Chapter 25, Problem 1P

Suppose an H1200 supercomputer has a cost of $200,000 and will have a residual market value of $60,000 in five years. The risk-free interest rate is 5% APR with monthly compounding.

1. a. What is the risk-free monthly lease rate for a five-year lease in a perfect market?
2. b. What would be the monthly payment for a five-year $200,000 risk-free loan to purchase the H1200?

Summary Introduction

To determine: The risk-free monthly lease rate payment in a perfect market.

Introduction: Lease is a contract between the lessee and lessor for the use of an asset. Lessee agrees to pay a specific amount as per contract to the lessor for the use of the lessor asset.

Answer to Problem 1P

The risk-free monthly lease rate payment in a perfect market is $2,880.02.

Explanation of Solution

Given information:

Supercomputer cost: $200,000.

Lease term: 5 years.

Lease payment period: $\text{Total}\text{year}\text{for}\text{lease}\text{×}\text{month}\text{in}\text{a year }=5\times 12=60$.

Residual market value: $60,000.

Risk free interest rate: 5.00%

Formula for the calculation of present value of the payment:

$\text{PV}\left(\text{Lease Payments}\right)=\text{Purchase Price}-\text{PV}\left(\text{Residual Value}\right)$

Formula for the calculation of the monthly lease payment in perfect market:

$\text{PV}\left(\text{Lease Payments}\right)=\text{Monthly}\text{lease}\left(1+\frac{1}{\text{Rate}/\text{Month}}\left(1-\frac{1}{{\left(1+\text{Rate}/\text{Month}\right)}^{\text{Period}-1}}\right)\right)$

Calculate the present value (PV) of the lease payment:

$\begin{array}{c}\text{PV}\left(\text{Lease Payments}\right)=\text{Purchase Price}-\text{PV}\left(\text{Residual Value}\right)\\ =\text{Purchase Price}-\frac{\text{Residual Value}}{{\left(1+\text{Rate}\right)}^{\text{period}}}\\ =\$200,000-\frac{\$60,000}{{\left(1+\frac{5.00\%}{12}\right)}^{5\times 12}}\\ =\$200,000-\$46,751.39\\ =\$153,\mathrm{248.60.}\end{array}$

Therefore, the PV of the lease payment is $153,248.60.

Calculate the monthly lease payment in a perfect market:

Lease payment remains for 59 payments that are paid as an annuity because the first lease payment is paid in upfront.

$\begin{array}{c}\text{PV}\left(\text{Lease Payments}\right)=\text{Monthly}\text{lease}\left(1+\frac{1}{\text{Rate}/\text{Month}}\left(1-\frac{1}{{\left(1+\text{Rate}/\text{Month}\right)}^{\text{Period}}}\right)\right)\\ \$153,247.68=\text{Monthly}\text{lease}\times \left[\left(1+\frac{1}{\text{5}\text{.00\%}/\text{12}}\right)\times \left(1-\frac{1}{{\left(1+\text{5}\text{.00\%}/\text{12}\right)}^{\text{59}}}\right)\right]\\ \$153,247.68=\text{Monthly}\text{lease}\times \left[53.211\right]\\ \text{Monthly}\text{lease}=\frac{\$153,247.68}{53.211}\\ =\$2,\mathrm{880.02.}\end{array}$

Therefore, the monthly lease payment in a perfect market is $2,880.02.

Summary Introduction

To determine: The monthly payment for a five-year risk free loan.

Answer to Problem 1P

The monthly payment for a five-year risk free loan is $3,774.

Explanation of Solution

Given information:

Lease term: 5 years.

Lease payment period: $\text{Total}\text{year}\text{for}\text{lease}\text{×}\text{Month}\text{in}\text{a year}=5\times 12=60$.

Interest rate: 5.00%.

Formula for the calculation of the monthly lease payment:

$\text{PV}\left(\text{Lease Payments}\right)=\text{Monthly}\text{lease}\left(1+\frac{1}{\text{Rate}/\text{Month}}\left(1-\frac{1}{{\left(1+\text{Rate}/\text{Month}\right)}^{\text{Period}}}\right)\right)$

Calculation of the monthly payment for a five-year risk free loan:

$\begin{array}{c}\text{PV}\left(\text{Lease Payments}\right)=\text{Monthly}\text{lease}\times \frac{1}{\text{Monthly}\text{rate}}\left(1-\frac{1}{{\left(1+\text{Monthly}\text{rate}\right)}^{\text{Period}}}\right)\\ \$153,247.68=\text{Monthly}\text{lease}\times \left[\frac{1}{\left(\text{5}\text{.00\%/12}\right)}\left(1-\frac{1}{{\left(1+\left(\text{5}\text{.00\%/12}\right)\right)}^{\text{60}}}\right)\right]\\ \$153,247.68=\text{Monthly}\text{lease}\times \left[52.99\right]\\ \text{Monthly}\text{lease}=\frac{\$3,774.247}{52.99}\\ =\$3,774.24\end{array}$

Therefore, the monthly payment for a five-year risk free loan is $3,774.24.