The price of $8,000 face value commercial paper is $7,930. If the annualized discount rate is 4%, when will the paper mature? If the annualized investment rate is 4%, when will the paper mature?

To solve the problem, we need to determine the maturity period under two different scenarios: 

1. **When the annualized discount rate is 4%.**
2. **When the annualized investment rate is 4%.**

We'll use different formulas to calculate the maturity period in each case.

### 1. Maturity with Annualized Discount Rate of 4%
The formula for the price of a commercial paper when using the discount rate is:

\[
P = F \times \left(1 - \frac{d \times t}{360}\right)
\]

Where:
- $ P $ = Price of the paper ($7,930)
- $ F $ = Face value of the paper ($8,000)
- $ d $ = Annualized discount rate (4% or 0.04)
- $ t $ = Time to maturity in days

We need to solve for $ t $. Rearranging the formula:

\[
t = \frac{360 \times (F - P)}{d \times F}
\]

Substitute the known values:

\[
t = \frac{360 \times (8,000 - 7,930)}{0.04 \times 8,000}
\]

\[
t = \frac{360 \times 70}{0.04 \times 8,000}
\]

\[
t = \frac{25,200}{320} = 78.75 \text{ days}
\]

So, the maturity period when the annualized discount rate is 4% is approximately **79 days**.

### 2. Maturity with Annualized Investment Rate of 4%
The formula for the price of a commercial paper when using the investment rate is:

\[
P = \frac{F}{\left(1 + \frac{r \times t}{360}\right)}
\]

Where:
- $ r $ = Annualized investment rate (4% or 0.04)
- $ t $ = Time to maturity in days

Rearranging the formula to solve for $ t $:

\[
t = \frac{360 \times \left(\frac{F}{P} - 1\right)}{r}
\]

Substitute the known values:

\[
t = \frac{360 \times \left(\frac{8,000}{7,930} - 1\right)}{0.04}
\]

First, calculate $\frac{8,000}{7,930}$:

\[
\frac{8,000}{7,930} \approx 1.0088
\]

Then:

\[
t = \frac{360 \times (1.0088 - 1)}{0.04}
\]

\[
t = \frac{360 \times 0.0088}{0.04} = \frac{3.168}{0.04} = 79.2 \text{ days}
\]

So, the maturity period when the annualized investment rate is 4% is approximately **79 days**.

### Conclusion:
In both cases (whether using the discount rate or the investment rate), the commercial paper will mature in approximately **79 days**.

---

Would you like more details or further clarification on this? Here are 5 related questions to consider:

1. How do the discount rate and investment rate affect the pricing of commercial paper?
2. What is the significance of using a 360-day year in these calculations?
3. How would the maturity period change if the discount rate were different?
4. What is the difference between discount yield and investment yield?
5. Can you apply similar calculations to other financial instruments like Treasury bills?

**Tip**: The 360-day convention is commonly used in finance to simplify interest rate calculations, especially for short-term instruments like commercial paper.

Learn how to calculate the maturity period for an $8,000 face value commercial paper priced at $7,930, using both a 4% annualized discount rate and a 4% investment rate. This problem uses key financial formulas for time to maturity and is suitable for undergraduate finance students.

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