Solved on Feb 07, 2024

Find the present value of a firm's 3-year cash flow stream with $5,000,$ 25,000, and $14,000 at 8% discount rate.

STEP 1

Assumptions
1. The cash flows occur at the end of each year for three years.
2. The cash flows are as follows: $5,000 at the end of year 1,$ 25,000 at the end of year 2, and $14,000 at the end of year 3.<br />3. The firm's opportunity cost is 8%.<br />4. The present value (PV) is calculated using the formula for the present value of a future cash flow, which is$ PV = \frac{CF}{(1 + r)^n} $, where$ CF $is the cash flow,$ r $is the discount rate (opportunity cost), and$ n$ is the number of periods until the cash flow occurs.

STEP 2

First, we will calculate the present value of the cash flow at the end of year 1.
$PV_1 = \frac{CF_1}{(1 + r)^1}$

STEP 3

Now, plug in the given values for the cash flow at the end of year 1 ( $CF_1$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 1.
$PV_1 = \frac{\$5,000}{(1 + 0.08)^1}$

STEP 4

Calculate the present value for the cash flow at the end of year 1.
$PV_1 = \frac{\$5,000}{1.08}$

STEP 5

Now, we will calculate the present value of the cash flow at the end of year 2.
$PV_2 = \frac{CF_2}{(1 + r)^2}$

STEP 6

Plug in the given values for the cash flow at the end of year 2 ( $CF_2$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 2.
$PV_2 = \frac{\$25,000}{(1 + 0.08)^2}$

STEP 7

Calculate the present value for the cash flow at the end of year 2.
$PV_2 = \frac{\$25,000}{1.08^2}$

STEP 8

Now, we will calculate the present value of the cash flow at the end of year 3.
$PV_3 = \frac{CF_3}{(1 + r)^3}$

STEP 9

Plug in the given values for the cash flow at the end of year 3 ( $CF_3$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 3.
$PV_3 = \frac{\$14,000}{(1 + 0.08)^3}$

STEP 10

Calculate the present value for the cash flow at the end of year 3.
$PV_3 = \frac{\$14,000}{1.08^3}$

STEP 11

Now, we will add the present values of all the cash flows to find the total present value of the stream of cash flows.
$Total\, PV = PV_1 + PV_2 + PV_3$

STEP 12

Plug in the calculated present values for each year to calculate the total present value.
$Total\, PV = \frac{\$5,000}{1.08} + \frac{\$25,000}{1.08^2} + \frac{\$14,000}{1.08^3}$

STEP 13

Calculate the total present value.
$Total\, PV = \frac{\$5,000}{1.08} + \frac{\$25,000}{(1.08)^2} + \frac{\$14,000}{(1.08)^3}$
$Total\, PV = \frac{\$5,000}{1.08} + \frac{\$25,000}{1.1664} + \frac{\$14,000}{1.259712}$
$Total\, PV = \$4,629.63 + \$21,433.47 + \$11,116.28$
$Total\, PV = \$37,179.38$
The present value of the firm's cash flow stream, assuming an opportunity cost of 8%, is approximately \$37,179.38.

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Find the present value of a firm's 3-year cash flow stream with $5,000,$ 25,000, and $14,000 at 8% discount rate.

STEP 1

STEP 2

First, we will calculate the present value of the cash flow at the end of year 1.
$PV_1 = \frac{CF_1}{(1 + r)^1}$

STEP 3

Now, plug in the given values for the cash flow at the end of year 1 ( $CF_1$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 1.
$PV_1 = \frac{\$5,000}{(1 + 0.08)^1}$

STEP 4

Calculate the present value for the cash flow at the end of year 1.
$PV_1 = \frac{\$5,000}{1.08}$

STEP 5

Now, we will calculate the present value of the cash flow at the end of year 2.
$PV_2 = \frac{CF_2}{(1 + r)^2}$

STEP 6

Plug in the given values for the cash flow at the end of year 2 ( $CF_2$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 2.
$PV_2 = \frac{\$25,000}{(1 + 0.08)^2}$

STEP 7

Calculate the present value for the cash flow at the end of year 2.
$PV_2 = \frac{\$25,000}{1.08^2}$

STEP 8

Now, we will calculate the present value of the cash flow at the end of year 3.
$PV_3 = \frac{CF_3}{(1 + r)^3}$

STEP 9

Plug in the given values for the cash flow at the end of year 3 ( $CF_3$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 3.
$PV_3 = \frac{\$14,000}{(1 + 0.08)^3}$

STEP 10

Calculate the present value for the cash flow at the end of year 3.
$PV_3 = \frac{\$14,000}{1.08^3}$

STEP 11

Now, we will add the present values of all the cash flows to find the total present value of the stream of cash flows.
$Total\, PV = PV_1 + PV_2 + PV_3$

STEP 12

Plug in the calculated present values for each year to calculate the total present value.
$Total\, PV = \frac{\$5,000}{1.08} + \frac{\$25,000}{1.08^2} + \frac{\$14,000}{1.08^3}$

STEP 13

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Find the present value of a firm's 3-year cash flow stream with 5,000,5,000, 5,000,25,000, and $14,000 at 8% discount rate.

STEP 1

STEP 2

First, we will calculate the present value of the cash flow at the end of year 1.PV1=CF1(1+r)1PV_1 = \frac{CF_1}{(1 + r)^1}PV1​=(1+r)1CF1​​

STEP 3

Now, plug in the given values for the cash flow at the end of year 1 (CF1CF_1CF1​) and the opportunity cost (rrr) to calculate the present value for year 1.PV1=$5,000(1+0.08)1PV_1 = \frac{\$5,000}{(1 + 0.08)^1}PV1​=(1+0.08)1$5,000​

STEP 4

Calculate the present value for the cash flow at the end of year 1.PV1=$5,0001.08PV_1 = \frac{\$5,000}{1.08}PV1​=1.08$5,000​

STEP 5

Now, we will calculate the present value of the cash flow at the end of year 2.PV2=CF2(1+r)2PV_2 = \frac{CF_2}{(1 + r)^2}PV2​=(1+r)2CF2​​

STEP 6

Plug in the given values for the cash flow at the end of year 2 (CF2CF_2CF2​) and the opportunity cost (rrr) to calculate the present value for year 2.PV2=$25,000(1+0.08)2PV_2 = \frac{\$25,000}{(1 + 0.08)^2}PV2​=(1+0.08)2$25,000​

STEP 7

Calculate the present value for the cash flow at the end of year 2.PV2=$25,0001.082PV_2 = \frac{\$25,000}{1.08^2}PV2​=1.082$25,000​

STEP 8

Now, we will calculate the present value of the cash flow at the end of year 3.PV3=CF3(1+r)3PV_3 = \frac{CF_3}{(1 + r)^3}PV3​=(1+r)3CF3​​

STEP 9

Plug in the given values for the cash flow at the end of year 3 (CF3CF_3CF3​) and the opportunity cost (rrr) to calculate the present value for year 3.PV3=$14,000(1+0.08)3PV_3 = \frac{\$14,000}{(1 + 0.08)^3}PV3​=(1+0.08)3$14,000​

STEP 10

Calculate the present value for the cash flow at the end of year 3.PV3=$14,0001.083PV_3 = \frac{\$14,000}{1.08^3}PV3​=1.083$14,000​

STEP 11

Now, we will add the present values of all the cash flows to find the total present value of the stream of cash flows.Total PV=PV1+PV2+PV3Total\, PV = PV_1 + PV_2 + PV_3TotalPV=PV1​+PV2​+PV3​

STEP 12

Plug in the calculated present values for each year to calculate the total present value.Total PV=$5,0001.08+$25,0001.082+$14,0001.083Total\, PV = \frac{\$5,000}{1.08} + \frac{\$25,000}{1.08^2} + \frac{\$14,000}{1.08^3}TotalPV=1.08$5,000​+1.082$25,000​+1.083$14,000​

STEP 13

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Find the present value of a firm's 3-year cash flow stream with $5,000,$ 25,000, and $14,000 at 8% discount rate.

First, we will calculate the present value of the cash flow at the end of year 1.
$PV_1 = \frac{CF_1}{(1 + r)^1}$

Now, plug in the given values for the cash flow at the end of year 1 ( $CF_1$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 1.
$PV_1 = \frac{\$5,000}{(1 + 0.08)^1}$

Calculate the present value for the cash flow at the end of year 1.
$PV_1 = \frac{\$5,000}{1.08}$

Now, we will calculate the present value of the cash flow at the end of year 2.
$PV_2 = \frac{CF_2}{(1 + r)^2}$

Plug in the given values for the cash flow at the end of year 2 ( $CF_2$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 2.
$PV_2 = \frac{\$25,000}{(1 + 0.08)^2}$

Calculate the present value for the cash flow at the end of year 2.
$PV_2 = \frac{\$25,000}{1.08^2}$

Now, we will calculate the present value of the cash flow at the end of year 3.
$PV_3 = \frac{CF_3}{(1 + r)^3}$

Plug in the given values for the cash flow at the end of year 3 ( $CF_3$ ) and the opportunity cost ( $r$ ) to calculate the present value for year 3.
$PV_3 = \frac{\$14,000}{(1 + 0.08)^3}$

Calculate the present value for the cash flow at the end of year 3.
$PV_3 = \frac{\$14,000}{1.08^3}$

Now, we will add the present values of all the cash flows to find the total present value of the stream of cash flows.
$Total\, PV = PV_1 + PV_2 + PV_3$

Plug in the calculated present values for each year to calculate the total present value.
$Total\, PV = \frac{\$5,000}{1.08} + \frac{\$25,000}{1.08^2} + \frac{\$14,000}{1.08^3}$