## Present Value and Future Value Interest Factor Tables

### Why These Tables Exist

Time-value-of-money calculations reduce to repeated multiplication/division by discount or growth factors. Rather than computing `(1+r)^n` or `1/(1+r)^n` from scratch every time, exam bodies pre-compute these four standard factors and present them as look-up tables.

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### The Four Factors at a Glance

Symbol	Full Name	Formula	Reads as…
PVIF(r,n)	Present Value Interest Factor	`1 / (1+r)^n`	₹1 received n years from now is worth this much today at rate r
PVIFA(r,n)	PV Interest Factor of Annuity	`[1 − 1/(1+r)^n] / r`	₹1 received at the end of each year for n years is worth this much today
FVIF(r,n)	Future Value Interest Factor	`(1+r)^n`	₹1 invested today grows to this amount in n years at rate r
FVIFA(r,n)	FV Interest Factor of Annuity	`[(1+r)^n − 1] / r × (1+r)`	₹1 deposited at the beginning of each year accumulates to this amount

> Critical footnote distinction:

> PVIFA assumes payments at the end of each period (ordinary annuity / annuity-immediate).

> FVIFA assumes payments at the beginning of each period (annuity-due).

> This asymmetry is baked into ICAI's published tables — never assume both use the same convention.

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### How to Read the Tables

1. Locate the column for the interest rate `r` (e.g., 10%).

2. Locate the row for the number of periods `n` (e.g., 5 years).

3. The cell value is your factor.

4. Multiply that factor by the cash-flow amount.

Example read: PVIF(10%, 5) = 0.621 → ₹1 five years from now = ₹0.621 today.

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### Relationship Between Single-Period and Annuity Factors

$$\text{PVIFA}(r,n) = \sum_{t=1}^{n} \text{PVIF}(r,t)$$

Each PVIFA row value equals the cumulative sum of PVIF values up to that year — so PVIFA(10%, 3) = PVIF(10%,1) + PVIF(10%,2) + PVIF(10%,3) = 0.909 + 0.826 + 0.751 = 2.487 ✓

Similarly, FVIFA(r,n) is the accumulated sum of FVIF values (annuity-due convention).

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### Quick Sanity Checks

PVIF values are always < 1 (discounting shrinks value).
FVIF values are always > 1 (compounding grows value).
Higher `r` → lower PVIF, higher FVIF.
PVIF(r,n) × FVIF(r,n) = 1 always (they are reciprocals).
PVIFA(r,n) increases with more years; PVIFA(r,∞) = 1/r (perpetuity formula).

Worked example

### Example 1

Using PVIF — Single lump sum discounting

A company expects to receive ₹50,000 at the end of 7 years. Required return = 12%. What is the present value?

Step 1: Look up PVIF(12%, 7) = 0.452

Step 2: PV = ₹50,000 × 0.452 = ₹22,600

Interpretation: ₹50,000 seven years away is worth only ₹22,600 today at 12%.

### Example 2

Using PVIFA — Ordinary annuity (end-of-period payments)

A project generates ₹10,000 per year for 8 years (end of each year). Discount rate = 15%. Find NPV of cash inflows.

Step 1: Look up PVIFA(15%, 8) = 4.487

Step 2: PV of annuity = ₹10,000 × 4.487 = ₹44,870

Note: PVIFA table assumes end-of-period payments, matching this problem exactly.

### Example 3

Using FVIF — Lump sum compounding

₹25,000 is invested today at 9% p.a. for 10 years. What is the maturity value?

Step 1: Look up FVIF(9%, 10) = 2.367

Step 2: FV = ₹25,000 × 2.367 = ₹59,175

### Example 4

Using FVIFA — Annuity-due (beginning-of-period payments)

A sinking fund requires ₹5,000 deposited at the start of each year for 6 years at 8% p.a. What is the fund's value at the end of year 6?

Step 1: Look up FVIFA(8%, 6) = 7.923

Step 2: FV = ₹5,000 × 7.923 = ₹39,615

Warning: This is valid only because the FVIFA table already assumes beginning-of-period deposits. If deposits were at year-end instead, divide the table factor by (1+r) first.

### Example 5

Combining PVIF and PVIFA — Mixed cash flows

Project cash flows: ₹8,000 p.a. for years 1–5 (end of year), then a terminal salvage of ₹20,000 at end of year 5. Discount rate = 10%.

PV of annuity = ₹8,000 × PVIFA(10%, 5) = ₹8,000 × 3.791 = ₹30,328

PV of salvage = ₹20,000 × PVIF(10%, 5) = ₹20,000 × 0.621 = ₹12,420

Total PV = ₹30,328 + ₹12,420 = ₹42,748

⚠️ Common exam mistakes

Confusing PVIFA and FVIFA payment-timing conventions: PVIFA uses end-of-period (ordinary annuity) but FVIFA uses beginning-of-period (annuity-due) in ICAI's tables — using the wrong factor gives an answer off by a factor of (1+r).
Using PVIFA for a single lump sum instead of PVIF — PVIFA is only for a stream of equal periodic payments.
Looking up the wrong row or column — always double-check both r (column) and n (row) before reading the cell value.
Forgetting to multiply the factor by the actual cash-flow amount — the table factor alone is not the answer.
Assuming PVIF(r,n) × FVIF(r,n) ≠ 1 — they are exact reciprocals; if your answer violates this, recheck your table read.
Using a PVIFA factor for an annuity that starts at year 0 (annuity-due) without adjustment — the ordinary-annuity PVIFA must be multiplied by (1+r) to convert to an annuity-due.
Interpolating between table values incorrectly for rates not in the table (e.g., 10.5%) — linear interpolation is an approximation; use the formula for precision when the rate falls between columns.

Reference:

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PV and FV Interest Factor Tables (PVIF, PVIFA, FVIF, FVIFA)

Worked example

⚠️ Common exam mistakes