Final answer:
The present value of an annual stream of cash flows that grows at 7% per year with a 13% discount rate, starting with $49 today, is approximately $922.83.
Step-by-step explanation:
To calculate the present value of an annual stream of cash flows that grows at a rate of 7% each year with an annual discount rate of 13% and initial cash flows beginning today at $49, we need to use the formula for the present value of a growing annuity. However, since the first cash flow begins today, we have to treat it as a separate cash flow and then calculate the present value of the remaining cash flows starting from next period as a growing annuity.
To find the present value of the first cash flow which happens immediately, we simply take the value as it is, which is $49, since it does not need to be discounted. For the rest of the growing annuity, the formula is:
PV = C / (r - g),
where C is the cash flow in the first period, r is the discount rate, and g is the growth rate.
The first period cash flow is the cash flow that happens a year from now which will be $49 * 1.07 due to growth. Applying the formula:
PV = ($49 * 1.07) / (0.13 - 0.07)
PV = $52.43 / 0.06
PV = $873.83
Lastly, we add the present value of the immediate cash flow:
Total Present Value = $873.83 + $49 = $922.83
Therefore, the present value of this stream of growing cash flows is approximately $922.83.