I can calculate the motion of heavenly bodies but not the madness of people. -Isaac Newton

## Disaggregating Annual Losses into Each Quarter

In loss forecasting, it is often necessary to disaggregate annual losses into each quarter. The most simple method to convert low frequency to high frequency time series is interpolation, such as the one implemented in EXPAND procedure of SAS/ETS. In the example below, there is a series of annual loss projections from 2013 through 2016. An interpolation by the natural spline is used to convert the annual losses into quarterly ones.
SAS Code:

```data annual;
input loss year mmddyy8.;
format year mmddyy8.;
datalines;
19270175 12/31/13
18043897 12/31/14
17111193 12/31/15
17011107 12/31/16
;
run;

proc expand data = annual out = quarterly from = year to = quarter;
id year;
convert loss / observed = total method = spline(natural);
run;

proc sql;
select
year(year) as year,
sum(case when qtr(year) = 1 then loss else 0 end) as qtr1,
sum(case when qtr(year) = 2 then loss else 0 end) as qtr2,
sum(case when qtr(year) = 3 then loss else 0 end) as qtr3,
sum(case when qtr(year) = 4 then loss else 0 end) as qtr4,
sum(loss) as total
from
quarterly
group by
calculated year;
quit;
```

Output:

```    year      qtr1      qtr2      qtr3      qtr4     total

2013   4868536   4844486   4818223   4738931  19270175
2014   4560049   4535549   4510106   4438194  18043897
2015   4279674   4276480   4287373   4267666  17111193
2016   4215505   4220260   4279095   4296247  17011107
```

While the mathematical interpolation is easy to implement, it might be difficult to justify and interpret from the business standpoint. In reality, there might be an assumption that the loss trend would follow the movement of macro-economy. Therefore, it might be advantageous to disaggregate annual losses into quarterly ones with the inclusion of one or more economic indicators. This approach can be implemented in tempdisagg package of R language. Below is a demo with the same loss data used above. However, disaggregation of annual losses is accomplished based upon a macro-economic indicator.
R Code:

```library(tempdisagg)

loss <- c(19270175, 18043897, 17111193, 17011107)
loss.a <- ts(loss, frequency = 1, start = 2013)

econ <- c(7.74, 7.67, 7.62, 7.48, 7.32, 7.11, 6.88, 6.63, 6.41, 6.26, 6.12, 6.01, 5.93, 5.83, 5.72, 5.59)
econ.q <- ts(econ, frequency = 4, start = 2013)

summary(mdl <- td(loss.a ~ econ.q))
print(predict(mdl))
```

Output:

```Call:
td(formula = loss.a ~ econ.q)

Residuals:
Time Series:
Start = 2013
End = 2016
Frequency = 1
[1]  199753 -234384 -199257  233888

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  2416610     359064   6.730   0.0214 *
econ.q        308226      53724   5.737   0.0291 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

'chow-lin-maxlog' disaggregation with 'sum' conversion
4 low-freq. obs. converted to 16 high-freq. obs.
Adjusted R-squared: 0.9141      AR1-Parameter:     0 (truncated)
Qtr1    Qtr2    Qtr3    Qtr4
2013 4852219 4830643 4815232 4772080
2014 4614230 4549503 4478611 4401554
2015 4342526 4296292 4253140 4219235
2016 4302864 4272041 4238136 4198067
```

In practice, if a simple and flexible solution is desired without the need of interpretation, then the mathematical interpolation might be a good choice. On the other hand, if there is a strong belief that the macro-economy might drive the loss trend, then the regression-based method implemented in tempdisagg package might be preferred. However, in our example, both methods generate extremely similar results.