Streams of Cash Flows

Present Value of an Annuity

A stream of equal payments received is discounted term-by-term, then summed and reconciled to a cent.

highlighted = computed this step

Each payment’s discounted value

A payment of $1,000.00 is worth $909.09 one period out; in two periods it is $826.45; in three periods, it is $751.31.

PV1=$909.09, PV2=$826.45, PV3=$751.31\text{PV}_{1}=\$909.09,\ \text{PV}_{2}=\$826.45,\ \text{PV}_{3}=\$751.31
PV of annuityThree equal payments received at period ends discounted together. 0123$1,000.00$1,000.00$1,000.00PV: $2,486.85

Exact fraction vs rounded display

The exact present value is the fraction 331000000/1331 cents, i.e. 3310000/1331 dollars. Rounded half-up gives $2,486.85. The exact and rounded forms represent the same value, and this rule only affects display. Institutions and regulations can use different rules (for example, banker's rounding or round-half-to-even).

PVexact=331000000/1331 cents\text{PV}_{\text{exact}}=331000000/1331\text{ cents}
PV of annuityThree equal payments received at period ends discounted together. 0123$1,000.00$1,000.00$1,000.00PV: $2,486.85

Recompute and reconcile

Term-by-term, $909.09 + $826.45 + $751.31 equals $2,486.85; this is the rounded value of the exact fraction term.

PVstream=$909.09+$826.45+$751.31=$2,486.85\text{PV}_{\text{stream}} = \$909.09 + \$826.45 + \$751.31=\$2,486.85
PV of annuityThree equal payments received at period ends discounted together. 0123$1,000.00$1,000.00$1,000.00PV: $2,486.85