Effective Interest Rate On Loans

For the following three questions consider a 250K loan with a 8% nominal interest rate compounded weekly. The term of the loan is 10 years. Per our class contract please assume 12 months in a year, 52 weeks in a year and 4 weeks in a month.

1.) What is the effective annual interest rate of this loan?
a.) 8.00%
b.) 8.32%
c.) 0.67%
d.) None of the above

To find the effective annual interest Rate you begin by using the formula:

(1)
\begin{align} EIR=\left[1+\left(\frac{i}{p}\right)\right]^{n}-1 \end{align}

n = compounding periods in desired period = 52
p = number of compound periods = 52
i = nominal interest rate = 8%

Therefore

(2)
\begin{align} EIR=\left[1+\left(\frac{0.08}{52}\right)\right]^{52}-1 = 0.0832 = 8.32 Percent \end{align}

2.) What would your weekly payments be on this loan?
a.) $738.68
b.) $698.80
c.) $1721.02
d.) 20,000.00

To find the weekly payments you must solve the following equation.

A=(A\P,i,N)


Plugging in known values for P,i,n

A=$250000(A\P,.08/52,52*10)

(3)
\begin{eqnarray} {Weekly Payment Amount} & = & 250000\cdot \left[\frac{(\frac{.08}{52}) \cdot (1+\frac{.08}{52})^{520}}{(1+\frac{.08}{52})^{520}-1} \right]\\& = & \$698.80 \end{eqnarray}

3.) What would your monthly payments be?
a.) $2,083.33
b.) $2795.20
c.) $2,954.73
d.) $20,000.00

To find your monthy payments you must begin by recalculationg your Effective Interest Rate:

(4)
\begin{equation} EIR=(1+(i/p))^n-1$ \end{equation}

n = compounding periods in desired period = 4
p = number of compound periods = 52
i = nominal interest rate = 8%
Therefore

EIR = (1+.08/52)4-1 = .006168 = .6168%

Now to solve to find your montly payments you need to solve for the following:

A=(A\P,i,N)


Plugging in known values for P,i,n

A=$250000(A\P,.006168,12*10)

(5)
\begin{eqnarray} {Monthly Payment Amount} & = & 250000\cdot \left[\frac{\left. (.006168) \cdot (1+ .006168)^{120}\right.} {(1+ .006168)^{120}-1}\right]\\ & = & \$2954.73 \end{eqnarray}
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