Time Value of Money

• Topic 01

  Time Value of Money

  88 questions

A few questions from this unit, with the answer and a full explanation. The complete bank is available when you start practising.

1. A client wants to know the discounted payback period for a project with an initial cost of £100,000 and annual cash flows of £40,000, £40,000, and £40,000 at a 10% discount rate. After which year is the investment recovered in discounted terms?

   • A
     Year 3, but discounted payback and standard payback give identical results for this cash flow pattern
   • B
     The investment is not fully recovered within three years; discounted cash flows are approximately £36,364, £33,058, and £30,053 (cumulative: £99,475), just short of £100,000
     Correct answer
   • C
     Year 2, because £40,000 + £40,000 = £80,000 and with interest the balance is recovered by end of year 2
   • D
     Year 2.5, which is the same as the standard payback period since discounting makes no difference for equal cash flows
   Explanation

   Year 1 PV: 40,000/1.10 = £36,364. Year 2 PV: 40,000/1.21 = £33,058. Year 3 PV: 40,000/1.331 = £30,053. Cumulative: Year 1: £36,364; Year 2: £69,421; Year 3: £99,475. The initial £100,000 is not fully recovered in three years (£525 short), whereas the standard payback period would show full recovery at exactly 2.5 years (3 × £40,000 = £120,000). This illustrates how discounting extends the payback period, showing the true time to recovery in economic terms.

2. What is the precise real rate of return for an investor who earns 15% nominally in a year when inflation is 10%?

   • A
     6.5%, using the geometric mean of the nominal and inflation rates
   • B
     5%, using the approximation 15% − 10%
   • C
     13.6%, calculated as 15% × (1 − 0.10)
   • D
     Approximately 4.55%, calculated as (1.15/1.10) − 1
     Correct answer
   Explanation

   Precise Fisher equation: real rate = (1 + nominal) / (1 + inflation) − 1 = (1.15 / 1.10) − 1 = 1.04545 − 1 = 4.545% ≈ 4.55%. The simple approximation (15% − 10% = 5%) overstates the true real return by approximately 0.45%. At high inflation levels of 10%, the cross-product term (real × inflation ≈ 0.45%) is material and the precise formula must be used.

3. What is NPV additivity and why is it an advantage over IRR when evaluating multiple projects?

   • A
     NPV additivity means any project with a positive NPV will also have an IRR above the cost of capital, making them equivalent
   • B
     NPV additivity means the NPV of a project is directly proportional to its duration, simplifying multi-project comparisons
   • C
     NPV additivity means the NPV of a combined portfolio of projects equals the sum of individual project NPVs; IRR cannot be added across projects and can produce conflicting rankings
     Correct answer
   • D
     NPV additivity means all NPV calculations produce the same result regardless of the discount rate chosen
   Explanation

   A key advantage of NPV is the additivity property: NPV(A + B) = NPV(A) + NPV(B). This means a firm can sum NPVs across projects and select the combination that maximises total value. IRR cannot be summed — the combined IRR of two projects is not the average of their individual IRRs. NPV additivity also means that when comparing mutually exclusive projects of different scale or cash flow timing, NPV gives the correct ranking, unlike IRR which may disagree.