NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to "at least" $5$ decimal places.
Round your final answer to $2$ decimal places.
Make sure you include your unit.
Solve all questions.
Use at least two methods where applicable.
Show all work.
For NSC Students For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits
from
behind.
Any comma included in a number indicates a decimal point. For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.
(1.) Determine the effective rate of interest for a nominal rate of $5\%$ per year compounded:
(a.) Annually
(b.) Semiannually
(c.) Quarterly
(d.) Monthly
(e.) Continuously
(2.) Three banks listed the following money market accounts: East Bank: $4.36\%$ compounded monthly South Bank: $4.35\%$ compounded daily North Bank: $4.31\%$ compounded continuously
(a.) Determine the annual percentage yield.
(b.) Which of the banks has the highest return?
(3.) A loan company wants to offer a CD (Certificate of deposit) with a monthly company rate
that has an APY of $7.5\%$.
What annual nominal rate compounded monthly should they use?
(4.) Mark bought a thousand shares for twenty eight thousand dollars including commissions.
Two years later, he sold the shares for thirty four thousand and two hundred dollars after deducting
commissions.
Calculate the effective rate of return on his investment over the two-year period.
$
r = ? \\[3ex]
APY = ? \\[3ex]
\underline{Compound\:\:Interest} \\[3ex]
m = 1 \\[3ex]
A = \$34200 \\[3ex]
P = \$28000 \\[3ex]
t = 2\:years \\[3ex]
r = m\left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{mt}} - 1\right] \\[7ex]
r = 1 * \left[\left(\dfrac{34200}{28000}\right)^{\dfrac{1}{1 * 2}} - 1\right] \\[7ex]
r = \left[(1.221428571)^{\dfrac{1}{2}} - 1\right] \\[5ex]
r = \left[1.221428571^{0.5} - 1\right] \\[5ex]
r = 1.105182596 - 1 \\[3ex]
r = 0.105182596 \\[3ex]
\underline{APY} \\[3ex]
r = 0.105182596 \\[3ex]
m = 1 \\[3ex]
APY = \left(1 + \dfrac{r}{m}\right)^m - 1 \\[5ex]
APY = \left(1 + \dfrac{0.105182596}{1}\right)^1 - 1 \\[5ex]
APY = (1 + 0.105182596)^1 - 1 \\[3ex]
APY = (1.105182596)^1 - 1 \\[3ex]
APY = 1.105182596 - 1 \\[3ex]
APY = 0.105182596 \\[3ex]
to\:\:percent = 0.105182596 * 100 \\[3ex]
= 10.5182596\% \\[3ex]
APY \approx 10.5183\%
$
(5.) Rita invested $\$10,000$ in a mutual fund that pays interest on a daily basis.
The balance in her account at the end of $8$ months ($245$ days) was $\$10,475.25$
Find the effective rate at which Rita's account earned interest over this period.
Assume a $365-day$ year.
$
r = ? \\[3ex]
APY = ? \\[3ex]
\underline{Compound\:\:Interest} \\[3ex]
m = 365 \\[3ex]
A = \$10475.25 \\[3ex]
P = \$10000 \\[3ex]
t = \dfrac{245}{365}\:years \\[5ex]
r = m\left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{mt}} - 1\right] \\[7ex]
r = 365 * \left[\left(\dfrac{10475.25}{10000}\right)^{\dfrac{1}{365 * \dfrac{245}{365}}} - 1\right]
\\[10ex]
r = 365 * \left[(1.047525)^{\dfrac{1}{245}} - 1\right] \\[5ex]
r = 365 * \left[1.047525^{0.004081632653} - 1\right] \\[5ex]
r = 365 * [1.000189529 - 1] \\[5ex]
r = 365 * 0.0001895291375 \\[3ex]
r = 0.06917813518 \\[3ex]
\underline{APY} \\[3ex]
r = 0.06917813518 \\[3ex]
m = 365 \\[3ex]
APY = \left(1 + \dfrac{r}{m}\right)^m - 1 \\[5ex]
APY = \left(1 + \dfrac{0.06917813518}{365}\right)^{365} - 1 \\[5ex]
APY = (1 + 0.0001895291375)^{365} - 1 \\[3ex]
APY = (1.000189529)^{365} - 1 \\[3ex]
APY = 1.071620062 - 1 \\[3ex]
APY = 0.07162006234 \\[3ex]
to\:\:percent = 0.07162006234 * 100 \\[3ex]
= 7.162006234\% \\[3ex]
APY \approx 7.1620\%
$
Formulas Used: Mathematics of Finance