Present Value Annuity Factor
Also written PVA · Present Value Annuity Factor (PVA) · Present Value Annuity factor (PVA)
The number that converts a level stream of equal payments for n periods into its present value at rate r: PVA(n, r) = (1 − (1 + r)^−n) / r. Used with the PV factor to value bonds.
In plain language
A bond pays the same coupon every year and then returns its face value at the end. Discounting each coupon one by one works, but it is slow. The present value annuity factor does all the coupon discounting in one multiplication.
The workbook defines it as "the annuity factor for 'n' time periods at a rate of return 'r'". Multiply the annual payment by it and you have the present value of the whole stream of payments. Its partner is the plain PV factor, 1 / (1 + r)^n, which brings a single future amount — such as the principal repaid at maturity — back to today.
Both rest on the time value of money, which the workbook states simply: money is worth more today than it will be tomorrow, so a future sum is multiplied by a present value factor to find what it is worth now.
How it works
Bond value (section 11.2):
Value = I × PVA(r, n) + F × PV(r, n)
where I is the annual interest, F the principal (par value), r the required return and n the maturity period.
Illustration 11.1. Rs 100 par bond, 12% coupon, 8 years, required return 14%.
PVA(14%, 8) = (1 − 1.14^−8) / 0.14 = 4.639
PV(14%, 8) = 1 / 1.14^8 = 0.351
Value = 12 × 4.639 + 100 × 0.351 = Rs 90.77
Illustration 11.2. Rs 1,000 par bond, 14% coupon, 5 years, required return 13%.
Value = 140 × 3.517 + 1,000 × 0.543 = Rs 1,035.4
In a spreadsheet the workbook uses =PV(rate, nper, pmt, fv), for example =PV(13%,5,140,1000). Factors can also be read from annuity and discounting tables.
Why an AIF distributor needs it. Fixed income valuation applies in an AIF context for three reasons the workbook lists: AIFs floated as debt funds invest in debt securities of investee companies, SPVs, InvITs and REITs; AIFs use debt structures alongside equity or as convertibles that become equity after set milestones; and AIFs may hold debt securities to manage the risk profile and liquidity of the scheme. Under the IPEV Guidelines, debt investments with predictable periodic cash flows are valued by DCF of those cash flows — which, for a level coupon, is exactly this calculation.
The formula
PVA(n, r) = (1 − (1 + r)^−n) / r
PV factor = 1 / (1 + r)^n
Bond value = I × PVA(r, n) + F × PV(r, n)
A worked example
Illustrative figures. A Category II debt AIF holds unlisted NCDs of a Hyderabad hospital chain: face value Rs 10 lakh, 11% annual coupon, 3 years to maturity. The valuer judges the required return for this credit at 13%.
Annual coupon I = 11% × 10,00,000 = Rs 1,10,000
PV(13%, 1) = 0.885
PV(13%, 2) = 0.783
PV(13%, 3) = 0.693
PVA(13%, 3) = 0.885 + 0.783 + 0.693 = 2.361
= (1 − 1.13^−3) / 0.13 = 2.361 (same answer)
Value = 1,10,000 × 2.361 + 10,00,000 × 0.693
= 2,59,710 + 6,93,000
= Rs 9,52,710
The NCD is worth about Rs 9.53 lakh, below its Rs 10 lakh face value, because the market now wants 13% from a paper that pays 11%. If the required return fell to 11%, the value would return to par; if it fell below 11%, the NCD would be worth more than Rs 10 lakh — as in Illustration 11.2, where a 14% coupon discounted at 13% gives Rs 1,035.4 on Rs 1,000 par.
Why NISM asks about it
Chapter 11 (Valuation) opens its technical content at 11.2 with fixed income valuation and defines the PVA and PV factors in the footnote to Illustration 11.1; Chapter 6 (Risk and Return) introduces the underlying present value idea. Expect a bond-value computation with the factors given, and a question on which factor applies to the coupons and which to the principal.
Common exam traps
- PVA applies to the coupon stream; the PV factor applies to the principal. Multiplying the principal by the annuity factor is the classic wrong answer.
- Discount at the required return, not at the coupon rate. Discounting at the coupon rate always gives par.
- Required return above coupon → value below par; below coupon → above par. Illustration 11.1 (14% required on 12% coupon) gives Rs 90.77; Illustration 11.2 (13% required on 14% coupon) gives Rs 1,035.4.
- PVA(n, r) equals the sum of the individual PV factors for years 1 to n — a handy check if a question gives only the single-year factors.
- The workbook writes the arguments in both orders — PVA(n, r) in the footnote and PVA(r, n) in the bond formula. They mean the same factor.
Where this is taught
- Series XIX-A · Chapter 11: Valuationintroduced here
- Series XIX-C · Chapter 14: Valuationintroduced here
- Series XIX-E · Chapter 12: Valuationintroduced here
- Series XIX-D · Chapter 11: Valuationintroduced here
Related terms
- Discounted Cash FlowA valuation method that estimates the cash a business will generate in future years and converts each year back to what it is worth today.
- Yield to MaturityThe single discount rate at which a bond's future coupons and redemption amount add up to exactly its market price today — the return you actually earn if you hold it to maturity.
- AnnuityAn insurance contract in which a corpus paid as a lump sum or in instalments buys a periodic payment from the insurer, at a rate fixed at purchase and guaranteed for life or the contracted term.
- Coupon rateThe rate of interest a bond pays, applied to its face value and never to its market price — which is why the coupon tells you the cash flow but not the return.
- Time value of moneyThe principle that the same sum of money is worth different amounts at different points on a timeline, because money held today can be invested and because inflation erodes what it will buy.
- Milestone ValuationAn investor protection under which a venture investor pegs the company's valuation to technical or commercial targets in the business plan, so missed milestones can mean a lower valuation or delayed funding.
- Present Value (PV) factorThe multiplier that converts a single future amount into today's value: 1 ÷ (1 + r)^n. At 14% for 8 years it is 0.351, so Rs 100 due then is worth Rs 35.10 now.