Final answer:
The student's question involves determining the present value of annuity payments from a lottery winning, which increase over time and are discounted by a particular rate. The present value is calculated using a formula for a growing annuity, taking into account factors like the growth rate of the payments, the discount rate, and the number of periods. An example with a simple bond issue is provided to illustrate the process of calculating present value at different discount rates.
Step-by-step explanation:
The student's question pertains to calculating the present value of annuity payments that increase at a fixed percentage rate over time, using a specified discount rate. This is a common exercise in finance and economics, especially related to determining the value of future cash flows given a particular rate of return (or discount rate). To solve the problem, we use the present value formula for a growing annuity.
For the lottery winnings scenario, we would calculate the present value of each individual payment and then sum them up. The first payment of $1,000,000 is received in one year, and it increases by 3 percent each year for 30 years. The discount rate is 7 percent. To analyze this, we must consider the effects of inflation and the time value of money, which is why the payments are 'discounted' back to their present value at the prevailing discount rate.
The general formula for the present value of a growing annuity is:
PV = P * [(1 - ((1 + g)/(1 + r))^n) / (r - g)]
where PV is the present value, P is the first payment, g is the growth rate of the payments, r is the discount rate, and n is the number of periods.
Using the simple two-year bond example provided, the present value calculations at an 8% discount rate are:
$240/(1+0.08) = $222.22
$3,240/(1+0.08)^2 = $2,777.78
Total present value = $222.22 + $2,777.78 = $3,000
If the discount rate increases to 11%, the new present values would be calculated using the higher rate, resulting in a lower total present value for the bond. The calculations reflect the principle that if the cost of borrowing money increases (higher discount rate), the present value of future payments decreases.