MMHA 6400 Week 8 Time Value Problem Set Example

Pay Now or Pay Later: Six Time Value Problems From Alder Bend Community Hospital

Student Name

Walden University

MMHA 6400: Healthcare Finance

Instructor Name

Month Day, Year

Pay Now or Pay Later: Six Time Value Problems From Alder Bend Community Hospital

Introduction

The six problems below all take place at Alder Bend, the course's made-up community hospital, and each one closes on the decision its number informs. Three conventions hold unless a problem says otherwise: payments land on the last day of each year, money costs the hospital 6% a year, and as a nonprofit it owes no income tax. Formulas follow standard healthcare finance texts (Reiter & Song, 2021; Pink & Song, 2020). Every problem is laid out the same way so the method is visible: a timeline showing when money moves, the formula written out, the inputs with their units, the result and a sentence on what the result means for a decision. Timelines matter more than they seem. Most errors in time value work come from placing a cash flow in the wrong year, especially payments made at the start of a period, and a drawn timeline catches them before any arithmetic is done. Capital decisions in hospitals rarely turn on a single calculation, but every one of them rests on getting this arithmetic right (Cleverley & Cleverley, 2018). Results are rounded to the dollar, and all figures are illustrative.

Problem 1: Compounding a Replacement Reserve

The board has designated $4,000,000 as a reserve for equipment replacement and expects it to earn 5% a year. How much will it hold in five years?

Table 1

Problem 1 Timeline

Formula: FV = PV × (1 + r)^n. Inputs: PV = $4,000,000, r = 0.05, n = 5. Calculation: $4,000,000 × 1.27628 = $5,105,126.

Decision: the reserve will hold about $5.1 million in five years, which will cover the planned replacement of the CT scanner and part of a second MRI, so the board can plan the MRI to require borrowing only the difference.

Problem 2: Annual Service Payments or One Upfront Price

A vendor offers a five-year service contract for the CT scanner at $180,000 a year, paid at the end of each year, or $760,000 paid today. Which costs less in today's dollars?

Table 2

Problem 2 Timeline

Formula: PV of an ordinary annuity = PMT × [1 − (1 + r)^−n] / r. Inputs: PMT = $180,000, r = 0.06, n = 5. The annuity factor is 4.21236, so PV = $758,226.

Decision: the annual option costs $1,774 less in present value terms, a difference small enough that the hospital should choose on other grounds. Paying annually keeps $760,000 of cash available today and lets the hospital renegotiate if the scanner is replaced early, so the annual option is preferred.

Problem 3: Payment on an Equipment Loan

Alder Bend borrows $1,500,000 for imaging equipment at 6.5% a year and will retire the loan with seven identical yearly installments. What is each payment?

Table 3

Problem 3 Timeline

Formula: PMT = PV × r / [1 − (1 + r)^−n]. Inputs: PV = $1,500,000, r = 0.065, n = 7. (1.065)^−7 = 0.64351, so PMT = $1,500,000 × 0.065 / 0.35649 = $273,497.

Decision: payments total $1,914,478 over seven years, of which $414,478 is interest. The operating budget must absorb about $273,500 a year, and the debt service coverage ratio should be recalculated with this payment added before the loan is approved.

Problem 4: Monthly Rate to Effective Annual Rate

A lender quotes a line of credit at 0.55% per month. What is the effective annual rate, and why does it differ from 12 times the monthly rate?

Table 4

Problem 4 Inputs

Formula: EAR = (1 + monthly rate)^12 − 1. Calculation: (1.0055)^12 − 1 = 1.06803 − 1 = 6.80%.

The effective rate is higher because interest compounds monthly: each month's interest earns interest in the months that follow. Decision: when comparing this line of credit with a bank quoting 6.75% compounded annually, the bank's offer is slightly cheaper, even though 6.60% looks lower at first glance.

Problem 5: Lease or Buy an Ultrasound Unit

Buying an ultrasound unit costs $210,000 today, plus $12,000 a year in maintenance for five years, with an expected resale value of $20,000 at the end of year 5. The lease alternative charges $52,000 per year for five years, due on the first day of each year, with maintenance covered by the lessor. Which option costs less?

Table 5

Problem 5 Timeline

Buy: PV = $210,000 + $12,000 × 4.21236 − $20,000 / (1.06)^5 = $210,000 + $50,548 − $14,945 = $245,603. Lease: because each installment is paid in advance, the stream is an annuity due, PV = $52,000 × 4.21236 × 1.06 = $232,186.

Decision: leasing costs $13,417 less in present value. Because Alder Bend pays no income tax, the depreciation tax shield that often favors buying in for-profit firms does not apply, so the lease is preferred unless the hospital expects to use the unit well beyond five years.

Problem 6: An Uneven Stream of Savings

A new supply chain system is expected to save $90,000, $120,000, $140,000, $140,000 and $110,000 in years 1 through 5. It costs $480,000 today. What is the present value of the savings, and is the project worth doing?

Table 6

Problem 6 Present Value of Each Year's Savings

Each year's savings is discounted separately, because no annuity formula fits an uneven stream. Net present value = $502,344 − $480,000 = $22,344.

Decision: the system clears the 6% hurdle, but not by much; a 5% shortfall in projected savings would nearly erase it. This year-by-year method is the one the Week 9 capital request will use for a second MRI scanner.

References

Cleverley, W. O., & Cleverley, J. O. (2018). Essentials of health care finance (8th ed.). Jones & Bartlett Learning.

Pink, G. H., & Song, P. H. (2020). Gapenski's fundamentals of healthcare finance (3rd ed.). Health Administration Press.

Reiter, K. L., & Song, P. H. (2021). Gapenski's healthcare finance: An introduction to accounting and financial management (7th ed.). Health Administration Press.