challenging questions.

Equations

We start with some equations, for reference.

Balance sheet equations

Definition of Bank Capital: $\text{Bank Capital} = \text{Assets} - \text{Liabilities}$
With algebra, this implies that the left and right of balance sheet are equal: (must balance ⚖)
$\text{Assets}= \text{Liabilities} + \text{Bank Capital}$

Reserves equations

$\text{Total Reserves} = \text{Required Reserves} + \text{Excess Reserves}$
Interpretation: Some reserves are required, but the total reserves may exceed that. The additional reserves are called excess reserves.
$\text{Total Reserves}=\text{Vault Cash} + \text{Deposits at Fed}$
Both Vault Cash and cash deposited at the Fed count as legal reserves.

Main equations:
$\text{\$Total Reserves} = (\text{R}+\text{E}) \times \text{Deposits}$
  $\frac{\text{\$Total Reserves}}{\text{Deposits}} = \text{R} + \text{E} = \text{Total Reserve Ratio}$

Related equations:
$\text{\$Required Reserves} = \text{R} \times \text{Deposits}$
  $\frac{\text{\$Required Reserves}}{\text{Deposits}} = \text{R} = \text{Required Reserve Ratio}$
$\text{\$Excess Reserves} = \text{E} \times \text{Deposits}$
$\frac{\text{\$Excess Reserves}}{\text{Deposits}} =\text{E}$

Analyzing balance sheets

✏️ Ie, “filling in the blanks”

⚖: $\text{Assets} = \text{Liabilities} + \text{Bank Capital}$

Question 1

Fill in the blanks in the following balance sheet.

Assets	Liabilities
____ Reserves  $.1M Vault Cash  $___ Deposits at Fed $4.5M other assets	$5M Deposits
	$0.5M

Hint: calculate the total Assets, Total Liabilitie and Bank Capital. Then, use $\text{Assets} = \text{Liabilities} + \text{Bank Capital}$ to fill in one of the blanks.

✔ Click here to view answer

Assets (4.5M+Reserves)	Liabilities (L+BC=5.5M)
$1M Reserves  $.1M Vault Cash  $.9M Deposits at Fed $4.5M other assets	$5M Deposits
	$0.5M

Question 2

Fill in the blanks in the following balance sheet.

Assets ()	Liabilities ()
$___M Reserves  $.1M Vault Cash  $.9M Deposits at Fed $4.5M other assets	$5M Deposits
	$___M

✔ Click here to view answer

$\text{Total Reserves} = \text{Vault Cash} + \text{Deposits at Fed}$
Both Vault Cash and cash deposited at the Fed count as legal reserves.

Assets	Liabilities
$1M Reserves  $.1M Vault Cash  $.9M Deposits at Fed $4.5M other assets	$5M Deposits
	$0.5M

Analyzing balance sheets when R and E are given

✏️ Q: Balance sheet with R and E…
The fed requires that banks hold 8% of their deposits as reserves and banks decide to hold excess reserves equal to 7% of their deposits.
$\text{x}= \$20 + \$.01$
Fill in the blank in the following balance sheet.

Assets ()	Liabilities ()
.6M Reserves  $.1M Vault Cash  $.5M Deposits at Fed $4.5M other assets	$4M Deposits
	$___M

✔ Click here to view answer

R=8% and E=7%, so R+E = 15%
You are holding 15% of your deposits as reserves.
Reserves = $15\% \times \$4\text{M}$ = $\$.6\text{M}$
This bank is required to hold $4\text{M} \times 8\%=\$320\text{K}$ of required reserves.
This bank has $4 \times 7 = \$280\text{K}$ of excess reserves.

Assets	Liabilities
$.6M Reserves  $.1M Vault Cash  $.5M Deposits at Fed $4.5M other assets	$4M Deposits
	$1.1M

Bank capital is ALWAYS changing. Remember that Bank Capital is the Net Worth of a bank. Therefore, anything that is profitable increases bank capital. Anything that is unprofitable decreases bank capital.

How much bank capital?

Consider the following transactions occur. How will this affect the balance sheet?

Calculating R or E from a balance sheet

✏️ Suppose $\text{R}=4\%$. What is E?

Assets	Liabilities
$1M Reserves  $.1M Vault Cash  $.9M Deposits at Fed $4.M other assets	$5M Deposits
	$.5M

You want to figure out $\text{R} + \text{E}$. Here is your equation from “Key Topics”
$\frac{(\text{\$Total Reserves})}{\text{Deposits}} = \text{R+E}$
$\frac{1}{5} = 0.2 = 20\% = \text{R} + \text{E}$
But $\text{R}=4\%$, so $\text{E}$ is $16\%$.

What is the money multiplier?

$\text{MM}=\frac{1}{\text{R} + \text{E}}=\frac{1}{4\%+16\%}=5$

Side note:
Suppose you weren’t told that $\text{R}=4\%$.

Is there a way to find the MM?
Recall that we know that $\text{R} + \text{E}=20\%$

Of course!
$\text{MM} = \frac{1}{\text{R} + \text{E}} = \frac{1}{20\%} = 5$

Calculating R or E from a balance sheet

✏️ Banks elect to hold $5\%$ of their deposits as excess reserves.

Here is the balance sheet a First National Bank.

What is the required reserve ratio?

Fill in the blank in the following balance sheet.

Assets ()	Liabilities ()
___ Reserves  $5M Vault Cash  $30M Deposits at Fed $120M other assets	___ Deposits
	$5M

✔ Click here to view answer

Start with the balance sheet:

Assets	Liabilities
$35M Reserves  $5M Vault Cash  $30M Deposits at Fed $120M other assets	$150M Deposits
	$5M

Now we use our favorite equation:
$\text{R} + \text{E} = \frac{\$35\text{M}}{\$150M} = 23.333\%$
$\text{E}$ is given as $5\%$ in the question, so
$R = 18.3333\%$

Additional Practice Problems

✏️ Suppose that Metro Credit Union has the following balance sheet:

Assets	Liabilities
DaF 2m VC ? Loans 40m Other 5m	Checking 40m Bank Capital 15m
Suppose that the required reserve ratio is 10%. What percentage of the banks deposits does it keep as excess reserves?

✔ First, we need to find out what quantity are we trying to find. The question asks for E. It also leaves Vault Cash blank, so we may need to figure that out along the way.

We know Assets = Liabilities + Bank Capital, so the left side of the balance sheet must be equally as large as the right. The right side adds up to $55m, so the left must also. This implies that VC=$8m.

From this, we recall that VC+DaF=Total Reserves, so Total Reserves = $2m+$8m = $10m

This allows us to use the “total reserves as a % of deposits” equation:
$R+E=\frac{\$TotalReserves}{Deposits}=\frac{\$10M}{\$40M}=25\%$ Because R+E is 25% and because R=10%, E must be 15%.

We can check our math using the “Fish Diagram:”

Another practice problem

✏️

Assets	Liabilities
DaF ? VC 5m Loans 40m Other 5m	Checking 40m Other Liabilities? Bank Capital 3m
Suppose that R = 10% and E=20%. How many other assets does the bank hold?

✔We are holding R+E=30% of our deposits as reserves. This is $40m × 30% = $12m of total reserves. (We used Deposits × (R+E) = $TotalReserves to find this.)

VC + DaF = TotalReserves, so $5m + DaF = $12m Therefore, DaF = $7m.

Assets = Liabilities + BankCapital Assets = 7m + 5m + 40m + 5m = 57m Therefore Liabilities = Assets - BC = 57m - 3m = 54m Other liabilities must therefore be $14m, because 40m + 14m = 54m = total liabilities.