Solve the problem.
-A note is due in 4 years, with interest of 4% compounded semiannually. The face value of the note is $26,000. Find the maturity value of the note and find the minimum sale price of the note if money can be deposited at 8% compounded quarterly. Use the compound interest table.

Question

Solve the problem. 
-A note is due in 4 years, with interest of 4% compounded semiannually. The face value of the note is $26,000. Find the maturity value of the note and find the minimum sale price of the note if money can be deposited at 8% compounded quarterly. Use the compound interest table.

Quizplus · Accepted Answer

To find the maturity value of the note, we use the formula $A=P(1+\frac{r}{n})^{nt}$ where $A$ is the maturity value, $P$ is the principal or face value of the note, $r$ is the interest rate, $n$ is the number of times the interest is compounded per year and $t$ is the time period in years.

For this problem, $P=26000$, $r=0.04$ (since the interest rate is given as 4%), $n=2$ (since the interest is compounded semiannually or twice a year), and $t=4$. Plugging these values into the formula, we get:

$A=26000(1+\frac{0.04}{2})^{2(4)}=30463.160148$

So the maturity value of the note is $30463.16$.

To find the minimum sale price of the note, we use the formula $P=\frac{A}{(1+\frac{r}{m})^{mt}}$ where $P$ is the sale price, $A$ is the maturity value, $r$ is the interest rate at which we can invest the money, $m$ is the number of times the interest is compounded per year and $t$ is the time period in years.

For this problem, $A=30463.16$, $r=0.08$ (since the money can be deposited at 8%, compounded quarterly), $m=4$ (since the interest is compounded quarterly or 4 times a year), and $t=4$. Plugging these values into the formula, we get:

$P=\frac{30463.16}{(1+\frac{0.08}{4})^{4(4)}}=22190.890217$

So the minimum sale price of the note is $22190.89$.

Therefore, the answer is C, $30,463.16, $22,190.89. Using the compound interest formula, we can find the maturity value of the note:
$M = P\left(1+\frac{r}{n}ight)^{nt}$
$M = 26000\left(1+\frac{0.04}{2}ight)^{2(4)}$
$M = 30463.16$
Now we need to find the minimum sale price of the note. We can use the compound interest table for this:
First, we need to find the quarterly interest rate:
$i = \sqrt[4]{1+\frac{r}{n}} - 1$
$i = \sqrt[4]{1+\frac{0.08}{4}} - 1$
$i = 0.0196$
Next, we need to find the number of quarters in 4 years:
$t = 4	imes4 = 16$
Using the compound interest table, we can find the factor for 16 quarters at 1.96% quarterly interest:
factor = 1.4594
Finally, we can calculate the minimum sale price of the note:
sale\ price = \frac{maturity\ value}{factor}
sale\ price = \frac{30463.16}{1.4594}
sale\ price = 22190.89
Therefore, the best choice is C, with a maturity value of $30,463.16 and a minimum sale price of $22,190.89.

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