GB550 Unit 2 Time Value of Money Problems Example

Six Decisions Priced in Time: A Unit 2 Time Value of Money Problem Set for an Inland Barge Operator

[Student Name]

Purdue University Global

GB550: Financial Management

Unit 2 Time Value of Money Problems

[Instructor Name]

[Date]

The barge operator and every amount, rate and offer are composite and illustrative; the methods and sources cited are real.

Problem 1: Repowering a Towboat

Setup. A new engine package costs $4,600,000 and saves $560,000 a year in fuel for 15 years. The firm discounts at 8 percent.

Formula. Present value of an ordinary annuity: PV = PMT × [1 − (1 + r)^−n] / r.

Solution. PV = 560,000 × [1 − 1.08^−15] / 0.08 = 560,000 × 8.5595 = $4,793,308. Net of the $4,600,000 cost, the repower is worth about $193,308.

Spreadsheet. PV(8%, 15, −560000).

Reading. The repower clears its cost, but by only about 4 percent, so a modest drop in fuel prices would erase the gain.

Problem 2: Financing Sixteen Barges

Setup. Sixteen covered hopper barges cost $815,000 each, $13,040,000 in all. The lender finances 80 percent, $10,432,000, over ten years at 7.2 percent with monthly payments.

Formula. Loan payment: PMT = PV × i / [1 − (1 + i)^−n], with i = 0.072 / 12 = 0.006 and n = 120.

Solution. PMT = 10,432,000 × 0.006 / [1 − 1.006^−120] = $122,202.40 a month. Total payments are $14,664,288, so interest comes to about $4,232,288.

Spreadsheet. PMT(0.6%, 120, −10432000).

Reading. Interest adds about 41 percent to the financed amount, which is the price of keeping $10.4 million of cash free for other uses.

Problem 3: Two Lenders' Quotes

Setup. One lender quotes 7.10 percent compounded monthly; another quotes 7.25 percent compounded semiannually.

Formula. Effective annual rate: EAR = (1 + APR / m)^m − 1.

Solution. Monthly: (1 + 0.071 / 12)^12 − 1 = 7.336 percent. Semiannual: (1 + 0.0725 / 2)^2 − 1 = 7.381 percent.

Spreadsheet. EFFECT(7.10%, 12) and EFFECT(7.25%, 2).

Reading. The lower quoted rate is also the cheaper loan, by about 4.5 basis points, but only after converting both to the same basis.

Two quotes can be compared only after both are converted to an effective annual rate, because compounding frequency changes the true cost.

Problem 4: An Escalating Shipping Contract

Setup. A fertilizer shipper offers a seven-year contract whose first annual payment, $3,200,000, arrives in twelve months and then rises 3 percent a year. The firm discounts at 9 percent.

Formula. Growing annuity: PV = C / (r − g) × [1 − ((1 + g) / (1 + r))^n], valid when r exceeds g.

Solution. PV = 3,200,000 / 0.06 × [1 − (1.03 / 1.09)^7] = $17,451,567.

Check. Discounting the seven payments individually, $3,200,000, $3,296,000, $3,394,880 and so on, at 9 percent gives the same total.

Spreadsheet. No single function; NPV(9%, payments) over the seven listed payments.

Reading. Treating the payments as level at $3,200,000 would understate the contract's value by about $1.35 million.

Problem 5: Saving for a Shipyard Overhaul

Setup. A towboat needs a $3,000,000 shipyard overhaul in six years. The firm plans a level annual deposit, made at each year's close, into an account earning 4.5 percent.

Formula. Sinking fund payment: PMT = FV × r / [(1 + r)^n − 1].

Solution. PMT = 3,000,000 × 0.045 / [1.045^6 − 1] = $446,635 a year.

Spreadsheet. PMT(4.5%, 6, 0, −3000000).

Reading. Setting aside about $447,000 a year avoids borrowing for the overhaul at roughly 6.6 percent when it falls due.

Problem 6: A Customer's Offer to Prepay

Setup. A grain shipper owes five annual payments of $2,100,000 under a contract and offers to pay $8,750,000 now instead.

Formula. Solve for the rate r at which 2,100,000 × [1 − (1 + r)^−5] / r = 8,750,000.

Solution. By trial or a financial calculator, r is about 6.40 percent.

Spreadsheet. RATE(5, 2100000, −8750000).

Reading. Accepting is equivalent to borrowing $8.75 million at about 6.40 percent, slightly cheaper than the firm's 6.60 percent bank loan. Whether to accept also depends on the customer's credit and the contract's terms. A shipper with weak credit who offers cash now is also reducing the firm's exposure to a later default, which would make the offer more attractive than the rate alone suggests.

What the Set Shows

Brealey et al. (2023) describe discounting as the common language of financial decisions, and the six problems show it working across very different choices: an investment, a loan, a comparison of quotes, a contract, a savings plan and a financing offer. Brigham and Ehrhardt (2020) emphasize that the setup, not the calculation, is where most errors occur, which is why each problem states its formula and inputs before solving. Damodaran (2012) adds that a discount rate must match the risk and timing of the cash flows it is applied to, so the 8 and 9 percent rates used here are placeholders until Unit 6 estimates the firm's own cost of capital. The repower and the prepayment are the two results the firm will revisit: the first when the cost of capital is estimated in Unit 6, and the second when financing is chosen in Unit 10.

References

Brealey, R. A., Myers, S. C., & Allen, F. (2023). Principles of corporate finance (14th ed.). McGraw Hill.

Brigham, E. F., & Ehrhardt, M. C. (2020). Financial management: Theory and practice (16th ed.). Cengage.

Damodaran, A. (2012). Investment valuation: Tools and techniques for determining the value of any asset (3rd ed.). Wiley.