Before spending money on a market survey, a geological report or a forecaster, a manager should ask what the information is worth. If it would never change the decision, it is worth nothing at any price. If it would sometimes steer the company away from a loss or towards a gain, it is worth the average benefit of that guidance.
The upper limit on what to pay for information that removes all uncertainty is the value of perfect information, which is easy to calculate once a decision tree has been built; the value of imperfect information, the more realistic case of a forecaster who is sometimes wrong, takes a little more work.
This topic explains how to calculate both, using the new-product decision of the last topic, and what to do with the answer. The next topic covers simulation, risk-adjusted discount rates and discounted payback.
The syllabus asks for the expected value criterion “including the value of information”. Read expected value and decision trees first, because the calculation builds directly on a tree.
In 30 seconds
The value of perfect information = expected value with perfect information (choose the best action for each state, weighted by the probability of each state) − expected value without information (the best action on the prior probabilities). It is the most that a decision maker should pay for information that removes all uncertainty.
Imperfect information (a forecast that may be wrong) is worth less: find the probability of each forecast, revise the probabilities of the states given each forecast, choose the best action after each forecast, and compare the expected value with that without information. Information has value only if it could change the decision; pay for it only if its value exceeds its cost.
The value of perfect information
Perfect information tells the decision maker for certain which state will occur before the decision is made. The value of perfect information is:
Value = expected value with perfect information − expected value without perfect information
With perfect information the decision maker would pick the best action for each state and the expected value is the average of the best payoffs, weighted by the probabilities of the states. Without it, the decision maker picks the single action with the best expected value.
A simple payoff table
A company chooses between actions X, Y and Z, and the state of the market is either S1 (probability 0.4) or S2 (probability 0.6). The payoffs (₦m) are: X, +50 in S1 and −10 in S2; Y, +20 and +25; Z, 0 and 0.
- Without information:
expected value (EV) of X = 0.4 × 50 + 0.6 × (−10) = 14
Y = 0.4 × 20 + 0.6 × 25 = 23
Z = 0.
Choose Y, EV 23. - With perfect information:
in S1 choose X (50), in S2 choose Y (25): EV = 0.4 × 50 + 0.6 × 25 = 35. - Value of perfect information.
35 − 23 = ₦12m: the most the company should pay for a perfect forecast.
From the decision tree
In the new-product tree, development costs ₦30m and succeeds with probability 0.7. After success the company chooses between launching (extra cost ₦15m; demand high with probability 0.6, present value ₦100m, or low with probability 0.4, present value ₦30m) and selling the rights for ₦35m. Without information the best choice after success is to launch, with a value of 0.6 × 100 + 0.4 × 30 − 15 = 57; the expected net present value (NPV) of developing is 0.7 × 57 − 30 = +9.9.
Suppose a perfect forecast of the level of demand could be obtained after the development succeeds and before the launch decision. Then: if demand will be high (probability 0.6) launch, worth 100 − 15 = 85; if it will be low (probability 0.4), do not launch but sell the rights for 35, which beats launching (30 − 15 = 15). The expected value after success becomes 0.6 × 85 + 0.4 × 35 = 65, against 57 without the forecast, a gain of 8.
This applies only if development succeeds (probability 0.7), so at the start the value of the forecast is 0.7 × 8 = ₦5.6m (the expected NPV becomes 0.7 × 65 − 30 = +15.5 against +9.9). A forecasting service costing more than ₦5.6m should not be bought, whatever it promises.
The value of imperfect information
Information is rarely perfect. A forecaster may be right 80% of the time. To value the forecast:
- Find the probability of each forecast: the forecast “high” is given with probability 0.6 × 0.8 (high demand, correct) + 0.4 × 0.2 (low demand, wrong) = 0.56; “low” with probability 0.44.
- Revise the probabilities of the states given each forecast: given “high”, P(high demand) = 0.48 ÷ 0.56 = 0.857
given “low”, P(high demand) = 0.12 ÷ 0.44 = 0.273. - Find the best action after each forecast: after “high”, launching is worth 0.857 × 100 + 0.143 × 30 − 15 = 75.0 (better than selling at 35)
after “low”, launching is worth 0.273 × 100 + 0.727 × 30 − 15 = 34.1, which is less than the 35 from selling, so sell. - Weight by the forecast probabilities:
0.56 × 75.0 + 0.44 × 35 = 57.4.
The value of the imperfect forecast is 57.4 − 57 = 0.4 after success, or 0.28 at the start. It is far below the value of perfect information, because the forecast is not good enough to change the decision very often: it only changes the decision after a “low” forecast, and then the gain is small. Such a forecast would be worth buying only if it cost less than ₦0.3m.
The ideas in more depth
Why the value of perfect information is an upper limit
No real source of information can do better than telling you the state with certainty, so no real source can be worth more than the value of perfect information. If a service costs more than that, it is not worth buying even if it were perfect. Imperfect sources are worth less, and sometimes nothing.
Why information can be worthless
If the best action is the same whatever the forecast says, information does not change what the company does, so it adds nothing. The company in the example would still launch after a “high” forecast. The only change is to sell after a “low” forecast; if selling were worth much less than launching even in the low state, the forecast would be worthless.
Sources and references
- No accounting standard governs the calculation. The revision of probabilities is an application of Bayes’ theorem, taught in the course on quantitative methods.
These points are stated as at 5 November 2026.
Worked examples
Example 1: perfect information. EV without information 23; best payoffs 50 (S1, probability 0.4) and 25 (S2, probability 0.6). Required: the value.
- EV with information.
0.4 × 50 + 0.6 × 25 = 35. - Value.
35 − 23 = ₦12m.
Example 2: the probability of a forecast. Demand high 0.6, low 0.4; the forecaster is right 80% of the time. Required: the probability of a “high” forecast.
- Probability.
0.6 × 0.8 + 0.4 × 0.2 = 0.56.
Example 3: revised probability. Required: the probability that demand is high given a “high” forecast.
- Revised.
(0.6 × 0.8) ÷ 0.56 = 0.48 ÷ 0.56 = 0.857.
How to answer in the exam
- Calculate the EV without information first from the prior probabilities.
- For perfect information, pick the best action for each state and weight by the state probabilities.
- For imperfect information, show the forecast probabilities, the revised probabilities and the action after each forecast.
- State the maximum price and compare it with the cost of the information.
Common mistakes in the exam
- Forgetting the initial outlay when valuing the options.
- Using the probabilities of the forecast as the probabilities of the states.
- Valuing information after the decision has been taken.
- Adding the value to the NPV instead of treating it as a ceiling on the price.
- Not adjusting for the probability that the stage at which the information is used is reached.
- Treating a forecast that cannot change the decision as valuable.
Quick revision
- Value of perfect information = EV with it − EV without it.
- It is the maximum to pay; imperfect information is worth less.
- Imperfect: forecast probabilities, revised probabilities, best action after each.
- Worthless if the decision would not change.
Practice questions
These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.
Recall check: multiple choice
Question 1. The value of perfect information is:
- The cost of the survey
- The best payoff
- The expected value with perfect information less the expected value without it
- The probability of success
Show the answer
C. The gain from knowing the state in advance.
Question 2. It represents:
- The minimum to pay
- The maximum a decision maker should pay for perfect information
- The profit
- The tax saving
Show the answer
B. Paying more would not be worthwhile.
Question 3. Perfect information tells the decision maker:
- Which state will occur, before the decision
- The average outcome
- The cost only
- Nothing
Show the answer
A. So the best action can be chosen for each state.
Question 4. Compared with perfect information, imperfect information is:
- Worth more
- Worth the same
- Never available
- Worth less or the same
Show the answer
D. It cannot be better than certainty.
Question 5. With perfect information, for each state the decision maker:
- Chooses the worst action
- Chooses the best action for that state
- Averages all actions
- Does nothing
Show the answer
B. And weights by the probability of the state.
Question 6. The value of information can never be:
- Negative
- Zero
- Positive
- Large
Show the answer
A. At worst, the decision is unchanged and the value is zero.
Question 7. Information should be bought if:
- It is free of error
- It is popular
- Its value exceeds its cost
- It is new
Show the answer
C. Otherwise it destroys value.
Question 8. The probability of a forecast uses:
- Only the forecaster’s name
- Only the outcome
- Only the cost
- The prior probabilities of the states and the forecaster’s accuracy
Show the answer
D. For example 0.6 × 0.8 + 0.4 × 0.2.
Question 9. Information is worthless if it:
- Is expensive
- Is perfect
- Would not change the decision
- Is early
Show the answer
C. Nothing is gained by knowing.
Question 10. The state probabilities used for the value of perfect information are:
- The prior probabilities
- Zero
- The revised probabilities
- Always 0.5
Show the answer
A. Perfect information is always right, so no revision is needed.
Section B style: written questions
Question 11 (10 marks). (a) Define the value of perfect information. (3 marks) (b) State how it is calculated. (3 marks) (c) State how imperfect information differs and when information is worthless. (4 marks)
Question 12 (15 marks). A company chooses between X (+50 in state S1, −10 in S2), Y (+20 and +25) and Z (0 and 0). S1 has probability 0.4 and S2 0.6. (a) Calculate the EV of each action and state the best without information. (6 marks) (b) Calculate the EV with perfect information. (5 marks) (c) State the maximum to pay for a perfect forecast. (4 marks)
A further written question
Question 13 (20 marks). A project costs ₦25m. If the economy is good (probability 0.55) the present value of the inflows is ₦70m; if poor (probability 0.45) it is ₦10m. A forecaster is right 90% of the time.
(a) Calculate the expected NPV without information. (4 marks) (b) Calculate the expected NPV with perfect information and its value. (6 marks) (c) Calculate the probability of a “good” forecast. (3 marks) (d) Calculate the value of the imperfect forecast, given that the revised probabilities are 0.917 good after a “good” forecast and 0.120 good after a “poor” forecast. (7 marks)
Model answers
Show the model answer to Question 11
(a) The maximum amount a decision maker should pay for information that removes all uncertainty about the state (3).
(b) Expected value with perfect information minus expected value without it (3).
(c) Imperfect information may be wrong, so its value is lower (2). Information is worthless if it would not change the decision (2).
Show the model answer to Question 12
(a)
- X: 0.4 × 50 + 0.6 × (−10) = 14; Y: 0.4 × 20 + 0.6 × 25 = 23; Z: 0 (4).
- Best: Y, EV 23 (2).
(b) State S1: choose X, 50; state S2: choose Y, 25; EV = 0.4 × 50 + 0.6 × 25 = 35 (5).
(c) 35 − 23 = ₦12m (4).
Show the model answer to Question 13
(a)
- 0.55 × 70 + 0.45 × 10 = 43
- less 25 = +₦18m (4).
(b)
- If good, invest.
70 − 25 = 45; if poor, do not invest: 0. - EV = 0.55 × 45 = ₦24.75m (4)
- value = 24.75 − 18 = ₦6.75m (2).
(c) 0.55 × 0.9 + 0.45 × 0.1 = 0.495 + 0.045 = 0.54 (3).
(d)
- After “good” (0.54): 0.917 × 70 + 0.083 × 10 − 25 = +40.0, invest (3).
- After “poor” (0.46): 0.120 × 70 + 0.880 × 10 − 25 = −7.8, do not invest (2).
- Expected = 0.54 × 40.0 = 21.6
- value = 21.6 − 18 = ₦3.6m (2).
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