QuestionFind the present value of a $\$ 5,000$ bonus in 5 years at $5\%$ interest, compounded quarterly. A. $\$ 3,896$ B. $\$ 3,900$ C. $\$ 3,918$ D. $\$ 6,381$

Studdy Solution

STEP 1

Assumptions1. The future value (V) of the bonus is $5,000. The interest rate is5%
3. The time until receipt is5 years4. The interest is compounded quarterly5. We need to find the present value (PV) of the bonus

STEP 2

The formula for calculating the present value (PV) when the interest is compounded quarterly is given by $PV = \frac{V}{(1 + r/n)^{nt}}$ where- FV is the future value- r is the interest rate- n is the number of times interest is compounded per year- t is the time in years

STEP 3

Now, plug in the given values into the formula. The future value (V) is $5,000, the interest rate (r) is5%, the number of times interest is compounded per year (n) is (since it's quarterly), and the time (t) is5 years.
$PV = \frac{\$5,000}{(1 +0.05/)^{ \times5}}$

STEP 4

Convert the percentage to a decimal value.
$\% =0.05$ So the formula becomes $PV = \frac{\$,000}{(1 +0.05/4)^{4 \times}}$

STEP 5

Calculate the value inside the parentheses.
$1 +0.05/4 =1.0125$ So the formula becomes $PV = \frac{\$5,000}{(1.0125)^{4 \times5}}$

STEP 6

Calculate the exponent.
$4 \times5 =20$ So the formula becomes $PV = \frac{\$5,000}{(1.0125)^{20}}$

STEP 7

Calculate the denominator.
$(1.0125)^{20} \approx1.282006737$ So the formula becomes $PV = \frac{\$5,000}{1.282006737}$

STEP 8

Calculate the present value.
$PV = \frac{\$5,000}{1.282006737} \approx \$3,900$ So, the present value of the bonus is approximately $3,900. The closest answer is B.$ \ $3,900$ .

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QuestionFind the present value of a $\$ 5,000$ bonus in 5 years at $5\%$ interest, compounded quarterly. A. $\$ 3,896$ B. $\$ 3,900$ C. $\$ 3,918$ D. $\$ 6,381$

Studdy Solution

STEP 1

Assumptions1. The future value (V) of the bonus is $5,000. The interest rate is5%
3. The time until receipt is5 years4. The interest is compounded quarterly5. We need to find the present value (PV) of the bonus

STEP 2

The formula for calculating the present value (PV) when the interest is compounded quarterly is given by $PV = \frac{V}{(1 + r/n)^{nt}}$ where- FV is the future value- r is the interest rate- n is the number of times interest is compounded per year- t is the time in years

STEP 3

Now, plug in the given values into the formula. The future value (V) is $5,000, the interest rate (r) is5%, the number of times interest is compounded per year (n) is (since it's quarterly), and the time (t) is5 years.
$PV = \frac{\$5,000}{(1 +0.05/)^{ \times5}}$

STEP 4

Convert the percentage to a decimal value.
$\% =0.05$ So the formula becomes $PV = \frac{\$,000}{(1 +0.05/4)^{4 \times}}$

STEP 5

Calculate the value inside the parentheses.
$1 +0.05/4 =1.0125$ So the formula becomes $PV = \frac{\$5,000}{(1.0125)^{4 \times5}}$

STEP 6

Calculate the exponent.
$4 \times5 =20$ So the formula becomes $PV = \frac{\$5,000}{(1.0125)^{20}}$

STEP 7

Calculate the denominator.
$(1.0125)^{20} \approx1.282006737$ So the formula becomes $PV = \frac{\$5,000}{1.282006737}$

STEP 8

Calculate the present value.
$PV = \frac{\$5,000}{1.282006737} \approx \$3,900$ So, the present value of the bonus is approximately $3,900. The closest answer is B.$ \ $3,900$ .

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QuestionFind the present value of a $5,000\$ 5,000$5,000 bonus in 5 years at 5%5\%5% interest, compounded quarterly. A. $3,896\$ 3,896$3,896 B. $3,900\$ 3,900$3,900 C. $3,918\$ 3,918$3,918 D. $6,381\$ 6,381$6,381

Studdy Solution

STEP 1

Assumptions1. The future value (V) of the bonus is $5,000. The interest rate is5%3. The time until receipt is5 years4. The interest is compounded quarterly5. We need to find the present value (PV) of the bonus

STEP 2

The formula for calculating the present value (PV) when the interest is compounded quarterly is given byPV=V(1+r/n)ntPV = \frac{V}{(1 + r/n)^{nt}}PV=(1+r/n)ntV​where- FV is the future value- r is the interest rate- n is the number of times interest is compounded per year- t is the time in years

STEP 3

STEP 4

Convert the percentage to a decimal value.%=0.05\% =0.05%=0.05So the formula becomesPV=$,000(1+0.05/4)4×PV = \frac{\$,000}{(1 +0.05/4)^{4 \times}}PV=(1+0.05/4)4×$,000​

STEP 5

Calculate the value inside the parentheses.1+0.05/4=1.01251 +0.05/4 =1.01251+0.05/4=1.0125So the formula becomesPV=$5,000(1.0125)4×5PV = \frac{\$5,000}{(1.0125)^{4 \times5}}PV=(1.0125)4×5$5,000​

STEP 6

Calculate the exponent.4×5=204 \times5 =204×5=20So the formula becomesPV=$5,000(1.0125)20PV = \frac{\$5,000}{(1.0125)^{20}}PV=(1.0125)20$5,000​

STEP 7

Calculate the denominator.(1.0125)20≈1.282006737(1.0125)^{20} \approx1.282006737(1.0125)20≈1.282006737So the formula becomesPV=$5,0001.282006737PV = \frac{\$5,000}{1.282006737}PV=1.282006737$5,000​

STEP 8

Calculate the present value.PV=$5,0001.282006737≈$3,900PV = \frac{\$5,000}{1.282006737} \approx \$3,900PV=1.282006737$5,000​≈$3,900So, the present value of the bonus is approximately 3,900.TheclosestanswerisB.3,900. The closest answer is B. 3,900.TheclosestanswerisB.\3,9003,9003,900.

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QuestionFind the present value of a $\$ 5,000$ bonus in 5 years at $5\%$ interest, compounded quarterly. A. $\$ 3,896$ B. $\$ 3,900$ C. $\$ 3,918$ D. $\$ 6,381$

Assumptions1. The future value (V) of the bonus is $5,000. The interest rate is5%
3. The time until receipt is5 years4. The interest is compounded quarterly5. We need to find the present value (PV) of the bonus

The formula for calculating the present value (PV) when the interest is compounded quarterly is given by $PV = \frac{V}{(1 + r/n)^{nt}}$ where- FV is the future value- r is the interest rate- n is the number of times interest is compounded per year- t is the time in years

Convert the percentage to a decimal value.
$\% =0.05$ So the formula becomes $PV = \frac{\$,000}{(1 +0.05/4)^{4 \times}}$

Calculate the value inside the parentheses.
$1 +0.05/4 =1.0125$ So the formula becomes $PV = \frac{\$5,000}{(1.0125)^{4 \times5}}$

Calculate the exponent.
$4 \times5 =20$ So the formula becomes $PV = \frac{\$5,000}{(1.0125)^{20}}$

Calculate the denominator.
$(1.0125)^{20} \approx1.282006737$ So the formula becomes $PV = \frac{\$5,000}{1.282006737}$

Calculate the present value.
$PV = \frac{\$5,000}{1.282006737} \approx \$3,900$ So, the present value of the bonus is approximately $3,900. The closest answer is B.$ \ $3,900$ .