A bank with total assets of $271 million and equity of $31 million has a leverage adjusted duration gap of +0.21 years.One-year maturity notes are currently priced at par and are paying 4.5 percent annually.Two-year maturity notes are currently priced at par and are paying 5 percent annually.The terms of a swap of $100 million notional value of liabilities' payments are 4.95 percent annual fixed payments in exchange for floating rate payments tied to the annual discount yield. What is the forward one-year discount yield expected next year?
A)5.013 percent.
B)5.530 percent.
C)4.500 percent.
D)5.000 percent.
E)4.950 percent.

Question

Quizplus · Accepted Answer

First, we need to calculate the bank's assets and liabilities gap:

Assets gap = (0.21 x equity) / (1 + (0.01 x equity)) = (0.21 x 31) / (1 + (0.01 x 31)) = 6.18 million

Liabilities gap = Total assets - Equity = 271 - 31 = 240 million

Next, we calculate the duration of each note:

One-year note duration = 1 year

Two-year note duration = 2 x (1 + 0.05) / (1 + 0.045)^2 = 1.921 years

Then we calculate the dollar duration of each note:

One-year note dollar duration = (4.5 / (1 + 0.045)) x 100 million = 4.305 million

Two-year note dollar duration = (5 / (1 + 0.05))^2 x 100 million = 8.667 million

Using the dollar duration and duration of each note, we can calculate the sensitivity of assets and liabilities to interest rate changes:

Assets sensitivity = 6.18 million x 0.21 = 1.298 million

Liabilities sensitivity = (4.305 million x 1) + (8.667 million x 1.921) = 21.802 million

Now we can calculate the bank's net sensitivity:

Net sensitivity = Assets sensitivity - Liabilities sensitivity = -20.504 million

Since the net sensitivity is negative, the bank is liability-sensitive, meaning that a decrease in interest rates would increase the bank's net income.

Finally, we can use the information from the swap to calculate the forward one-year discount yield expected next year:

4.95% fixed rate - annual discount yield = expected one-year discount yield

Solving for the annual discount yield, we get:

Annual discount yield = 4.95% - expected one-year discount yield

We know that the bank is liability-sensitive and hence benefits from lower interest rates. Since the fixed rate on the swap is higher than both the current one-year and two-year note rates, we can assume that the expected one-year discount yield is lower than 4.5%. Therefore, we can eliminate choices C and D.

To determine whether the answer is A, B or E, we need to consider the impact of the swap on the bank's net interest income. Since the notional value of the swap is $100 million and the fixed rate is 4.95%, the bank will receive $4.95 million in fixed rate payments and will have to pay floating rate payments based on the annual discount yield. If we assume that the annual discount yield on the swap is the same as the expected one-year discount yield next year, then the net effect on the bank's net interest income will be:

Net effect on net interest income = $4.95 million - (100 million x expected one-year discount yield)

If the net effect is positive, then the bank will benefit from the swap and we can eliminate choice E.

For choice A, we calculate the expected one-year discount yield as:

$4.95 million - (100 million x 5.013%) = -$486,000

The negative sign indicates that the net effect on net interest income is negative, meaning that the bank would lose money on the swap. Therefore, we can eliminate choice A and select choice B as the correct answer.

For choice B, we calculate the expected one-year discount yield as:

$4.95 million - (100 million x 5.53%) = $570,000

The positive sign indicates that the net effect on net interest income is positive, meaning that the bank would benefit from the swap. Therefore, we select choice B as the correct answer.

A Bank with Total Assets of $271 Million and Equity