# 4.7: Appendix - Forecasting

- Page ID
- 33079

## Forecasting of Frequency and Severity

When insurers or risk managers use frequency and severity to
project the future, they use trending techniques that apply to the
loss distributions known to them.Forecasting is part of the
Associate Risk Manager designation under the Risk Assessment course
using the book: Baranoff Etti, Scott Harrington, and Greg Niehaus,
*Risk Assessment* (Malvern, PA: American Institute for
Chartered Property Casualty Underwriters/Insurance Institute of
America, 2005). Regressions are the most commonly used tools to
predict future losses and claims based on the past. In this
textbook, we introduce linear regression using the data featured in
"2: Risk Measurement and Metrics". The scientific notations for
the regressions are discussed later in this appendix.

Year | Actual Fire Claims | Linear Trend For Claims | Actual Fire Losses | Linear Trend For Losses |
---|---|---|---|---|

1 | 11 | 8.80 | $16,500 | $10,900.00 |

2 | 9 | 9.50 | $40,000 | $36,900.00 |

3 | 7 | 10.20 | $30,000 | $62,900.00 |

4 | 10 | 10.90 | $123,000 | $88,900.00 |

5 | 14 | 11.60 | $105,000 | $114,900.00 |

## Using Linear Regression

Linear regression attempts to explain the relationship among observed values by applying a straight line fit to the data. The linear regression model postulates that

\[Y= b+mX+e\]

,where the “residual” *e* is a random variable with mean
of zero. The coefficients *a* and *b* are determined
by the condition that the sum of the square residuals is as small
as possible. For our purposes, we do not discuss the error term. We
use the frequency and severity data of A for 5 years. Here, we
provide the scientific notation that is behind Figure
\(\PageIndex{1}\) and Figure \(\PageIndex{2}\).

In order to determine the intercept of the line on the y-axis
and the slope, we use *m* (slope) and *b*
(y-intercept) in the equation.

Given a set of data with *n* data points, ** the
slope (m) and the y-intercept (b)** are determined
using:

m= nΣ(xy)−ΣxΣy nΣ( x 2 )− (Σx) 2

b= Σy−mΣx n

*The graph is provided by Chris D. Odom, with
permission.*

Most commonly, practitioners use various software applications to obtain the trends. The student is invited to experiment with Microsoft Excel spreadsheets. Table 4.6 provides the formulas and calculations for the intercept and slope of the claims to construct the trend line.

(1) | (2) | (3) = (1) × (2) | (4) = (1)2 | ||
---|---|---|---|---|---|

Year | Claims | ||||

X | Y | XY | X2 | ||

1 | 11 | 11.00 | 1 | ||

2 | 9 | 18.00 | 4 | ||

3 | 7 | 21.00 | 9 | ||

4 | 10 | 40.00 | 16 | ||

n=5 | 14 | 70.00 | 25 | ||

Total | 15 | 51 | 160 | 55 | |

M = Slope = 0.7 |
= m= nΣ(xy)−ΣxΣy nΣ( x 2 )− (Σx) 2 | = (5×160)−(15×51) (5×55)−(15×15) | |||

b = Intercept = 8.1 |
b= Σy−mΣx n | = 51−(0.7×15) 5 |

Future Forecasts using the Slopes and Intercepts for A:

- Future claims = \(Intercept + Slope × (X)\)
- In year 6, the forecast of the number of claims is projected to be: \(8.1 + (0.7 × 6) = 12.3\) claims

- Future losses = \(Intercept + Slope × (X)\)
- In year 6, the forecast of the losses in dollars is projected to be: \(−15, 100 + (26,000 × 6) = $140,900\) in losses

The in-depth statistical explanation of the linear regression model is beyond the scope of this course. Interested students are invited to explore statistical models in elementary statistics textbooks. This first exposure to the world of forecasting, however, is critical to a student seeking further study in the fields of insurance and risk management.