2.1 Time Value of Money

Time Value of Money in FMVA Models

The time value of money (TVM) is the rule that a dollar in the firm's account today is worth more than the same dollar delivered later, because the dollar in hand can be invested at a required return. CFI's FMVA path treats this as core Finance skill, not a side formula sheet. Finance is about 10% of the final exam, and the prep courses Math for Finance and Corporate Finance Fundamentals both feed TVM directly into NPV, IRR, bond math, and the discounted cash flow (DCF) models you will build in the core. If you reverse a discount factor incorrectly, every enterprise-value output downstream is wrong.

Two identities cover lump sums. Present value brings a future amount back to today:

PV = FV / (1 + r)^n

Future value pushes a present amount forward:

FV = PV × (1 + r)^n

r is the periodic rate and n is the number of periods. They must be consistent. If r is 8% per year, n is years. If you convert an 8% annual percentage rate into monthly compounding, the periodic rate is 0.08/12 and n is the number of months. Mixing an annual rate with a monthly period count is one of the highest-frequency arithmetic errors on applied Excel case studies.

Lump-Sum Worked Example (Dollars, Not Symbols)

Northline Packaging will receive a $10,000 performance bonus from a customer in 5 years. Northline's opportunity cost of capital is 8% compounded annually.

PV = 10,000 / (1.08)^5

(1.08)^2 = 1.1664, (1.08)^4 = 1.360489, (1.08)^5 = 1.469328

PV = 10,000 / 1.469328 = $6,805.83

The bonus is worth $6,805.83 today. Suppose a factoring firm offers $7,000 cash now for the same receivable. $7,000 is greater than $6,805.83, so Northline should take the $7,000. Waiting for the nominally larger $10,000 destroys $194.17 of value relative to the factor's offer, and $3,194.17 relative to having $10,000 in hand today.

Now compound the other direction. The treasurer invests $5,000 today at 6% for 4 years.

FV = 5,000 × (1.06)^4

(1.06)^2 = 1.1236, (1.06)^4 = 1.262477

FV = $6,312.38

If the firm instead has a contractual need for $6,500 in four years, this $5,000 deposit is $187.62 short. The rate that would hit $6,500 exactly is:

r = (FV / PV)^(1/n) − 1 = (6,500 / 5,000)^(1/4) − 1 = 1.30^0.25 − 1 ≈ 6.78%

Excel: =RATE(4, 0, -5000, 6500) returns 6.78%. The negative $5,000 is the outflow you invest; the positive $6,500 is the inflow you need. This is the same comparison you will later make in a DCF: is the present value of future unlevered free cash flow greater than the capital you must deploy today?

A plant manager who says "we will get $10,000 either now or in five years, so it does not matter" is ignoring the 8% the firm could earn on the early cash. Five years of missed compounding is not a rounding difference. FMVA case studies bury that comparison inside a longer model; the math does not change.

Compounding Frequency and EAR versus APR

A brochure rate is often an annual percentage rate (APR) that compounds m times per year. The future-value identity becomes:

FV = PV × (1 + r/m)^(n × m)

Take $10,000 at an 8% APR for 3 years:

The effective annual rate (EAR) restates the same quote as a once-a-year rate you can place next to annual cash flows:

EAR = (1 + APR/m)^m − 1

A loan that quotes 12% APR compounded monthly has EAR = (1 + 0.12/12)^12 − 1 = (1.01)^12 − 1 = 12.68%. The same 12% APR compounded quarterly has EAR = (1.03)^4 − 1 = 12.55%. If you compare a monthly-pay credit line with a project weighted average cost of capital (WACC) quoted as an annual effective rate, convert first. Discounting annual cash flows at 12% when the true opportunity cost is a 12.68% EAR understates the charge for capital and overstates NPV.

Continuous compounding, EAR = e^(APR) − 1, is the mathematical upper limit as m goes to infinity. At 12%, e^0.12 − 1 = 12.75%. Monthly compounding at 12.68% is already close to that ceiling. Core FMVA items almost always use discrete compounding; know the direction of the effect so you can sanity-check a model rather than derive e^r under timed pressure.

Annuities, Excel Functions, and DCF Timing

The bar chart above is the lump-sum identity in pictures: at 8% year-end discounting, each extra year of delay knocks the $10,000 claim down by another 1/1.08. Year 1 is worth $9,259.26; year 5 is worth $6,805.83. Capital budgeting and DCF simply sum many of those bars, one per dated cash flow.

Ordinary Annuities

Most project, lease, and bond problems are not one lump sum. An ordinary annuity pays a constant PMT at the end of each period for n periods:

PV_annuity = PMT × [1 − (1 + r)^(−n)] / r

The bracketed term is the annuity factor, the present value of $1 paid at each year-end. It is also what Excel's PV function computes when you supply a non-zero pmt and a zero fv.

Worked example: a maintenance contract pays $2,000 at the end of each year for 5 years. The discount rate is 10%.

(1.10)^5 = 1.610510, so (1.10)^(−5) = 0.620921

Annuity factor = (1 − 0.620921) / 0.10 = 3.790787

PV = 2,000 × 3.790787 = $7,581.57

Excel: =PV(0.10, 5, -2000, 0) returns 7,581.57. The payment is entered negative if you are the party who will pay; reverse the sign if you are receiving the contract. Five isolated lump-sum PVs of $2,000 at 10% (years 1 through 5) add to the same $7,581.57 — the annuity factor is only a shortcut for that sum.

An annuity due pays at the beginning of each period. PV_due = PV_ordinary × (1 + r). In Excel, the optional type argument is 0 for end-of-period (ordinary) and 1 for beginning (due). A growing annuity with first end-of-year payment PMT and growth g, when r ≠ g, is:

PV = PMT × [1 − ((1 + g)/(1 + r))^n] / (r − g)

CFI forecast models use a close cousin of this when revenue or free cash flow grows at a constant rate over a finite explicit period before a Gordon growth or exit-multiple terminal value is applied. If r = g, the formula collapses to PV = n × PMT / (1 + r); do not divide by zero.

Excel PV, FV, NPER, and RATE

CFI modeling standards expect you to let Excel's financial functions do the arithmetic, with a consistent sign convention: money that leaves the firm is negative; money that enters is positive.

=PV(0.08, 5, 0, 10000) returns −6,805.83. Excel is telling you that you would pay $6,805.83 today to receive $10,000 in five years. If your cash-flow schedule needs a positive present value of an inflow, either enter the future value as −10000 or take the absolute value after you have checked the economics. Mixing signs is a frequent #NUM! cause in RATE and NPER, and a silent audit fail when a negative PV is summed into a DCF total.

NPER and RATE iterate from a guess. They are not as transparent as the closed-form PV identity, so sanity-check against the identity you already know: NPER at 8% between −6,805.83 and +10,000 must be 5, not 50. RATE between those same two cash amounts over 5 years must be 8%, not 0.8% (a percent-versus-decimal trap) and not 8% monthly unless you also changed nper to 60.

Mid-Year versus Year-End Discounting

Capital-budgeting problems in this chapter, and most textbook TVM drills, assume year-end cash flows: year 1 is discounted by (1 + r)^1, year 2 by (1 + r)^2. That matches a bond coupon or a contract that pays on 31 December.

An operating business earns cash throughout the year. CFI's DCF courses therefore often apply a mid-year convention, treating each forecast year's cash as arriving at the midpoint:

Year-end: PV of CF_t = CF_t / (1 + r)^t

Mid-year: PV of CF_t = CF_t / (1 + r)^(t − 0.5)

A shortcut used in many CFI-style DCF templates: compute the year-end present value of the explicit forecast cash flows, then multiply by (1 + r)^0.5.

Worked comparison: $1,000 of cash associated with year 1, r = 10%.

Year-end PV = 1,000 / 1.10 = $909.09

Mid-year PV = 1,000 / (1.10)^0.5 = 1,000 / 1.048809 = $953.46

Check: $909.09 × 1.048809 = $953.46. Mid-year discounting increases present value because you assume the cash arrives six months earlier. Terminal value still needs its own timing decision — placing TV at the end of the last explicit year is common, then optionally applying the same half-year adjustment to the whole DCF. You will apply this in the DCF chapters. The TVM lesson is that the exponent is a timing assumption, not a physical law. If an FMVA case study says project cash flows occur at year-end, do not switch to mid-year. If it says cash is earned evenly through the year, mid-year is the better match.

Traps That Cost Easy Points

• Discounting a monthly equipment lease with an annual WACC while leaving n in years. Divide the rate by 12 and multiply n by 12, or convert the lease to an annual EAR first.
• Using a 12% APR as if it were an EAR when cash flows are annual. The matching annual rate is 12.68% if compounding is monthly.
• Dropping Excel's negative PV into an inflow row and then wondering why enterprise value went negative.
• Applying mid-year to a capital-budgeting timeline that is specified as year-end, or skipping mid-year in a DCF that states cash is earned evenly.

TVM does not decide whether a project is good. It only translates dated cash into a common unit — today's dollars. Section 2.2 uses that unit to apply NPV and IRR. Section 2.3 decides which cash flows belong on the timeline.