Question 1166021: In 2008 a major bank in the US borrowed 10 million dollars for the US government at an annual rate of 0.50% compounded annually. They then used that money to purchase US government 30 year bonds (investment) that yields 4.65% compounded annually? How much did this cost the US taxpayers in the first year? [It will be the profit the bank makes] (use your financial application and fill in the appropriate inputs)
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! This is an interesting question about the financial spread (profit) a bank makes by borrowing money at a low rate and investing it at a higher rate. We can calculate the profit by finding the difference between the interest earned and the interest paid over the first year.
## 💰 Bank Profit Calculation (First Year)
The cost to U.S. taxpayers is the **profit** the bank makes on the \$10 million borrowed, specifically the difference between the interest the bank *earns* and the interest the bank *pays* in the first year.
### Given Inputs
* **Principal ($P$):** \$10,000,000
* **Time ($t$):** 1 year
* **Borrowing Rate ($r_b$):** 0.50% compounded annually
* **Investment Rate ($r_i$):** 4.65% compounded annually
Since the compounding period is annual, the interest calculated is the final interest for the first year.
### 1. Interest Paid (Cost of Borrowing)
The bank pays interest to the U.S. government at $0.50\%$.
$$\text{Interest Paid} = P \times r_b \times t$$
$$\text{Interest Paid} = \$10,000,000 \times 0.0050 \times 1$$
$$\text{Interest Paid} = \mathbf{\$50,000}$$
### 2. Interest Earned (Return from Investment)
The bank earns interest from the 30-year U.S. bonds at $4.65\%$.
$$\text{Interest Earned} = P \times r_i \times t$$
$$\text{Interest Earned} = \$10,000,000 \times 0.0465 \times 1$$
$$\text{Interest Earned} = \mathbf{\$465,000}$$
### 3. Cost to U.S. Taxpayers (Bank Profit)
The profit (or the cost to taxpayers) is the difference between the interest earned and the interest paid.
$$\text{Profit} = \text{Interest Earned} - \text{Interest Paid}$$
$$\text{Profit} = \$465,000 - \$50,000$$
$$\text{Profit} = \mathbf{\$415,000}$$
The cost to U.S. taxpayers in the first year was **\$415,000**.
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