Question

Suppose an H1200 supercomputer has a cost of 300,000 and will have a residual marked value of 45,000 in 7 years. The risk-free interest rate is 6.2% APR with monthly compounding.

A. What is the risk-free monthly lease rate for a 7-year lease in a perfect market?

B. What would be the monthly payment for a 7-year $300,000 risk-free loan to purchase the H1200?

Note: Round the monthly interest rate to at least six decimal places

Answer 1

Q.A)

The initial investment for the lease is $300,000.

Residual marked value= $45,000 in 7 years

Risk free interest rate is 6.2% APR monthly compunding.

In order to calculate the PV of the lease payments, we calculate the PV of the residual value at an annual risk free interest rate and subtract it from the initial investment:

Hence the PV of lease payments is given by: 300,000-45,00(1+0.062/12)^(7*12) = $230,623.71

The term (1+0.062/12) denotes the transition between monthly compunding and annual compounding since the risk free interest rate is given in monthly compounding. The term is raised to the power 7*12 which denotes the duration of 7 years comprising of 12 months each.

In case of lease payments, the traditional methodology states that the first payment is an upfront down payment and the leases are then distributed accross the remaining months. Hence, 7*12-1= 83 months.

Let the lease payment be determined by A. The lease payment can be calculated as :

230,623.71= A (1+ [1/(0.062/12)]* (1-1/(0.062/12)⁸³)

A= $1186.466864

Q.B)

For a monthly payment of 7-year risk free loan of $300,000, the equation remains the same, the lease payment has to be replaced by the principal of the risk free loan. Let the monthly payments be B:

300,000= B (1+ [1/(0.062/12)]* (1-1/(0.062/12)⁸³)

B= $1543.380163