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CFA Level II Exam · Residual Income Valuation

Residual Income Persistence Factor and Terminal Value Explained

Updated 7 October 2026 · Fact-checked

The persistence factor (ω) is the share of this year's residual income that carries into next year. Residual income after the forecast horizon is assumed to fade by ω each year. Terminal value at time T is RI(T+1) ÷ (1 + r − ω), added to book value and discounted back.

Understand Residual Income Persistence and Terminal Value

Residual income (RI) is earnings above the charge for equity capital: RI = net income − (r × beginning book value). A firm creates value only when RI is positive. The model values equity as book value today plus the present value of all future RI.

You cannot forecast RI forever. So you forecast for a set horizon and then estimate what happens after it. Competition usually erodes high returns, and weak firms often improve. RI therefore tends to drift toward zero over time. This drift is called fade.

The persistence factor ω captures the fade. It is a number between 0 and 1. If ω = 0, RI vanishes at once after the horizon. If ω = 1, RI continues forever at the same level. A value such as 0.6 means each year's RI is 60% of the year before. Higher ω means slower fade and a higher terminal value.

The exam also describes ω in words. High persistence suits firms with durable advantages. Low persistence suits firms in competitive industries or with accounting-driven earnings. Research suggests persistence is higher for firms with low accruals and lower for firms with large one-time items. Treat that as a tendency, not a rule.

The terminal value is the present value, at the end of the horizon, of all RI after it. With RI declining by ω each year, that value is a growing perpetuity with a growth rate of ω. You then discount it back to today, along with the explicit-period RI.

Key formulas to remember

Residual income
RI(t) = NI(t) − r × B(t−1)
Equivalent: RI(t) = (ROE(t) − r) × B(t−1). B is book value of equity.
Persistence fade
RI(T+n) = ω^n × RI(T)
Also RI(t+1) = ω × RI(t). ω is between 0 and 1 in the usual exam case.
Terminal value of continuing RI at time T
PV at T of RI after T = RI(T+1) ÷ (1 + r − ω)
Here RI(T+1) = ω × RI(T). Check the sign: the denominator is 1 + r − ω, not r − ω.
Multistage RI value
V0 = B0 + Σ RI(t) ÷ (1 + r)^t + [RI(T+1) ÷ (1 + r − ω)] ÷ (1 + r)^T
Sum runs t = 1 to T. Discount the terminal value by T years.
Special cases of the terminal value
ω = 1: persistent RI, PV = RI(T+1) ÷ r. ω = 0: PV = 0
With ω = 1 the denominator 1 + r − ω becomes r, so it is a perpetuity.

How to solve Residual Income Persistence and Terminal Value questions

Use the same sequence for any vignette that asks for a multistage RI value or a terminal value with fade.

  1. 1Find r, B0, the horizon T, and ω in the vignette or exhibits. Note whether ω is given or must be judged from the description.
  2. 2Compute RI for each forecast year as NI − r × beginning book value. Check you use beginning, not ending, book value.
  3. 3Find the last explicit RI, RI(T). Compute RI(T+1) = ω × RI(T).
  4. 4Compute the terminal value at time T as RI(T+1) ÷ (1 + r − ω).
  5. 5Discount each explicit RI and the terminal value to today at r. The terminal value is discounted T years.
  6. 6Add everything to B0 to get intrinsic equity value. Divide by shares if per-share value is asked.
  7. 7Sense-check: a higher ω should raise value, and negative RI with ω below 1 should shrink toward zero.

Quickest way: Single-shortcut terminal value

When to use it: When the question gives RI in the final forecast year and ω, and asks for the PV of continuing RI or the total value.

  1. Multiply final RI by ω to get RI(T+1).
  2. Divide by (1 + r − ω) to get the terminal value at T.
  3. Divide by (1 + r)^T for the present value.
  4. Add PV of explicit RI and B0. Skip any step the question has already done.

Common mistakes in Residual Income Persistence and Terminal Value

  • Using r − ω in the denominator instead of 1 + r − ω.

    It looks like the Gordon growth formula, r − g.

    Fix: RI declines by factor ω, so the growth rate is ω − 1. That gives r − (ω − 1) = 1 + r − ω.

  • Forgetting to multiply RI(T) by ω before applying the formula.

    Students plug in the last explicit RI as if it were the first terminal-period RI.

    Fix: The numerator is RI(T+1) = ω × RI(T). Write that line explicitly.

  • Discounting the terminal value by the wrong number of years.

    Confusion between T and T+1 after dividing.

    Fix: The formula already values at time T. Discount by (1 + r)^T.

  • Computing RI with ending book value.

    Both book values appear in the exhibit.

    Fix: Equity charge is r × beginning-of-year book value.

  • Treating ω as a growth rate and allowing RI to grow.

    The word persistence is confused with growth.

    Fix: ω is a retention share of RI. At ω = 1, RI is flat, not growing. Fade means ω below 1.

  • Adding B0 twice or leaving it out.

    Focus on the RI stream hides the starting term.

    Fix: Always finish with V0 = B0 + PV of all RI.

Worked examples

Example 1

Vignette: A firm has B0 = 100, required return r = 10%, and forecast RI of 8 in year 1 and 6 in year 2. After year 2, RI fades with persistence factor ω = 0.5. Q1: What is RI in year 3? Q2: What is the terminal value at the end of year 2? Q3: What is the intrinsic equity value today?

Show the solution
  1. Q1: RI(3) = ω × RI(2) = 0.5 × 6 = 3.
  2. Q2: TV at year 2 = 3 ÷ (1 + 0.10 − 0.5) = 3 ÷ 0.6 = 5.
  3. Q3: PV of RI(1) = 8 ÷ 1.10 = 7.2727. PV of RI(2) = 6 ÷ 1.21 = 4.9587.
  4. PV of TV = 5 ÷ 1.21 = 4.1322.
  5. V0 = 100 + 7.2727 + 4.9587 + 4.1322 = 116.36.

Answer: RI(3) = 3; terminal value at year 2 = 5; intrinsic equity value ≈ 116.36.

Example 2

Vignette: An analyst values a company with B0 = 50 and r = 8%. Net income is forecast at 6.00 in year 1 and 6.60 in year 2. Book value at end of year 1 is 54. After year 2, RI persists with ω = 0.7. Q1: What is RI in year 2? Q2: What is the PV today of RI after year 2? Q3: What happens to the PV of continuing RI if ω rises to 1?

Show the solution
  1. Year 1 RI = 6.00 − 0.08 × 50 = 2.00.
  2. Q1: RI(2) = 6.60 − 0.08 × 54 = 6.60 − 4.32 = 2.28.
  3. Q2: RI(3) = 0.7 × 2.28 = 1.596.
  4. TV at year 2 = 1.596 ÷ (1 + 0.08 − 0.7) = 1.596 ÷ 0.38 = 4.20.
  5. PV today = 4.20 ÷ 1.08² = 4.20 ÷ 1.1664 = 3.60.
  6. Q3: With ω = 1, RI(3) = 2.28 and TV = 2.28 ÷ 0.08 = 28.50, far higher than 4.20. Value rises.

Answer: RI(2) = 2.28; PV today of continuing RI ≈ 3.60; raising ω to 1 increases the PV of continuing RI sharply (terminal value 28.50 at year 2).

Exam tips

  • Read the vignette for ω clues. Phrases like intense competition or large one-off gains point to a lower ω; durable franchise points to a higher one.
  • Write RI(T+1) as its own line before dividing. Most lost marks come from skipping it.
  • Questions often ask direction only. Higher ω raises terminal value, and for negative RI it makes value lower because losses persist.
  • There is no penalty for wrong answers, so answer every question even if time is short.
  • Keep four decimals through the discounting and round only at the end, since options can be close.

Residual Income Persistence and Terminal Value: frequently asked questions

What is the persistence factor in residual income valuation?

It is the proportion of one year's residual income that continues into the next year. It lies between 0 and 1 in the usual case. A value of 1 means RI never fades and 0 means it disappears immediately.

How do you calculate terminal value in the residual income model?

Multiply the last forecast RI by ω to get RI for the first year after the horizon. Divide by (1 + r − ω) to get the value at the horizon. Then discount it back by the number of forecast years.

Why is the denominator 1 + r − ω?

RI shrinks by a factor ω each year, so it behaves like a perpetuity growing at ω − 1. Applying the growth perpetuity formula r − g with g = ω − 1 gives 1 + r − ω.

What if the persistence factor is 1?

Then RI continues at the same level forever. The terminal value becomes RI(T+1) ÷ r, a simple perpetuity. This is an aggressive assumption and the exam may ask you to judge it.

Does a high persistence factor always raise value?

Only when continuing RI is positive. If RI is negative, higher persistence means losses last longer and value falls.