# Back to basics – uncertainty and sensitivity

## October 2012

**This is the second article in a two part feature by Bob Scarlett on risk, probability, uncertainty and sensitivity, of use to students studying for or exempt from papers C03, C04, P1 and P2.**

Conditions of uncertainty often arise where different alternative outcomes are possible but the decision maker cannot assign discrete probabilities to those alternatives. A different range of techniques then becomes appropriate.

Part one of this feature considered situations where the decision maker is confronted by conditions of uncertainty but where those conditions can be expressed in the form of a limited number of discrete alternative outcomes, each capable of having a known probability assigned to it.

However, conditions of uncertainty sometimes arise where there is not a limited number of discrete outcomes and it is not possible to assign known probabilities. In such cases the ‘sensitivity analysis approach’ may be appropriate. **Example**

Two alternative designs for the unit are under consideration. 100 unit sales are forecast at a selling price of GBP40 each. Cost details of the designs are:

Design A – 1 kilo flubber per unit + GBP1500 fixed costs

Design B – 2 kilos flubber per unit + GBP500 fixed costs

Flubber can cost between GBP0 and GBP30 per kilo depending on market conditions which are highly volatile and unpredictable. **Required: **Compare the two alternative designs using sensitivity analysis.

The alternatives can be tested for sensitivity to the flubber price, which is the critical variable. The profit generated by the two designs across the whole range of possible flubber prices is as follows:

Flubber |
Design A |
Design B |

GBP per kilo |
Profit GBP |
Profit GBP |

0 | 2500 | 3500 |

5 | 2000 | 2500 |

10 | 1500 | 1500 |

15 | 1000 | 500 |

20 | 500 | -500 |

25 | 0 | -1500 |

30 | -500 | -2500 |

Let’s take the profit generated by design A at a flubber price of GBP10 per kilo to illustrate the workings of the above figures:

Sales | GBP | 4000 (100 units x GBP40) |

Flubber cost | 1000 (100 units x 1 kilo x GBP10) | |

Fixed costs | 1500 | |

Profit | 1500 |

This sensitivity analysis situation may be represented graphically as follows :

The chart indicates that with a break even flubber price of GBP25 per kilo, design A offers a profit over 83% (25/30) of possible outcomes. A also offers a better return than B at all flubber prices over GBP10 per kilo, that being 66% (20/30) of possible outcomes.

The profit outcome from adopting design A is less sensitive to the flubber price than is that of design B as evidenced by the shallower slope of the line representing design A on the above chart. However, design B offers the highest best-case outcome at a flubber price of GBP0 per kilo.

The decision maker’s choice between designs A and B is not clear cut and involves behavioural factors in the assessment of risk and reward:

- The risk averse investor would probably prefer design A because of its relatively low risk profile. The chart illustrates design A’s lower sensitivity to movement in the flubber price and less exposure to actually suffering a loss.

The risk seeker might favour design B because of its high best-case outcome regardless of the higher exposure to uncertainty involved.

In some situations, a number of factors impacting on the outcome of a decision may involve uncertainty. The sensitivity analysis approach can be used to test each of these uncertain factors in turn in order to identify which of them is most likely to cause a problem or offer an opportunity.

**Example two**

Proposed design C for the unit is under consideration.

The base case for the appraisal of design C is that 100 unit sales are forecast at a selling price of GBP40 each with cost details as follows: 1 kilo flubber (cost GBP10 per kilo) per unit + GBP1500 fixed costs

It is considered that unit sales volume, fixed costs and the price of flubber are all uncertain and capable of a ± 25% variation in each case.

**Required:** appraise design C using sensitivity analysis.

The base case gives a profit of GBP1500, being GBP4000 sales, less GBP1000 flubber cost, less GBP1500 fixed costs. A ± variation in each of the uncertain factors may be tested for independently with the following result:

Sensitivity cases ±25% |
|||||||

Design C |
Base case |
Units (fav) |
Units (adv) |
Flubber (fav) |
Flubber (adv) |
Fixed c. fav) |
fixed c. (adv) |

GBP profit |
GBP1500 | GBP2250 | GBP750 | GBP1750 | GBP1250 | GBP1875 | GBP1125 |

% movement | ± 50% | ± 17% | ±25% |

Let us take the profit generated by design C in the sensitivity case of a 25% favourable variance in the unit sales volume in order to illustrate the workings of the above figures:

Sales | GBP | 5000 (125 units x GBP40) |

Flubber cost | 1250 (125 units x 1 kilo x GBP10) | |

Fixed costs | 1500 | |

Profit | 2250 |

The above analysis does not give any conclusive evidence concerning the viability of the design. However, it identifies those factors to which the economics of the design are most sensitive.

It can be seen that a ± 25% movement in unit sales produces a ±50% movement in profit whereas a ±25% movement in the flubber price produces only a ±17% movement in profit. So, in appraising design C, attention should focus more closely on the reliability of the unit sales forecast than on the flubber price.

The management accountant often has to incorporate an allowance for risk and uncertainty into decision support and planning activities. The techniques that may be used rarely provide definitive answers to the questions that may be asked. Allowing for risk and uncertainty is more of an art than a science.

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