jane doe plans to make thirteen end - of - month payments of $12,000 each on a short term investment…

jane doe plans to make thirteen end - of - month payments of $12,000 each on a short term investment account. the account earns a monthly interest rate of 3%. a. what is the present worth (i.e., $p_0$) of these payments? b. repeat part (a) but assuming that they are beginning - of - month payments. a. the present equivalent of the payments is $\\square$. (round to the nearest dollar.) b. the present equivalent of the payments is $\\square$. (round to the nearest dollar.)

jane doe plans to make thirteen end - of - month payments of $12,000 each on a short term investment account. the account earns a monthly interest rate of 3%. a. what is the present worth (i.e., $p_0$) of these payments? b. repeat part (a) but assuming that they are beginning - of - month payments. a. the present equivalent of the payments is $\\square$. (round to the nearest dollar.) b. the present equivalent of the payments is $\\square$. (round to the nearest dollar.)

Answer

Explanation:

Step1: Identify the formula for end - of - month payments (ordinary annuity)

The formula for the present worth $P$ of an ordinary annuity is $P = A\times\frac{1-(1 + i)^{-n}}{i}$, where $A$ is the annuity payment, $i$ is the interest rate per period, and $n$ is the number of periods. Here, $A=$12000$, $i = 0.03$, and $n = 13$.

Step2: Calculate the present worth for end - of - month payments

Substitute the values into the formula: [ \begin{align*} P&=12000\times\frac{1-(1 + 0.03)^{-13}}{0.03}\ &=12000\times\frac{1 - 1.03^{-13}}{0.03}\ \end{align*} ] First, calculate $1.03^{-13}\approx0.68093$. Then $1-0.68093 = 0.31907$. And $\frac{0.31907}{0.03}\approx10.6357$. So $P = 12000\times10.6357=$127628.4\approx$127628$.

Step3: Identify the formula for beginning - of - month payments (annuity due)

The formula for the present worth $P$ of an annuity due is $P = A\times(1 + i)\times\frac{1-(1 + i)^{-n}}{i}$.

Step4: Calculate the present worth for beginning - of - month payments

Substitute $A = 12000$, $i=0.03$, and $n = 13$ into the formula: [ \begin{align*} P&=12000\times(1 + 0.03)\times\frac{1-(1 + 0.03)^{-13}}{0.03}\ &=12000\times1.03\times\frac{1 - 1.03^{-13}}{0.03}\ \end{align*} ] We know from Step 2 that $\frac{1 - 1.03^{-13}}{0.03}\approx10.6357$. So $P=12000\times1.03\times10.6357=12000\times10.954871=$131458.452\approx$131458$.

Answer:

a. $127628$ b. $131458$