. Suppose you just inherited an gold mine. This gold mine is believed to have three years worth of gold deposit. Here is how much income this gold mine is projected to bring you each year for the next three years:

Year 1: $49,000,000
Year 2: $61,000,000
Year 3: $85,000,000
Compute the present value of this stream of income at a discount rate of 7%. Remember, you are calculating the present value for a whole stream of income, i.e. the total value of receiving all three payments (how much you would pay right now to receive these three payments in the future). Your answer should be one number - the present value for this gold mine at a 7% discount rate but you have to show how you got to this number.
Now compute the present value of the income stream from the gold mine at a discount rate of 5%, and at a discount rate of 3%. Compare the present values of the income stream under the three discount rates and write a short paragraph with conclusions from the computations.

To calculate the present value of an income stream, we need to discount each future cash flow back to its present value. The formula to calculate the present value (PV) of a future cash flow is:

PV = CF / (1 + r)^n

where CF is the cash flow, r is the discount rate, and n is the number of years in the future.

Using this formula, let's calculate the present value of the income stream at a discount rate of 7%:

Year 1:
PV1 = $49,000,000 / (1 + 0.07)^1 = $45,794,393.86

Year 2:
PV2 = $61,000,000 / (1 + 0.07)^2 = $53,183,364.84

Year 3:
PV3 = $85,000,000 / (1 + 0.07)^3 = $70,406,006.75

Total present value:
PV = PV1 + PV2 + PV3 = $45,794,393.86 + $53,183,364.84 + $70,406,006.75 = $169,383,765.45

Now, let's calculate the present value of the income stream at a discount rate of 5%:

Year 1:
PV1 = $49,000,000 / (1 + 0.05)^1 = $46,666,666.67

Year 2:
PV2 = $61,000,000 / (1 + 0.05)^2 = $53,832,255.64

Year 3:
PV3 = $85,000,000 / (1 + 0.05)^3 = $72,016,460.91

Total present value:
PV = PV1 + PV2 + PV3 = $46,666,666.67 + $53,832,255.64 + $72,016,460.91 = $172,515,383.22

Finally, let's calculate the present value of the income stream at a discount rate of 3%:

Year 1:
PV1 = $49,000,000 / (1 + 0.03)^1 = $47,572,815.53

Year 2:
PV2 = $61,000,000 / (1 + 0.03)^2 = $55,315,516.16

Year 3:
PV3 = $85,000,000 / (1 + 0.03)^3 = $77,037,801.67

Total present value:
PV = PV1 + PV2 + PV3 = $47,572,815.53 + $55,315,516.16 + $77,037,801.67 = $179,926,133.36

Conclusions:
- As the discount rate increases, the present value of the income stream decreases. This is because the higher the discount rate, the less valuable future cash flows are considered to be.
- At a discount rate of 7%, the present value of the income stream is $169,383,765.45.
- At a discount rate of 5%, the present value of the income stream is $172,515,383.22.
- At a discount rate of 3%, the present value of the income stream is $179,926,133.36.
- The higher the discount rate, the lower the present value. Therefore, a higher discount rate results in a lower value for the gold mine.

To compute the present value of the income stream from the gold mine at a discount rate of 7%, we need to determine the present value (PV) of each annual cash flow and sum them up.

Using the formula for calculating present value:

PV = CF / (1 + r)^n

Where PV is the present value, CF is the cash flow, r is the discount rate, and n is the year.

For Year 1: PV = $49,000,000 / (1 + 0.07)^1 = $45,794,392.52
For Year 2: PV = $61,000,000 / (1 + 0.07)^2 = $52,445,077.74
For Year 3: PV = $85,000,000 / (1 + 0.07)^3 = $67,189,542.13

Now we sum up the present values of each cash flow:

Present value at a discount rate of 7% = $45,794,392.52 + $52,445,077.74 + $67,189,542.13 = $165,429,012.39

To compute the present value of the income stream at discount rates of 5% and 3%, we repeat the same process using the respective discount rates.

For discount rate of 5%:
Year 1: PV = $49,000,000 / (1 + 0.05)^1 = $46,666,666.67
Year 2: PV = $61,000,000 / (1 + 0.05)^2 = $54,567,901.23
Year 3: PV = $85,000,000 / (1 + 0.05)^3 = $72,255,639.10

Present value at a discount rate of 5% = $46,666,666.67 + $54,567,901.23 + $72,255,639.10 = $173,490,206

For discount rate of 3%:
Year 1: PV = $49,000,000 / (1 + 0.03)^1 = $47,572,815.53
Year 2: PV = $61,000,000 / (1 + 0.03)^2 = $56,221,964.62
Year 3: PV = $85,000,000 / (1 + 0.03)^3 = $76,685,076.68

Present value at a discount rate of 3% = $47,572,815.53 + $56,221,964.62 + $76,685,076.68 = $180,479,856.83

Comparing the present values:
- Present value at 7% discount rate: $165,429,012.39
- Present value at 5% discount rate: $173,490,206
- Present value at 3% discount rate: $180,479,856.83

From the computations, we observe that as the discount rate decreases:
- The present value of the income stream increases.
- The expected value of receiving future cash flows increases when discounting at a lower rate.
- This suggests that a lower discount rate increases the value placed on future cash flows, indicating a higher valuation for the gold mine's income stream.