A dollar today can be invested — that is the whole reason

◈ 7 cards

Explain why money has time value, compute a present-value factor as 1 ÷ (1 + r)ⁿ, read a 4-decimal table, and predict how PV moves with the rate and the horizon.

Why 100,000 today beats 100,000 in five years

Offered 100,000 today or 100,000 in five years, take it today — not because prices will rise, but because 100,000 today can be invested. At 8 % a year it grows to $100, 000 \times 1.0 8^{5} = 146, 933$ by the time the other 100,000 arrives. The time value of money is that investment opportunity. Inflation is a separate matter: it changes what the dollars buy, and an analyst allows for it by adjusting the rate or the flows, but money would have time value in a world with no inflation at all, so long as it could earn a return.

Capital budgeting runs the same idea backwards. Lakehead’s boat lift pays 270,000 in year 4; what is that worth today? The amount that, invested now at the required return, would grow to 270,000 by year 4 — its present value.

The factor

If 1 today grows to $(1 + r)^{n}$ in $n$ years, then 1 received in $n$ years is worth

$PV factor = \frac{1}{( 1 + r ) ^{n}}$

today. At 8 % for 4 years: $1 \div 1.0 8^{4} = 1 \div 1.3605 = 0.7350$ . A present-value table lists these factors, rounded to four decimals, for each rate and year — the exam gives you one, so the skill is reading the row for $n$ and the column for $r$ . Multiplying by the factor is discounting.

Cobalt Ridge Mining expects to collect 250,000 in four years from a customer whose contract pays at the end of the mine’s development. At Cobalt Ridge’s 8 % required return, that receivable is worth

$250, 000 \times 0.7350 = 183, 750$

today. The other 66,250 is the return the money would have earned over the four years had Cobalt Ridge held it now.

The two sensitivities

The factor is a fraction with $(1 + r)^{n}$ underneath, so it shrinks when either $r$ or $n$ grows. Take 100,000 five years out:

Rate   Factor (5 years)   PV of 100,000
 4 %        0.8219             82,190
 8 %        0.6806             68,060
12 %        0.5674             56,740

A higher rate gives a lower present value — the more the money could earn elsewhere, the less a fixed future sum is worth now. And at a fixed 8 %:

Years   Factor (8 %)   PV of 100,000
  3        0.7938          79,380
  5        0.6806          68,060
 10        0.4632          46,320

A longer wait gives a lower present value — ten years out, 100,000 is worth less than half of itself. Both rules are what makes lesson 13.6’s relocation reverse when discounted: the savings arrive late, the costs arrive now.

Table or calculator

A calculator or a line of Python gives $100, 000 \div 1.0 4^{5} = 82, 193$ where the table gives 82,190; the table’s factor was rounded at four decimals. The gap is a few dollars in a hundred thousand and is not an error. Every graded worksheet in this module uses the factors the prompt supplies, so the expected value is the table’s. The first code block below prints the table so you can see where each factor comes from; the second prints the calculator answers so you can see the size of the rounding.

Discounting moves a future sum to t0 by multiplying by the factor: 250,000 × 0.7350 = 183,750. The other 66,250 is four years of return at 8 %.