A 6-year, $1650.00 note bearing interest at 9.51% compounded annually was discounted at 11.2% compounded semi-annually yielding proceeds of $1916.75. How many months before the due date does this discounting happen ? (As explained in class, find the Future Value or Maturity Value of the promissory note FIRST before you calculate its  Present Discounted Value; the discounting factor is 11.2% compounded semi-annually).

To solve this problem, we need to calculate two things:

1. The **Maturity Value (Future Value, FV)** of the promissory note after 6 years at 9.51% interest compounded annually.
2. The **Present Discounted Value (PDV)**, given that it was discounted at 11.2% interest compounded semi-annually, which will tell us how many months before the due date the discounting occurred.

### Step 1: Calculate the Maturity Value (FV)

Given:
- Principal (P): \$1650.00
- Interest rate: 9.51% compounded annually
- Time (t): 6 years

The formula for Future Value (FV) compounded annually is:

\[
FV = P \times \left(1 + \frac{r}{100}\right)^t
\]

Where:
- $P = 1650$
- $r = 9.51$
- $t = 6$

\[
FV = 1650 \times \left(1 + \frac{9.51}{100}\right)^6
\]

Let's calculate this first:

\[
FV = 1650 \times (1.0951)^6
\]The Maturity Value (Future Value, $FV$) of the promissory note after 6 years is approximately **\$2845.81**.

### Step 2: Calculate the Present Discounted Value (PDV)

The Present Discounted Value (PDV) is given by the formula:

\[
PDV = FV \times \left(1 + \frac{i}{n}\right)^{-nt}
\]

Where:
- $FV = 2845.81$
- Discount rate $i = 11.2\%$
- Compounding frequency $n = 2$ (semi-annually)
- $nt$ represents the number of compounding periods remaining until the note is due

Given that the proceeds are $1916.75$, we set the equation for the proceeds equal to the PDV:

\[
1916.75 = 2845.81 \times \left(1 + \frac{11.2}{2 \times 100}\right)^{-2t}
\]

Let's solve for $t$ (in years) first and then convert it to months:

\[
1916.75 = 2845.81 \times \left(1 + \frac{11.2}{200}\right)^{-2t}
\]

\[
\frac{1916.75}{2845.81} = \left(1.056\right)^{-2t}
\]

Taking the logarithm to solve for $t$:

\[
\log\left(\frac{1916.75}{2845.81}\right) = -2t \cdot \log(1.056)
\]

\[
t = \frac{\log\left(\frac{1916.75}{2845.81}\right)}{-2 \cdot \log(1.056)}
\]

Let's calculate the value of $t$.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, I can guide you through the steps manually if you'd like to continue solving this problem together. Here’s a summary of where we are:

1. We found the Future Value (FV) of the note to be approximately \$2845.81 after 6 years.
2. Next, to find the number of months before the due date when the discounting occurs, we need to solve for $t$ in the equation:

\[
\frac{1916.75}{2845.81} = (1.056)^{-2t}
\]

Taking logarithms:

\[
t = \frac{\log\left(\frac{1916.75}{2845.81}\right)}{-2 \cdot \log(1.056)}
\]

To compute this manually, you'll need to use a calculator for logarithms.

Would you like a step-by-step explanation for calculating this manually, or do you have any questions about the steps involved? 

### Follow-Up Questions:
1. How do we convert compounding rates to effective annual rates?
2. How can you find the present value of a sum with continuous compounding?
3. What are some common real-world applications of discounting future cash flows?
4. How does changing the compounding frequency affect the present discounted value?
5. What is the difference between simple interest and compound interest?

**Tip:** When dealing with logarithmic equations, always remember to check the base of the logarithm you're using.

A 6-year, $1650.00 note bearing interest at 9.51% compounded annually was discounted at 11.2% compounded semi-annually yielding proceeds of $1916.75. How many months before the due date does this discounting happen?

Solve how many months before the due date a 6-year, $1650.00 promissory note bearing 9.51% interest compounded annually was discounted at 11.2% compounded semi-annually, yielding proceeds of $1916.75. Includes step-by-step calculations for Future Value and Present Discounted Value using logarithms.

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