13.1 Time Value of Money, Future Value & Present Value

Mine projects exchange large early expenditures for uncertain cash flows years later. The time value of money makes those amounts comparable at one valuation date. A peso today can be invested, avoids financing cost, and is exposed to less waiting risk than a peso received later. Valuation therefore compounds money forward or discounts it backward.

Future Value

For present capital $PV$, periodic rate $i$, and $n$ periods:

$FV = PV(1+i)^n$

If PHP 80 million is committed today and the applicable annual financing rate is 8%, its future-value equivalent after three years is:

$FV = 80(1.08)^3 = PHP100.78\text{ million}$

This does not mean the equipment will be worth PHP 100.78 million. It measures the time-value equivalent of the capital. If compounding occurs monthly, convert the annual convention correctly; do not divide an effective annual rate by 12 without checking its definition.

Present Value

A future cash flow is discounted as:

$PV = \frac{FV}{(1+i)^n}$

A PHP 150 million net cash flow expected in Year 4 has a present value at 10% of:

$PV = \frac{150}{(1.10)^4}=PHP102.45\text{ million}$

Higher discount rates or later receipt reduce present value. Use end-of-period timing unless the problem states otherwise. A Year 0 cost is not discounted; a Year 1 cash flow is divided by $(1+i)^1$.

Uniform Series

For an equal end-of-period payment $A$ over $n$ periods, the present value of an ordinary annuity is:

$PV=A\left[\frac{1-(1+i)^{-n}}{i}\right]$

This can value equal lease payments or simplified closure-monitoring costs. A perpetuity $A/i$ is inappropriate where the obligation ends or changes materially.

Discounted Cash Flow and NPV

A mining cash-flow model generally includes development capital, sustaining capital, production revenue, mining and processing cost, royalties and taxes, working capital, reclamation and closure cost, salvage, and financing treatment according to the valuation purpose.

$NPV = \sum_{t=0}^{n}\frac{CF_t}{(1+i)^t}$

Consider cash flows in PHP millions of -120 at Year 0, +50 at Year 1, +55 at Year 2, and +60 at Year 3. At 10%:

$NPV=-120+\frac{50}{1.10}+\frac{55}{1.10^2}+\frac{60}{1.10^3}$

$NPV=-120+45.45+45.45+45.08=PHP15.98\text{ million}$

The positive value means the modeled project earns more than the 10% hurdle. It is not proof that the geological model or price forecast is correct. Internal rate of return (IRR) is the rate at which NPV equals zero; unusual cash-flow sign changes can create multiple IRRs, so examine NPV and the cash-flow profile.

Real and Nominal Consistency

A nominal rate includes inflation; a real rate removes it. The relationship is:

$(1+i_n)=(1+i_r)(1+f)$

If real return is 6% and inflation is 4%, nominal rate is $(1.06)(1.04)-1=10.24%$. Discount inflated nominal cash flows with the nominal rate. Discount constant-price real cash flows with the real rate. Also maintain currency consistency: a US-dollar metal price combined with Philippine-peso costs requires explicit exchange-rate assumptions.

Risk and Sensitivity

The discount rate cannot repair biased inputs. Test variables that drive value:

• head grade and ore tonnes;
• dilution and mining recovery;
• metallurgical recovery and payability;
• commodity price and exchange rate;
• development schedule and ramp-up;
• operating and capital cost; and
• closure, social, permitting, and environmental obligations.

A tornado chart varies one input at a time; scenarios change coherent sets, such as low price plus delayed ramp-up. State whether values are pre-tax or after-tax, levered or unlevered, nominal or real, and the valuation date. An apparently precise NPV without those conventions is not auditable.

Equivalent Annual Cost Example

Two dewatering alternatives with unequal lives cannot be compared from purchase price alone. Convert each alternative's discounted life-cycle cost to an equivalent annual cost (EAC):

$EAC = PV_{cost}\left[\frac{i(1+i)^n}{(1+i)^n-1}\right]$

If Alternative A has present cost PHP 50 million, five-year life, and 10% discount rate, its capital-recovery factor is about 0.2638 and EAC is PHP 13.19 million per year before common operating items. Repeat consistently for the other alternative, including replacement, downtime, energy, maintenance, salvage, and tax treatment where the problem specifies them.

Common Calculation Checks

Draw a time line, label Year 0, state whether each flow occurs at the beginning or end of a period, preserve signs, and show units. Verify that a positive discount rate makes a future receipt smaller in present-value terms. Recalculate one result using a spreadsheet function or factor table, but confirm the function's timing convention.

Final reasonableness test: estimate an order of magnitude without discounting. A present value of a positive future receipt should remain positive and below its undiscounted amount when the rate is positive.

Timing Convention Check

State whether cash flows occur at period end, period start, or continuously; a one-period timing shift changes value. Match monthly or annual periods to the discount rate, and convert rates consistently rather than dividing an effective annual rate by twelve without checking the convention. Separate valuation date, base date, escalation date and exchange-rate date so an apparently correct formula does not discount mismatched cash flows.