Asset Allocation using Economic Indicators
August 24, 2010
Appendix B – Calculation Details
Forward Rate Ratio
FRR2-10 is the rate at which one can lock in borrowing for the eight year period starting two years from now, divided by the ten-year rate itself. The FRR2-10 is indicative of the slope of the yield curve between the two-year and the ten-year note yields; a FRR2-10 greater than 1.00 indicates a positively sloped yield curve (ten-year note yields are higher than two-year note yields); a FRR2-10 less than 1.00 indicates an inversion of the yield curve (two-year note yields are higher than ten-year note yields).
The formula used for calculating FRR2-10 is
where i10 and i2 are the 10 year and 2 year U.S. Treasury Note yields respectively.
Exponential Moving Average
The exponential moving average (EMA) is a type of filter that applies weighting factors to the observed data values. The weighting for each older data point decreases exponentially, never reaching zero. For a detailed description see http://en.wikipedia.org/wiki/Moving_average
The formula used for calculating the EMA is
EMAtoday = EMAyesterday + a × (valuetoday - EMAyesterday)
The coefficient α represents the degree of weighting decrease, a constant smoothing factor between 0 and 1. A higher α discounts older observations faster. Alternatively, α may be expressed in terms of N time periods, where α = 2/(N+1). For example, N = 199 for the long EMA of WLI with a smoothing factor of 0.01, and N = 19 for the short EMA of WLI with a smoothing factor of 0.10. The “long” and “short” refers to the N time periods used in the calculation of the smoothing factor for the EMA.
Sell Signals
Sell signal type A
An initial basic sell signal is generated when the [short EMA of WLIg] moves below the [long EMA of WLIg] while the [long EMA of WLIg]still shows positive growth but which is steadily declining and theEMA of the FRR2-10 is at or below 1.00 (indicating an inversion of the yield curve), or is above 1.175. Also the Federal Funds Rate must be greater than the previous maximum rate minus 0.50%, the previous maximum rate being considered over the preceding six months period. (Later basic sell signals are eliminated if they appear within 100 days of an initial basic sell signal.)
A Type A sell signal is generated 13 weeks after a basic sell signal occurs, because it was found by inspection that this selling delay achieved on average the highest selling values.
Sell signal type B
A type B sell signal is generated when a basic sell signal has previously been generated and the 5-day slope of the [long EMA of WLIg +1] drops below the 20-day slope of the [long EMA of WLIg +1].
Sell signal type C
A type C sell signal is generated when the 120-day slope of the [long EMA of WLIg +1] slopes downward and initially passes through 1.015. The sell signal is generated if within a 1-year period this slope subsequently passes through 0.980. Just before the signal date the market must still be in a modestly rising trend, which requires the 35 day slope of the [EMA of S&P500] to be greater than 0.005 x [EMA of S&P500].
Buy signals
Buy signal type A
A type A buy signal is generated when the [short EMA of WLIg], after having formed a bottom, moves from negative growth to positive growth and moves above the [long EMA of WLIg]. The [short EMA of WLIg] must be less than 1.000 and the 20-day slope of the [long EMA of WLIg +1] must be greater than 1.000. Further, the 20-day slope of the [long EMA of WLI] must still be negative or flat and the [short EMA of WLI],after having bottomed out, moves upwards again. Also the Federal Funds Rate must have dropped at least 2.50% from a previous high, the previous high being considered over the preceding 20 months period. (Later type A buy signals are eliminated if they appear within 100 days of an initial type A buy signal.)
Buy signal type B
A type B buy signal is generated when the conditions for a type A buy signal have been met, but with the Federal Funds Rate condition omitted and theEMA of the Forward Rate Ratio is greater than 1.050. Also the 20-day slope of the [long EMA of WLI +1] must be greater than 1.000.
Buy signal type C
A type C buy signal is generated when the 120-day slope of the [long EMA of WLIg +1] is greater than the [long EMA of WLIg] and is between 1.010 and 0.975. Also the [long EMA of WLIg] minus the [short EMA of WLIg] must be greater than 0.005.
Appendix C - Signal Dates and S&P 500 values
Buy Signal Type A |
Buy Signal Type B |
Buy Signal Type C |
|||
08/18/70 |
76.20 |
04/10/03 |
871.58 |
03/26/82 |
111.94 |
02/12/75 |
79.92 |
11/18/04 |
1183.55 |
01/23/85 |
177.30 |
07/03/80 |
117.46 |
|
|
05/25/88 |
253.77 |
01/15/82 |
116.33 |
|
|
03/24/99 |
1268.59 |
12/20/90 |
330.12 |
|
|
04/03/03 |
876.45 |
03/12/03 |
804.19 |
|
|
03/14/05 |
1206.83 |
03/30/09 |
787.53 |
|
|
|
|
Sell Signal Type A |
Sell Signal Type B |
Sell Signal Type C |
|||
04/17/69 |
100.78 |
10/03/69 |
93.19 |
06/10/69 |
100.42 |
09/23/69 |
95.63 |
02/07/74 |
93.30 |
11/01/83 |
163.66 |
10/16/73 |
110.19 |
03/13/79 |
99.84 |
10/02/87 |
328.07 |
02/09/79 |
97.87 |
09/19/79 |
108.28 |
04/09/90 |
341.37 |
07/17/79 |
101.83 |
02/14/80 |
116.72 |
06/29/00 |
1442.39 |
01/09/80 |
109.05 |
04/01/81 |
136.57 |
11/05/07 |
1502.17 |
03/31/81 |
136.00 |
08/10/89 |
348.25 |
03/30/10 |
1173.27 |
08/28/89 |
352.09 |
09/27/00 |
1426.57 |
|
|
08/07/00 |
1479.32 |
|
|
|
|
05/22/07 |
1524.12 |
|
|
|
|
04/20/10 |
1207.17 |
|
|
|
|
Buy / Sell signals sorted by date
The initial 1966 buy signal was obtained from a trend following model. Obsolete signals were omitted, i.e. buy signals not preceded by a sell signal and occurring after a previous buy signal, or sell signals not preceded by a buy signal and occurring after a previous sell signal were removed.
| 10/20/66 | 77.84 | buy |
| 04/17/69 | 100.78 | sell A |
| 08/18/70 | 76.20 | buy A |
| 10/16/73 | 110.19 | sell A |
| 02/12/75 | 79.92 | buy A |
| 02/09/79 | 97.87 | sell A |
| 07/03/80 | 117.46 | buy A |
| 03/31/81 | 136.00 | sell A |
| 01/15/82 | 116.33 | buy A |
| 11/01/83 | 163.66 | sell C |
| 01/23/85 | 177.30 | buy C |
| 10/02/87 | 328.07 | sell C |
| 05/25/88 | 253.77 | buy C |
| 08/10/89 | 348.25 | sell B |
| 12/20/90 | 330.12 | buy A |
| 06/29/00 | 1442.39 | sell C |
| 03/12/03 | 804.19 | buy A |
| 05/22/07 | 1524.12 | sell A |
| 03/30/09 | 787.53 | buy A |
| 03/30/10 | 1173.27 | sell C |
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