Question

Consider a bank with the following balance sheet (M means million):

Assets	Value	Duration of the Asset	Convexity of the Asset
5yr bondbought at a yield of 3.4% (lending money)	$550M	4.562	12.026
12yr bondbought at a yield of 4% (lending money)	$800M	9.453	53.565

Liabilities	Value	Duration of the Liability	Convexity of the Liability
2yr bondsold at a yield of 2.4% (borrowing money)	$300M	1.941	2.384
4yr bondsold at a yield of 2.8% (borrowing money)	$500M	3.759	8.206

1. Calculate the equity (total asset – total liability) to asset ratio of the bank

(Hint: equity to asset ratio = total equity/total asset)

2. Calculate the duration and convexity of the both asset and liability sides;
3. If the interest rates go up by 1%, using the duration and convexity rule to determine the net worth of the bank and the equity to asset ratio;
4. In c)’s scenario, to maintain the equity to asset ratio at 40% which is required by the regulation, the bank decides to raise cash (zero duration and zero convexity) from the equity holders. How much cash does the bank need to raise?
5. Do you agree with the following statement? Explain why.

“The information about a bond’s duration and convexity adjustment is sufficient to quantify interest rate risk exposure.”

Answer 1

Ans a) Equity to Total Asset Ratio :

Equity = Total Assets - Total Liability  

Total Assets = Sum of Value of All Bonds Bought = 550 + 800 = $ 1350 M

Total Liability = Sum of Value of All Bonds Sold = 300 + 500 = $ 800 M

Equity =1350 - 800 = $ 550 M

Now Equity to Total Asset Ratio = Equity /  Total Assets = 550 / 1350 = 0.4074

Equity to Total Asset Ratio 0.4074 (Ans)

Ans b)  duration and convexity of the both asset and liability sides :

Effective Duration = Weightage of Bonds in Assets/ Liability * Duration of Each Bonds

Effective Convexity = = Weightage of Bonds in Assets/ Liability * Convexity of Each Bonds

Weightage of Bonds in Assets/ Liability = Bond Value / Total Asset or  Total Liability

duration of Assets = Sum of Effective Duration of Bonds in Each Assets Side = 1.859 + 5.602 = 7.460 (Ans)

duration of liability = Sum of Effective Duration of Bonds in Each liability Side = 0.728 + 2.349 = 3.077 (Ans)

Convexity of Assets =  Sum of Effective Convexity of Bonds in Each Assets Side = 4.899 + 31.742 = 36.642 (Ans)

Convexity of liability = Sum of Effective Convexity of Bonds in Each liability Side = 0.894 + 5.129 = 6.023 (Ans)

Ans c)

Rate goes Up by 01% = Change in Interest Rate

% Change in Price of Bonds = - Duration of Assets * (Change in Interest Rate ) + 0.5 * Convexity of Asset * (Change in Interest Rate ) ^2

Value of New Bonds = Previous Value + % Change in Price of Bonds * Previous Value

Net Worth of Bank = New Total Assets - New Total Liability

= New Sum of Value of All Bonds Bought   - New Sum of Value of All Bonds Sold

= 1251.758 - 775.623 = 476.135

New Net Worth of Bank = 476.135  (Ans)

New Equity to Total Asset Ratio = New Equity / New Total Assets = 476.135 / 1251.758 = 0.3803 (Ans)

Ans d)

Now Equity to Assets Ratio Required is 40%.

Equity Value Required by the regulation =  Equity to Assets Ratio * Total Asset = 40% * 1251.758 = 500.703

The shortfall in Equity = Equity Value Required by the regulation - Current Equity / Net Worth = 500.703 - 476.135 = 24.567

Ans : the bank need to raise $ 24.567 M (Ans)