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# Calculating present value and annual payments

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1. Calculate the present value of \$90,000 to be received 14 years from now if the decision makers opportunity cost 10 percent.

2. Find the present value at 9 percent of each of the following five cash inflow streams. Assume that cash inflows occur at the end of the year.

Year A B
1 \$8,000 \$10,000
2 9,000 10,000
3 10,000 10,000
4 11,000 10,000
5 12,000 10,000

3. If an individual wishes to accumulate \$15,000 in six years by making equal annual end of year deposits into an account paying 7 percent interest, how large should the deposit be?

4. A lender wishes to to determine the size of the equal annual end of year payments necessary to fully amortize a \$40,000 loan at 11 percent interest over three years. How large should the payments be?

#### Solution Preview

1. Calculate the present value of \$90,000 to be received 14 years from now if the decision makers opportunity cost 10 percent.

Future Value=FV=\$90000
Number of periods=n=14
Opportunity cost=r=10%
Present Value=PV=FV/(1+r)^n=90000/(1+10%)^14=23699.81

2. Find the present value at 9 percent of each of the following five cash inflow streams. Assume that cash inflows occur at the end of the year.

year A B
1 \$8,000 ...

#### Solution Summary

There are four problems. Solutions to first two problems depict the steps to calculate present value of the given cash flows. Solutions to another two problems explain the methodology to calculate annual year end payments to meet the given financial objective.

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## Present Value Lease: Calculate Annual Payments for Robotics of Beverly Hills and Leases R Us, Inc.

Leases R Us, Inc. (LRU) has been contracted by Robotics of Beverly Hills (RBH) to provide lease financing for a machine that would assist in automating a large part of their current assembly line. Annual lease payments will start at the beginning of each year. The purchase price of this machine is \$200,000, and it will be leased by RBH for a period of 5 years. LRU will utilize straight line depreciation of \$40,000 per year with a zero book salvage value. However, salvage value is estimated to actually be \$35,000 at the end of 5 years. LRU is required to earn a 14%, after-tax rate of return on the lease. LRU uses a marginal tax rate of 40%. Calculate the annual lease payments. (Remember, these payments are to be considered at the beginning of each year?annuity due.)

Hints for students:

There are 3 major steps that need to be accomplished in order to calculate the annual lease payment.

Step A: You need to calculate the amount to be amortized. This would be the cost of the machine less the PV of the after tax salvage value of the machine and less the PV of the depreciation tax shield

Step B: You need to calculate the annual after-tax required lease income. (Remember, in this step, you need to calculate it as an annuity due?a beginning of the year payment.) Take your answer from Step A as a present value, and using the number of years and the required rate of return, calculate the payment.

Step C: Calculate the lease payment. You need to adjust for the appropriate tax rate. Therefore, take your answer in Step B and divide it by (1 - the tax rate). This will give you the required lease payment.

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