Advanced Capital Budgeting Decisions — Practice

Question 1

A project has expected cash flows of ₹60 lakh in Year 1 and ₹80 lakh in Year 2, with certainty equivalent coefficients of 0.90 and 0.80 respectively, reflecting increasing risk further into the future. The risk-free rate is 6%. Compute the present value of these cash flows using the certainty equivalent method.

Accepted Answer

Certain equivalent cash flow Year 1 = 0.90 × ₹60 lakh = ₹54 lakh. Certain equivalent cash flow Year 2 = 0.80 × ₹80 lakh = ₹64 lakh. Present value Year 1 = ₹54 lakh ÷ 1.06 = ₹50.94 lakh. Present value Year 2 = ₹64 lakh ÷ (1.06)² = ₹64 lakh ÷ 1.1236 = ₹56.96 lakh. Total present value = ₹50.94 lakh + ₹56.96 lakh = ₹107.90 lakh. Both years' certainty equivalents are discounted at the risk-free rate of 6%, not a risk-adjusted rate, because the risk adjustment has already been fully incorporated at the cash flow stage through the certainty equivalent coefficients, which decline from 0.90 to 0.80 reflecting the increasing risk (and therefore increasing gap between the expected and certain-equivalent cash flow) further into the future; applying a risk-adjusted discount rate on top of already risk-adjusted cash flows would double-count the risk adjustment.

Question 2

Explain why the risk-adjusted discount rate method is described as compounding its risk adjustment every year regardless of whether a project's risk is actually concentrated in a specific year, and how the certainty equivalent method avoids this issue.

Accepted Answer

The risk-adjusted discount rate method applies a single, elevated discount rate uniformly to every year's cash flow, and because discounting is compounding by construction, a higher rate applied consistently reduces later years' cash flows proportionately more than earlier years' cash flows, implicitly assuming that risk grows at a constant, compounding rate the further into the future a cash flow lies, regardless of whether the project's actual risk profile genuinely behaves this way; a project whose greatest uncertainty is concentrated in an early, uncertain launch phase, with cash flows becoming progressively more predictable and less risky in later, more established years, is poorly served by a method that mechanically assumes the opposite, ever-increasing risk pattern. The certainty equivalent method avoids this issue because the certainty equivalent coefficient for each individual year is chosen independently, based on the actual assessed risk of that specific year's cash flow, allowing a project with front-loaded risk to have a lower coefficient (more risk adjustment) in its early years and a higher coefficient (less risk adjustment, if that year's cash flow is genuinely more predictable) in later years, rather than being forced into the RADR method's implicit assumption that risk necessarily compounds uniformly over time.

Question 3

A project's Year 1 revenue in today's prices is ₹100 lakh, expected to inflate at 8% per year, and Year 1 operating cost in today's prices is ₹60 lakh, expected to inflate at 5% per year. The nominal discount rate is 13%. Compute the nominal net cash flow for Year 2 and its present value, and explain why a single blended inflation rate applied to the net cash flow figure would be unreliable here.

Accepted Answer

Nominal revenue in Year 2 = ₹100 lakh × (1.08)² = ₹100 lakh × 1.1664 = ₹116.64 lakh. Nominal operating cost in Year 2 = ₹60 lakh × (1.05)² = ₹60 lakh × 1.1025 = ₹66.15 lakh. Nominal net cash flow Year 2 = ₹116.64 lakh − ₹66.15 lakh = ₹50.49 lakh. Present value = ₹50.49 lakh ÷ (1.13)² = ₹50.49 lakh ÷ 1.2769 = ₹39.54 lakh. A single blended inflation rate applied to the net cash flow figure would be unreliable here because revenue and cost are inflating at genuinely different rates, 8% and 5% respectively, and the net cash flow's own effective inflation rate is not simply an average of the two component rates; it depends on the relative size and growth trajectory of each component, meaning any single blended rate applied to today's net cash flow of ₹40 lakh (₹100 lakh − ₹60 lakh) would need to somehow capture this asymmetric compounding effect correctly, which a single rate cannot do precisely — the only reliable approach is to inflate each component individually at its own specific rate and net the resulting nominal figures afterward, exactly as this computation does, rather than attempting to shortcut the process with one blended rate applied to the combined figure.

Question 4

Project X has an NPV of ₹60 lakh over a 4-year life; Project Y has an NPV of ₹85 lakh over a 7-year life. The relevant discount rate is 10%, with PV annuity factors of 3.170 for 4 years and 4.868 for 7 years. Determine which project should be selected using the equivalent annual annuity method.

Accepted Answer

EAA for Project X = NPV ÷ PV annuity factor (4 years) = ₹60 lakh ÷ 3.170 = ₹18.93 lakh per year. EAA for Project Y = NPV ÷ PV annuity factor (7 years) = ₹85 lakh ÷ 4.868 = ₹17.46 lakh per year. Since Project X's equivalent annual annuity of ₹18.93 lakh exceeds Project Y's ₹17.46 lakh, Project X should be selected, despite having the lower raw NPV of the two (₹60 lakh versus ₹85 lakh), because comparing raw NPVs directly is misleading when the two projects have different lives; Project Y's higher raw NPV partly reflects that it generates value over three additional years compared to Project X, and once each project's NPV is converted to an equivalent constant annual figure spread over its own specific life, Project X is shown to create more value per year of its life than Project Y does per year of its own longer life, which is the genuinely comparable basis for choosing between mutually exclusive projects of unequal duration.

Question 5

A firm has a capital budget of ₹100 lakh and four indivisible projects available: Project A requires ₹60 lakh, NPV ₹18 lakh; Project B requires ₹50 lakh, NPV ₹16 lakh; Project C requires ₹40 lakh, NPV ₹14 lakh; Project D requires ₹30 lakh, NPV ₹9 lakh. Determine the combination of projects that maximises total NPV within the budget, and explain why simply ranking by profitability index might not identify this combination.

Accepted Answer

Profitability index (PI = 1 + NPV/Investment) for each: Project A = 1 + 18/60 = 1.30; Project B = 1 + 16/50 = 1.32; Project C = 1 + 14/40 = 1.35; Project D = 1 + 9/30 = 1.30. Ranked by PI: C (1.35), B (1.32), A (1.30, tied), D (1.30, tied). Taking projects in PI order until the ₹100 lakh budget is exhausted: C (₹40 lakh) + B (₹50 lakh) = ₹90 lakh used, NPV = ₹14 + ₹16 = ₹30 lakh, with ₹10 lakh of budget left unused (insufficient for any remaining project). Testing other feasible combinations within the ₹100 lakh budget: A + C = ₹60 + ₹40 = ₹100 lakh, NPV = ₹18 + ₹14 = ₹32 lakh; A + D = ₹60 + ₹30 = ₹90 lakh, NPV = ₹18 + ₹9 = ₹27 lakh; B + D = ₹50 + ₹30 = ₹80 lakh, NPV = ₹16 + ₹9 = ₹25 lakh; C + D = ₹40 + ₹30 = ₹70 lakh, NPV = ₹14 + ₹9 = ₹23 lakh (with ₹30 lakh unused, insufficient for A or B). Comparing all feasible combinations, A + C, at ₹32 lakh total NPV, exceeds the PI-ranking-derived combination of C + B at ₹30 lakh, despite Project A ranking below Project B on PI; this demonstrates that with indivisible projects, since a partially funded project is not possible, the PI ranking approach can fail to identify the value-maximising combination, because it does not account for how efficiently the budget is actually used in full — B and C together leave ₹10 lakh of budget unused entirely, while A and C together use the full ₹100 lakh budget and thereby capture more total value, illustrating why every feasible combination must be explicitly tested rather than relying on PI ranking alone whenever projects are indivisible.

Question 6

A company is evaluating a project with an expected NPV, ignoring managerial flexibility, of −₹2 lakh, but the project includes machinery that could be sold for a known salvage value if early results are poor, allowing the company to abandon the project after one year rather than continuing to incur losses for its full remaining life. Explain how this abandonment option could change the accept/reject decision, and why ignoring it risks a suboptimal decision.

Accepted Answer

A conventional, static NPV computation of −₹2 lakh values the project's cash flows as though the company is committed to running it for its entire life regardless of how results actually unfold, but the genuine ability to abandon the project after one year, recovering a known salvage value if early results are indeed poor, means the company is not actually exposed to the full downside a static, full-life NPV computation assumes; if results turn out poor, the company can exercise the abandonment option and limit its losses to the first year plus the shortfall (if any) between the salvage value recovered and the original investment, rather than continuing to bear negative cash flows for the project's full remaining life as the static NPV calculation implicitly assumes it must. This abandonment option has genuine positive value, analogous to a put option limiting downside risk, and adding this option value to the static −₹2 lakh NPV could produce an overall positive expanded NPV, meaning the project may genuinely be worth undertaking once the value of the flexibility to abandon early is properly accounted for; ignoring this option and rejecting the project based purely on the static −₹2 lakh figure risks a suboptimal decision, turning down a project that, with its embedded flexibility properly valued, may actually create value for the firm.

Question 7

A project's expected real rate of return required by investors is 7%, and expected inflation over the project's life is 5%. Compute the nominal discount rate that should be applied to nominal cash flow projections for this project.

Accepted Answer

Using the Fisher relationship: (1 + Nominal rate) = (1 + Real rate) × (1 + Inflation rate) = (1.07) × (1.05) = 1.1235, so the nominal rate = 0.1235, or 12.35%. This nominal rate of 12.35% should be applied to nominal cash flow projections, that is, cash flows already expressed in the actual rupee amounts expected to be received or paid in each future year, reflecting the expected effect of inflation on those figures; note that simply adding the real rate and inflation rate together (7% + 5% = 12%) gives an approximation close to, but not exactly equal to, the precise Fisher-derived 12.35%, with the small difference arising from the cross-term (Real rate × Inflation rate) that the simple additive approximation omits, a difference that becomes more material as either the real rate or inflation rate rises to higher levels.

Question 8

A junior analyst computes a project's NPV by discounting nominal cash flows (already inflated at an assumed 6% annual rate) using a real discount rate of 10%, instead of the correct nominal rate of approximately 16.6%. Explain the direction and nature of the error this produces.

Accepted Answer

Discounting nominal (inflated) cash flows at a real (non-inflation-adjusted) discount rate of 10%, when the correct nominal rate is approximately 16.6% (using the Fisher relationship, (1.10)(1.06) ≈ 1.166), systematically understates the discount rate actually applied relative to what the nominal cash flows require, since the nominal cash flows already embed the effect of inflation growing them larger each year, but the discount rate used to bring them back to present value fails to account for that same inflation effect being compounded away. Using too low a discount rate for cash flows that are, in effect, overstated in nominal terms relative to the rate applied to them produces a present value, and therefore an NPV, that is materially overstated compared to the correct figure obtained using the proper nominal rate; this specific mismatch, nominal cash flows with a real rate, is precisely the trap this chapter identifies as the single most commonly tested inflation-related error, and it always biases the NPV in the direction of overstatement, since discounting a larger, already-inflated cash flow stream at an insufficiently high rate understates how much that future purchasing power should genuinely be reduced to reflect present value.

Question 9

Project M has an NPV of ₹40 lakh over a 2-year life; Project N has an NPV of ₹65 lakh over a 3-year life, with a discount rate of 12%. Using the replacement chain method over a 6-year common horizon, explain conceptually how each project's chain NPV would be constructed, without computing the exact figures.

Accepted Answer

Under the replacement chain method, each project is conceptually replicated identically at the end of its own life until both projects reach a common time horizon; since Project M has a 2-year life and Project N has a 3-year life, the lowest common multiple of 2 and 3 is 6 years, so Project M would be replicated three times in succession (occurring at years 0, 2, and 4, each cycle running 2 years) to reach the 6-year horizon, while Project N would be replicated twice in succession (occurring at years 0 and 3, each cycle running 3 years) to reach the same 6-year horizon. Each project's chain NPV is computed as the sum of the present values of each replicated cycle's own NPV, with each subsequent cycle's NPV itself discounted back from the point in time that cycle begins (year 2 and year 4 for Project M's second and third cycles; year 3 for Project N's second cycle) to time zero, and the project with the higher total chain NPV over this common 6-year horizon is preferred; this approach is mathematically equivalent to the EAA method under the same assumption that each project can genuinely be replicated identically at the end of its life, but requires this explicit extension and repeated discounting across the common horizon rather than the single division computation the EAA shortcut uses.

Question 10

Explain what a certainty equivalent coefficient of 0.75 for a given year's expected cash flow of ₹100 lakh actually represents, in terms of a risk-averse decision-maker's preferences.

Accepted Answer

A certainty equivalent coefficient of 0.75 applied to an expected cash flow of ₹100 lakh means the certain equivalent cash flow for that year is 0.75 × ₹100 lakh = ₹75 lakh, representing the amount a risk-averse decision-maker would consider equally desirable to receiving the genuinely uncertain, risky expected cash flow of ₹100 lakh; in other words, the decision-maker would be indifferent between a certain, guaranteed ₹75 lakh and a risky prospect whose expected value is ₹100 lakh but which carries genuine uncertainty about the actual amount that will be received. The gap between the certain equivalent (₹75 lakh) and the expected value (₹100 lakh), here ₹25 lakh, represents the risk premium the decision-maker implicitly demands to accept the underlying uncertainty rather than a guaranteed, smaller but certain amount, and a lower coefficient (further from 1) indicates greater risk aversion, or greater perceived risk in that specific cash flow, than a coefficient closer to 1, which would indicate the decision-maker views the cash flow as close to riskless and therefore requires little discount from its expected value to arrive at its certain equivalent.

Question 11

A company is evaluating a pilot manufacturing project with a static NPV of −₹5 lakh, but success in the pilot would allow the company to expand into a full-scale plant with an NPV, evaluated as of the expansion decision point two years from now, of ₹80 lakh, achievable only if the pilot succeeds, an outcome management assesses as having a 35% probability. Explain, in outline, how the option to expand should be valued and combined with the static NPV to reach an expanded NPV decision.

Accepted Answer

The option to expand should be valued by recognising that the company will only actually exercise the expansion, committing further capital to the full-scale plant, if the pilot succeeds and the expansion genuinely appears value-creating at that future decision point, meaning the ₹80 lakh expansion NPV is only realised in the successful, 35%-probability scenario, and is foregone entirely (at no further cost or loss to the company, since the option, unlike an obligation, need not be exercised) in the unsuccessful scenario. A simplified expected-value approach to valuing this option would compute the probability-weighted value of the expansion opportunity, 35% × ₹80 lakh = ₹28 lakh, and then discount this expected option value back to the present at an appropriate rate reflecting the risk of this contingent opportunity, to arrive at the option's present value; a more rigorous approach would apply an actual option-pricing framework recognising that the option's value also depends on the volatility of the underlying outcome and the time remaining until the expansion decision must be made, typically producing a higher option value than the simplified expected-value approach alone, since option value generally increases with volatility and time to expiry, factors the simplified approach does not explicitly capture. Either way, this option value, once computed, is added to the pilot project's static NPV of −₹5 lakh to arrive at an expanded NPV; if the resulting expanded NPV is positive, the pilot project, despite its own negative static NPV, is worth undertaking specifically because of the valuable follow-on expansion opportunity it creates and preserves.

Question 12

Explain why getting the nominal/real consistency rule right matters proportionately more for a 15-year infrastructure project than for a 2-year working capital-related project, even though the same underlying rule applies to both.

Accepted Answer

The compounding nature of both discounting and inflation means that any systematic error introduced by mismatching nominal and real figures compounds over the number of years the mismatch is applied, so an error that produces only a small distortion over a 2-year project's short life can produce a dramatically larger cumulative distortion over a 15-year project's much longer life, since the same percentage-point mismatch between the rate actually needed and the rate mistakenly used is being compounded across many more years for the longer project. For a 15-year infrastructure project, expected future inflation assumptions, and whether cash flows and the discount rate are handled on a consistently nominal or consistently real basis, materially shape the entire evaluation, potentially reversing an accept/reject conclusion if the mismatch is large enough and sustained over many years; for a short, 2-year project, even a genuine nominal/real inconsistency produces a comparatively smaller absolute distortion simply because there are fewer years over which the compounding error can accumulate, which is why particular care with inflation consistency is emphasised specifically for longer-lived capital projects, where the stakes of getting this specific consistency rule right are proportionately much higher.

Question 13

Explain why the profitability index ranking approach reliably identifies the value-maximising set of projects when projects are divisible, but can fail to do so when projects are indivisible, in one or two sentences of core reasoning.

Accepted Answer

When projects are divisible, a firm can fund any fraction of a project, meaning it can always use its entire capital budget productively by taking whole projects in PI order and then a partial fraction of the next-ranked project to exactly exhaust the remaining budget, guaranteeing that PI ranking allocates every rupee of the budget to its highest available marginal return use and therefore maximises total NPV. When projects are indivisible, a firm cannot take a fractional project, so PI ranking can leave a portion of the budget unused (because the next-ranked project doesn't fit within the remaining budget), and this unused capital represents lost value-creating opportunity that a different combination of lower-individually-ranked but better-budget-fitting projects might have captured instead, which is exactly why indivisible capital rationing requires testing complete feasible combinations for total NPV rather than relying on the simpler PI-ranking shortcut that works reliably only under divisibility.

Question 14

A company holds the rights to develop a piece of land, and its static NPV calculation, based on current market conditions, shows the development would be marginally value-destroying today. The company has no obligation to develop the land by any specific deadline. Explain why the option to delay the investment decision has value here, and what specifically drives that value.

Accepted Answer

The option to delay has value because the company is not forced to make an irreversible investment decision immediately based on current, marginally unfavourable market conditions; by waiting, the company preserves the ability to observe how market conditions evolve, and to proceed with development only if and when conditions improve sufficiently to make the project genuinely value-creating, while avoiding the loss that committing to the marginally negative-NPV project today would lock in. This value is driven specifically by the uncertainty in future market conditions (greater uncertainty generally increases the value of waiting, since it increases the range of possible future outcomes, including meaningfully favourable ones, without requiring the company to bear the downside of an unfavourable outcome since it need not proceed in that case) and by the absence of any deadline forcing an immediate decision, since a genuine option to delay only has value if the company can actually choose to wait rather than being compelled to decide now; a company facing a use-it-or-lose-it deadline, by contrast, would have no genuine option to delay and would need to evaluate the investment based on today's conditions and information alone, without the benefit of this additional flexibility value.

Question 15

Two mutually exclusive one-year projects each require an investment of ₹50 lakh. Project P has an expected cash flow of ₹70 lakh with a certainty equivalent coefficient of 0.85; Project Q has an expected cash flow of ₹75 lakh with a certainty equivalent coefficient of 0.78. The risk-free rate is 7%. Determine which project should be selected using the certainty equivalent method.

Accepted Answer

Certain equivalent cash flow for Project P = 0.85 × ₹70 lakh = ₹59.5 lakh; present value = ₹59.5 lakh ÷ 1.07 = ₹55.61 lakh; NPV = ₹55.61 lakh − ₹50 lakh = ₹5.61 lakh. Certain equivalent cash flow for Project Q = 0.78 × ₹75 lakh = ₹58.5 lakh; present value = ₹58.5 lakh ÷ 1.07 = ₹54.67 lakh; NPV = ₹54.67 lakh − ₹50 lakh = ₹4.67 lakh. Project P has the higher NPV, ₹5.61 lakh versus ₹4.67 lakh, and should be selected, despite Project Q having the higher raw expected cash flow (₹75 lakh versus ₹70 lakh); this reversal occurs precisely because Project Q's lower certainty equivalent coefficient (0.78 versus 0.85) reflects its comparatively greater risk, and once this greater risk is properly accounted for through the smaller certainty equivalent, Project Q's genuinely risk-adjusted value falls below Project P's, illustrating exactly why comparing raw expected cash flows or unadjusted NPVs across projects of differing risk, without applying a risk adjustment such as the certainty equivalent method, can lead to selecting the objectively inferior project on a genuinely risk-adjusted basis.

Practice — Advanced Capital Budgeting Decisions

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