Present Value (PV) factor

A rupee received later is worth less than a rupee today, because today's rupee could be earning a return in the meantime.

The PV factor is the number that does the conversion for one future amount. Multiply the future amount by the factor and you have its present value. The higher the required return, and the further away the cash, the smaller the factor.

Its close relative is the present value annuity factor (PVA), which does the same job for a series of equal amounts — such as a bond's annual coupons. Bond valuation uses both: PVA for the coupons, PV factor for the principal.

The formulas (Chapter 11, footnote 122).

• PV factor = 1 / (1 + r)^n, where n is the number of periods and r the rate of return. Multiplying any future amount by it gives its present value.
• PVA factor = (1 − (1 + r)^(−n)) / r — the annuity factor for n periods at r.

Both can be calculated on a calculator or spreadsheet, or looked up in annuity and discounting factor tables.

Bond value (section 11.2).

Value = I × PVA(r, n) + F × PV(r, n)

where I is annual interest, F the par value, r the required return and n the years to maturity.

When the required return is above the coupon (14% vs 12%) the bond is worth less than par; when it is below (13% vs 14%) it is worth more.

In DCF valuation (section 11.6). The value of an asset is the present value of its future cash flows. In the workbook's DCF computation, each year's free cash flow is multiplied by its PV factor at a WACC of 11.35%, and the terminal value of Rs 43,062.54 is multiplied by a terminal value factor of 0.5245; the sum gives enterprise value, from which debt is deducted. (0.5245 is 1 ÷ 1.1135^6.)

The workbook warns that DCF is extremely sensitive to the timing of each cash flow: receiving it at the start or end of a year changes the answer materially — which is simply the PV factor changing with n.

PV factor   = 1 / (1 + r)^n
PVA factor  = (1 - (1 + r)^(-n)) / r
Present value of a single amount = Future amount x PV factor

A founder buyback promise (illustrative). An AIF holds a contractual right to receive Rs 50 lakh from a promoter in 3 years. Its required return is 12%.

Timing matters. If the same Rs 50 lakh were due in 4 years, the factor would be 1 ÷ 1.12^4 = 1 ÷ 1.573519 = 0.6355, and the present value about Rs 31,78,000 — Rs 3.8 lakh less for a one-year delay.

Rate matters. At 14% for 3 years the factor is 1 ÷ 1.481544 = 0.6750, giving about Rs 33,75,000.

Back to the workbook's bond. Rs 100 of principal due in 8 years at 14% is worth 100 × 0.351 = Rs 35.10 today; the eight Rs 12 coupons are worth 12 × 4.639 = Rs 55.67. Together, Rs 90.77.

Chapter 11 (Valuation), section 11.2 on valuation basics for fixed income instruments uses the PV and PVA factors in Illustrations 11.1 and 11.2 and defines both in footnote 122; section 11.6 applies PV factors in the DCF computation. Expect a bond valuation question that gives both factors, a question asking for the PV factor formula, and interpretation of why a bond trades below or above par.

• PV factor is for a single amount; PVA factor is for a series of equal amounts. A bond needs both.
• 1 / (1 + r)^n, not (1 + r)^n. The factor is always below 1 for a positive rate.
• Required return above coupon → value below par; required return below coupon → value above par.
• The factor falls as n or r rises — a later or riskier cash flow is worth less today.
• Spreadsheet order: =PV(rate, nper, pmt, fv), as in =PV(14%,8,12,100).
• DCF is highly sensitive to timing because every cash flow carries its own PV factor.