Question

20 pm andmother would like to pay your first two years of rent while you're in college by giving you the money you'll need in one lump sum. You share an apartment with another student and currently spend 300 per month on your share of the rent. Assume you receive the lump sum today, and you will raw 5300 per month, starting one month from today and continuing every month thereafter for two years. (PIA, 170, 24) a (o points) If you invest the lump sum in an account that pays 12% per year, compounded money, T o ney should you receive today to fully cover your monthly expenses? rm = 12=1r P=F (PIF, Tin) P = 300 (08274) P=26682. b) (6 points) If you invest the lump sum in an account that pays 12% per year, compounded weekly (52 weeks in one year), how much money should you receive today to fully cover your monthly expenses? 1752373 23-09 lo P=PIA, 23.07%, 3122 {eff = ( c) (8 points) Assume you invested in the account described in part a, and received from your grandmother the amount you calculated in part a. However, beginning in your sophomore year, your rent increases to $600 per month. If you begin withdrawing $600 each month, starting in month 13. how many months will the account fully cover your rent payments? (In other words, you make $300 monthly withdrawals for 12 months, and then make $600 monthly withdrawals as long as possible.) Express your answer as an integer.

Answer 1

A. Let us assume you deposit an amount A today and withdraw $ 300 every month for 2 years.

A = 300(P/A, 1%, 24)

1 - 1.01-24 A = 300 x = 0.01

$\large \implies A = 300\times 21.243387$

$\large \implies A \approx \$ \ 6,373$

Lump sum amount = $ 6,373.

B. First of all calculate the effective rate.

ieft = (1 + inom ) - m

$\large \implies i_{eff} = (1 + \frac{0.12}{52})^{52} - 1$

$\large \implies i_{eff} = 0.12734$

Effective rate = 12.734% per annum

Monthly interest rate = (12.734/12) % = 1.06117% per month

Now calculating the amount required to withdraw $ 300 every month for 2 years

$\large A = 300(P/A,1.06117\%,24)$

$\large \implies A = 300\times \frac{1 - 1.0106117^{-24}}{0.0106117}$

$\large \implies A = 300\times 21.08947$

$\large \implies A \approx \$ \ 6,327$

C. Let us assume you can withdraw $ 600 starting from 13th month for n months.

$\large 6,373.0117 = 300(P/A,1\%,12) + 600(P/F,1\%,12)(P/A,1\%,n)$

$\large \implies 6,373.0117 = 300\times \frac{1-1.01^{-12}}{0.01}+ 600\times 1.01^{-12} \times \frac{1 - 1.01^{-n}}{0.01}$

$\large \implies 6,373.0117 = 300\times 11.25507 + 600\times 0.88744 \times \frac{1 - 1.01^{-n}}{0.01}$

$\large \implies 6,373.0117 = 3,376.5232 + 532.4695 \times \frac{1 - 1.01^{-n}}{0.01}$

$\large \implies 6,373.0117 - 3,376.5232 = 532.4695 \times \frac{1 - 1.01^{-n}}{0.01}$

$\large \implies 532.4695 \times \frac{1 - 1.01^{-n}}{0.01} = 2,996.4929$

$\large \implies 1 - 1.01^{-n}= \frac{2,996.4929\times 0.01}{532.4695}$

$\large \implies 1.01^{-n}= 1 - 0.0562753$

$\large \implies 1.01^{-n}= 0.943724$

$\large \implies 1.01^{n}= 0.943724^{-1}$

$\large \implies 1.01^{n}= 1.059631154$

Taking log on both sides we get

$\large \implies n\times log (1.01)= log(1.059631154)$

$\large \implies n=\frac{ log(1.059631154)}{log(1.01)}$

$\large \implies n= 5.82 \ months$

Problem 3. Loan amount = $ 150,000

The present value of the loan can be written as

150,000 = 25,000(P/A, 6%, 9)-1,000(P/G, 6%, 9)+X(P/F,6%, 10)

$\large \implies 150,000 = 25,000\times \frac{1-1.06^{-9}}{0.06}- 1,000\times \frac{1.06^9 -0.06\times 9 -1}{0.06^2\times 1.06^9}+ X\times 1.06^{-10}$

$\large \implies 150,000 = 25,000\times 6.80169 - 1,000\times 24.5767 + X\times 0.558394$

$\large \implies 150,000 = 170,042.3069 - 24,576.7684 + X\times 0.558394$

$\large \implies 150,000 = 145,465.5385 + X\times 0.558394$

$\large \implies 150,000 - 145,465.5385 = X\times 0.558394$

$\large \implies X\times 0.558394 = 4,534.4615$

$\large \implies X = \frac{4,534.4615}{0.558394}$

$\large \implies X \approx \$ \ 8,220.53$

B. The present value of first 9 years cash flow is

$\large PV = 25,000(P/A,6\%,9) - 1,000(P/G,6\%,9)$

$\large \implies PV = 25,000\times \frac{1-1.06^{-9}}{0.06}- 1,000\times \frac{1.06^9 -0.06\times 9 -1}{0.06^2\times 1.06^9}$

$\large \implies PV = 25,000\times 6.80169 - 1,000\times 24.5767$

$\large \implies PV = 170,042.3069 - 24,576.7684$

$\large \implies PV = 145,465.5385$

Now convert it to annual equivalent

AE = PV (A/P,6\%,9)

=> AE = 145,465.54 × 0.147022

=> AE = $ 21,386.66

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