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Math  /  Algebra

QuestionInvestment XX offers to pay you $5,400\$ 5,400 per year for nine years, whereas Investment YY offers to pay you $7,700\$ 7,700 per year for five years. a. Calculate the present value for Investments XX and YY if the discount rate is 6 percent. (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.) b. Calculate the present value for Investments XX and YY if the discount rate is 16 percent. (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 3216.) \begin{tabular}{|c|c|c|} \hline a. Investment XX & \ & 36,728.98 \\ \hline a. Investment Y & \$ & 32,434.23 \\ \hline b. Investment X & \$ & 25,179.82 \\ \hline b. Investment Y$ & \$ & 25,211.49 \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The present value (PV) calculation requires the use of the formula for the present value of an annuity.
2. The formula for the present value of an annuity is given by PV=PMT×(1−(1+r)−n)/r PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) / r , where PMT PMT is the annual payment, r r is the discount rate, and n n is the number of years.
3. All calculations should be done without rounding intermediate steps, and the final answers should be rounded to 2 decimal places.

STEP 2

1. Calculate the present value of Investment X X for a discount rate of 6%.
2. Calculate the present value of Investment Y Y for a discount rate of 6%.
3. Calculate the present value of Investment X X for a discount rate of 16%.
4. Calculate the present value of Investment Y Y for a discount rate of 16%.

STEP 3

Calculate the present value of Investment X X for a discount rate of 6%.
Using the formula PV=PMT×(1−(1+r)−n)/r PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) / r :
PMT=5400 PMT = 5400 r=0.06 r = 0.06 n=9 n = 9
PVX=5400×(1−(1+0.06)−9)/0.06 PV_X = 5400 \times \left(1 - \left(1 + 0.06\right)^{-9}\right) / 0.06

STEP 4

Calculate the expression inside the parentheses:
(1+0.06)−9=(1.06)−9 \left(1 + 0.06\right)^{-9} = \left(1.06\right)^{-9}

STEP 5

Evaluate (1.06)−9 \left(1.06\right)^{-9} :
(1.06)−9≈0.59345 \left(1.06\right)^{-9} \approx 0.59345

STEP 6

Substitute back into the formula:
PVX=5400×(1−0.59345)/0.06 PV_X = 5400 \times \left(1 - 0.59345\right) / 0.06

STEP 7

Simplify the expression:
PVX=5400×0.40655/0.06 PV_X = 5400 \times 0.40655 / 0.06

STEP 8

Calculate the final value:
PVX≈5400×6.77583≈36,728.98 PV_X \approx 5400 \times 6.77583 \approx 36,728.98

STEP 9

Calculate the present value of Investment Y Y for a discount rate of 6%.
Using the formula PV=PMT×(1−(1+r)−n)/r PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) / r :
PMT=7700 PMT = 7700 r=0.06 r = 0.06 n=5 n = 5
PVY=7700×(1−(1+0.06)−5)/0.06 PV_Y = 7700 \times \left(1 - \left(1 + 0.06\right)^{-5}\right) / 0.06

STEP 10

Calculate the expression inside the parentheses:
(1+0.06)−5=(1.06)−5 \left(1 + 0.06\right)^{-5} = \left(1.06\right)^{-5}

STEP 11

Evaluate (1.06)−5 \left(1.06\right)^{-5} :
(1.06)−5≈0.74726 \left(1.06\right)^{-5} \approx 0.74726

STEP 12

Substitute back into the formula:
PVY=7700×(1−0.74726)/0.06 PV_Y = 7700 \times \left(1 - 0.74726\right) / 0.06

STEP 13

Simplify the expression:
PVY=7700×0.25274/0.06 PV_Y = 7700 \times 0.25274 / 0.06

STEP 14

Calculate the final value:
PVY≈7700×4.21233≈32,434.23 PV_Y \approx 7700 \times 4.21233 \approx 32,434.23

STEP 15

Calculate the present value of Investment X X for a discount rate of 16%.
Using the formula PV=PMT×(1−(1+r)−n)/r PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) / r :
PMT=5400 PMT = 5400 r=0.16 r = 0.16 n=9 n = 9
PVX=5400×(1−(1+0.16)−9)/0.16 PV_X = 5400 \times \left(1 - \left(1 + 0.16\right)^{-9}\right) / 0.16

STEP 16

Calculate the expression inside the parentheses:
(1+0.16)−9=(1.16)−9 \left(1 + 0.16\right)^{-9} = \left(1.16\right)^{-9}

STEP 17

Evaluate (1.16)−9 \left(1.16\right)^{-9} :
(1.16)−9≈0.2327 \left(1.16\right)^{-9} \approx 0.2327

STEP 18

Substitute back into the formula:
PVX=5400×(1−0.2327)/0.16 PV_X = 5400 \times \left(1 - 0.2327\right) / 0.16

STEP 19

Simplify the expression:
PVX=5400×0.7673/0.16 PV_X = 5400 \times 0.7673 / 0.16

STEP 20

Calculate the final value:
PVX≈5400×4.79563≈25,179.82 PV_X \approx 5400 \times 4.79563 \approx 25,179.82

STEP 21

Calculate the present value of Investment Y Y for a discount rate of 16%.
Using the formula PV=PMT×(1−(1+r)−n)/r PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) / r :
PMT=7700 PMT = 7700 r=0.16 r = 0.16 n=5 n = 5
PVY=7700×(1−(1+0.16)−5)/0.16 PV_Y = 7700 \times \left(1 - \left(1 + 0.16\right)^{-5}\right) / 0.16

STEP 22

Calculate the expression inside the parentheses:
(1+0.16)−5=(1.16)−5 \left(1 + 0.16\right)^{-5} = \left(1.16\right)^{-5}

STEP 23

Evaluate (1.16)−5 \left(1.16\right)^{-5} :
(1.16)−5≈0.4761 \left(1.16\right)^{-5} \approx 0.4761

STEP 24

Substitute back into the formula:
PVY=7700×(1−0.4761)/0.16 PV_Y = 7700 \times \left(1 - 0.4761\right) / 0.16

STEP 25

Simplify the expression:
PVY=7700×0.5239/0.16 PV_Y = 7700 \times 0.5239 / 0.16

STEP 26

Calculate the final value:
PVY≈7700×3.27438≈25,211.49 PV_Y \approx 7700 \times 3.27438 \approx 25,211.49
\begin{tabular}{|c|c|c|} \hline a. Investment XX & \ & 36,728.98 \\ \hline a. Investment Y & \$ & 32,434.23 \\ \hline b. Investment X & \$ & 25,179.82 \\ \hline b. Investment Y$ & \$ & 25,211.49 \\ \hline \end{tabular}

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