MRP Finance Internship Unit 06

Unit 06 · Practice solutions

Practice solutions

Time Value of Money · Kenya · KSh

Worked answers to the optional practice questions. The required question is turned in to the TAs, and its answer is not published here.

Practice solutions

6.1

  1. (a) Opportunity cost — inflation is low and default risk is minimal, so almost all of the rate is compensation for waiting. (b) Risk — the promise may simply not be kept, which dwarfs the other two. (c) Inflation — there is no return being forgone on cash under a mattress and no counterparty to fail, so the entire loss is purchasing power.
  2. Because people prefer consumption sooner rather than later, and because capital put to work in the economy produces goods and services that did not exist before. Someone who gives up money now is giving up both the enjoyment and the productive use of it, and must be compensated to do so. That compensation is the real rate, and it would exist in a world with perfectly stable prices and perfectly reliable promises.
  3. The two promises have different probabilities of being honoured: a listed bank is regulated, capitalised, and has a long record, while a two-year-old logistics firm may not exist in two years. Value is the present value of expected cash flows at a rate reflecting their risk, so the less reliable promise is worth less today. The start-up's promise is discounted at the higher rate.

6.2

  1. 4,000,000 / 1.15⁶ = 4,000,000 / 2.31306 = KSh 1,729,310.
  2. Monthly rate = 24%/12 = 2%. EAR = (1.02)¹² − 1 = 26.82%. It exceeds 24% because each month's interest is charged on a balance that already includes previous months' interest — the borrower pays interest on interest, which the simple annual quotation ignores.
  3. (1+r)⁶ = 2, so 1+r = 2^(1/6) = 1.12246, giving r = 12.25%. Since doubling at roughly 12% takes about 6 years, the rule of thumb is that years to double ≈ 72 divided by the rate — at 12%, about 6 years, which matches.

6.3

  1. At 8%: factors 0.9259, 0.8573, 0.7938, 0.7350. PVs total 231,475 + 257,190 + 238,140 + 147,000 = KSh 873,805, so NPV = +KSh 73,805. At 20%: factors 0.8333, 0.6944, 0.5787, 0.4823. PVs total 208,325 + 208,320 + 173,610 + 96,460 = KSh 686,715, so NPV = −KSh 113,285. NPV crosses zero at roughly 12.2%.
  2. It means that after recovering the full cost of the investment, and after earning the required return on the money tied up in it for the whole period, there is KSh 500,000 of value left over in today's shillings. The required return is already inside the calculation — it is not additional profit on top of a normal return, it is the excess over a normal return. A zero NPV would therefore not be a failure; it would mean the project earned exactly what its capital could have earned elsewhere at the same risk.
  3. Adding seven years of shillings treats a shilling received in year seven as equal to one received today, which ignores everything in this week. It overstates the benefit. At 15%, the seven discount factors sum to about 4.16, so the true present value is roughly 4.16/7 = 59% of the undiscounted total — the colleague is overstating the benefit by about 70% of its true value.

6.4

  1. PV = 400,000 × [1 − 1.11⁻⁸]/0.11. Here 1.11⁻⁸ = 0.43393, so PV = KSh 2,058,449.
  2. Amount borrowed = the present value of the instalments = 620,000 × [1 − 1.16⁻⁵]/0.16. Here 1.16⁻⁵ = 0.47611, so the amount borrowed is KSh 2,030,062.
  3. Mathematically, if g equalled r the denominator would be zero and the value infinite, and if g exceeded r the value would be negative, which is meaningless for a stream of positive cash flows — the underlying series simply does not converge. Economically, nothing can grow faster than the discount rate forever: a business compounding above the growth rate of the economy would eventually become larger than the economy, which does not happen. A model with g above r has not been thought through.

6.5

  1. Nominal risk-free = 1.03 × 1.08 − 1 = 11.24%. Risk premium = 17% − 11.24% = 5.76 percentage points.
  2. 2,000,000 divided by: 1.05⁷ = 1.4071 → 1,421,435; 1.11⁷ = 2.0762 → 963,299; 1.16⁷ = 2.8262 → 707,663; 1.24⁷ = 4.5077 → 443,684. The same promise ranges from KSh 1.42M to KSh 0.44M — a factor of more than three — with no change to the amount or the date. Over seven years the rate does far more work than it does over one or two.
  3. The 23% is largely compensation for the cedi losing purchasing power. Real return = 1.23/1.20 − 1 = 2.5%. It is not literally risk-free: inflation can exceed the forecast, a foreign-currency investor bears exchange-rate risk, and sovereign debt can be restructured or default even when issued locally. “Same-currency government benchmark” is the disciplined description.

6.6

  1. Machinery replacement: 12–14%, below the firm's 15%, because it is the firm doing more of exactly what it already does with a known technology and predictable savings. Mobile app in a new market: 25–35%, well above 15%, because there is no operating history, no customer base, and the revenue is entirely a forecast. Using 15% for both overvalues the app and undervalues the machinery, so over time the firm approves speculative projects and rejects reliable ones — the reverse of what it intends.
  2. The contracted rent is a legal obligation of a known tenant over a defined period, so the main uncertainty is whether that tenant survives and pays. The sale price in year 10 depends on the property market a decade out, on whether a buyer exists at that price, and on whether the building is let at the time — none of which is contracted or knowable. The terminal value deserves the higher rate, often by several percentage points, and blending them understates the risk of the piece that usually contributes most of the value.
  3. Strategic importance describes how much the firm wants the project to succeed, which is a statement about preference rather than about the riskiness of the cash flows. Lowering the rate for it simply makes the project's NPV look larger without changing anything about what will actually happen. If the project is genuinely strategic, that value should be argued explicitly — as an option it creates or a cost it avoids elsewhere — and put into the cash flows, where it can be examined.
  4. I would tell the client that the project's NPV is positive across most of the plausible range but crosses zero at about 17%, so the recommendation depends on whether the right rate for these cash flows is below that — this is a fragile project rather than a robust one, and the decision is really an argument about risk rather than about the project itself. Before recommending, I would work on justifying the rate: identify comparable investments and what they yield, separate the more contracted cash flows from the more speculative ones and consider discounting them differently, and test whether the NPV is also fragile to the revenue and margin assumptions or only to the rate.