Suppose the borrowing rate r_B = 10\%r
B
=10% compounded annually. However,
the lending rate (or equivalently, the interest rate on deposits) is
only 8\%8% compounded annually. Compute the difference between the upper
and lower bounds on the price of an perpetuity that pays \(A = 10,000\\)$ per
year.
Please submit your answer rounded to the nearest dollar so if your answer is 23,456.78923,456.789 then you should submit an answer of 2345723457
25000
To find the difference between the upper and lower bounds on the price of a perpetuity, we need to calculate the present value of the perpetuity using the borrowing rate and the lending rate.
The upper bound can be calculated by assuming all future cash flows are discounted at the borrowing rate of 10% compounded annually. The formula for the present value of a perpetuity is:
PV = A / r
where PV is the present value, A is the annual payment, and r is the interest rate.
For the upper bound, we have:
PV_upper = 10,000 / (0.10)
PV_upper = $100,000
The lower bound can be calculated by assuming all future cash flows are discounted at the lending rate of 8% compounded annually. Using the same formula, we get:
PV_lower = 10,000 / (0.08)
PV_lower = $125,000
The difference between the upper and lower bounds is:
Difference = PV_upper - PV_lower
Difference = $100,000 - $125,000
Difference = -$25,000
Therefore, the difference between the upper and lower bounds on the price of the perpetuity is -$25,000.
To calculate the difference between the upper and lower bounds on the price of a perpetuity, we need to consider the concept of present value. The present value represents the current worth of future cash flows, taking into account the time value of money.
In this case, we need to find the upper and lower bounds for the price of a perpetuity that pays $10,000 per year. The perpetuity is an infinite series of cash flows, with each cash flow occurring annually.
To find the upper bound, we assume that the perpetuity is valued as if it were a perpetuity at the borrowing rate of 10%. The formula to calculate the present value of a perpetuity is:
Present Value = Cash Flow / Interest Rate
Upper Bound = $10,000 / 10% = $100,000
To find the lower bound, we assume that the perpetuity is valued as if it were a perpetuity at the lending rate of 8%. The formula remains the same:
Lower Bound = $10,000 / 8% = $125,000
The difference between the upper and lower bounds is:
Difference = Upper Bound - Lower Bound = $100,000 - $125,000 = -$25,000
Therefore, the difference between the upper and lower bounds on the price of the perpetuity is -$25,000.