Last updated 2018-06-10 by Muchen

# Lesson 7 Practice Problems1

You work for a company that provides maintenance for marine diesel engines in large container ships. The main product you sell is a five-year service contract for a specific type of engine. You charge $85,000 each year, payable at the end of the year. One of your customers wants to pre-pay the entire five-year contract. If your company’s interest rate is 12.5% compounded annually, how much should you charge them (to the nearest dollar)? Answer: We are interested in the present value of a uniform series for 5 years. Our known variables: $A=85000$, $i=12.5\%$, $n=5$. Apply present worth factor formula: ## Problem 2 With another customer, your company has included a riser to account for increasing costs over time. The first-year payment is$85,000, with the price increasing $2,500 each year thereafter. If the customer wanted to just pay a constant amount for the five years, how much should that be? Use the same 12.5% interest rate, round to the nearest dollar. Answer: This time, since the cost is increasing linearly each period, we have a arithmetic series. In this case, the arithmetic series follows $P_i=A+nG$, where $A=85000$ and $G=2500$. We could get the arithmetic gradient uniform series factor, which gives us the equivalent $A_{eq}$ and thus we can solve this like a uniform series.$4415 is the amount need to paid in addition to the $85000 base cost. Thus the equivalent total annuity is ## Problem 3 A third, and somewhat less helpful customer, has a seven-year contract. The first-year payment was$60,000, with a clause that payments would increase by 4% per year.  The contract has now just expired (the seven years are up), and they have not yet payed a single bill.  What amount should you claim is owing when you sue them?

Since payments increase by 4% each year, this is an indication of geometric series. Assuming 12.5% interest rate still. Here are the givens: $g=0.04$, $i=0.125$, $A_1=60,000$, $n=7$.

We are looking future values now, so we shall use the compound amount factor to compute $F$:

For your fourth customer, you estimate the present value of the work to be done is $1.2 million over six years. To avoid another situation like customer three, you are making this customer pay in advance. Still with 12.5% interest, what should their (uniform) annual payments be? Answer: We are back to working with uniform payments, and knowing present value $P=1.2\text M$, we shall work with the capital recover factor to solve for the annuity $A$ over the next $n=6$ years. Because we ask that they pay up front (annuity due), we append the $1+i$ term. Since this is capital recovery factor, it is inversed. ## Problem 5 Your fifth customer is a government operator. They expect to operate a fleet of five ships for a long time (assume forever), and want to capitalize the maintenance costs. If the expected costs for the fleet are a total of$463,000 per year, how much should the government pay you now to contract you to maintain their ships for their lifetime?

This is a perpetuity situation. Assuming interests are still 12.5% and the cost each period are uniform, then the annuity can be given simply:

## Problem 6

Your sixth and final customer is a small shop. Their contract is only for four years, and is \$20,000 per year.  You charge them 12.5% compounded annually, however, they want to make quarterly payments. How much should their payments be?

This problem can be split into two parts:

1. Find the present worth of all the costs.
2. Use present worth find payment per adjusted quarterly period.

The present worth of all costs:

Now we find the equivalent interest rate because payment period is different than compounding period: (where $c=1$ is the number of compounding periods per year and $p=4$ is the number of payment periods per year)

Applying capital recovery factor, we get the amount of payment per period, where $n=4\times 4$:

1 M. Hollett, MECH 431 001 Lesson 7 Practice Problems. Web. Accessed 2018-06-10.