EC 362    Fall 1998    Problem Set 2

1. Variance in spot price of sterling
2. Variance in futures price of sterling
3. Correlation between spot, futures price series

I will use Small Stata (available in SLSC) to do these computations. I have
stripped the first line from the data file.

. infile Yr Mo Da W Con Spot Fut using  ":Keewaydin:Desktop Folder:ec362.f
> 98.ps2.txt"
(60 observations read)

. summ Spot,detail

                            Spot
-------------------------------------------------------------
      Percentiles      Smallest
 1%       1.3902         1.3902
 5%      1.43475         1.4127
10%      1.45925          1.434       Obs                  60
25%      1.56065         1.4355       Sum of Wgt.          60

50%      1.64675                      Mean             1.6562
                        Largest       Std. Dev.      .1430842
75%       1.7584          1.881
90%       1.8726          1.888       Variance       .0204731  <===
95%       1.8845         1.9178       Skewness       .1332011
99%       1.9515         1.9515       Kurtosis       2.183159

. summ Fut,detail

                             Fut
-------------------------------------------------------------
      Percentiles      Smallest
 1%        1.362          1.362
 5%      1.39925          1.388
10%      1.43425          1.397       Obs                  60
25%      1.52285         1.4015       Sum of Wgt.          60

50%        1.616                      Mean           1.628887
                        Largest       Std. Dev.      .1434266
75%       1.7365          1.864
90%       1.8305          1.867       Variance       .0205712  <===
95%       1.8655          1.873       Skewness       .0964101
99%        1.898          1.898       Kurtosis       2.075545

. corr Spot Fut
(obs=60)

        |     Spot      Fut
--------+------------------
    Spot|   1.0000
     Fut|   0.9946   1.0000
     
The two price series are very highly correlated; the variance of futures
prices exceeds that of spot prices by a very small amount.

4. Calculate the optimal hedge ratio. First we must create new variables 
containing changes in the spot (dSpot) and futures (dFut) prices, then
regress dSpot on dFut:

. gen dSpot = Spot - Spot[_n-1]
(1 missing value generated)

. gen dFut = Fut - Fut[_n-1]
(1 missing value generated)

. regress dSpot dFut

  Source |       SS       df       MS                  Number of obs =     59
---------+------------------------------               F(  1,    57) =2666.25
   Model |  .183748702     1  .183748702               Prob > F      = 0.0000
Residual |   .00392824    57  .000068916               R-squared     = 0.9791
---------+------------------------------               Adj R-squared = 0.9787
   Total |  .187676942    58  .003235809               Root MSE      =  .0083

------------------------------------------------------------------------------
   dSpot |      Coef.   Std. Err.       t     P>|t|       [95% Conf.Interval]
---------+--------------------------------------------------------------------
    dFut |    .942536   .0182536     51.636   0.000       .9059839   .9790882
   _cons |   .0009068   .0010894      0.832   0.409      -.0012747   .0030882
------------------------------------------------------------------------------

This says that .9425 futures contracts should be sold for every 'unit' of
the spot commodity. Since one contract represents 62,500 pounds sterling (see 
Dubofsky p.349), a spot position of 625,000 pounds sterling would represent
10 contracts; this hedge ratio says that 9.425 (or 9) should be sold.

The R-squared for this regression is 0.979, indicating that over the entire
five-year period changes in futures prices explained 97.9 per cent of the 
variation in spot price changes. 

5. We must consider the prices at which the spot and futures positions would
be established and closed out:

. list Spot Fut in 1/1

          Spot        Fut  
  1.    1.4477      1.425  

. list Spot Fut in 60/60

          Spot        Fut  
 60.    1.9178      1.867  
 
Thus if we bought 625,000 pounds sterling at 1.4477 and sold them at 1.9178,
our gross profit would be ($1,198,625.00 - $904,812.50) = $293,812.50.

If we sold 9 futures contracts at 1.425 and bought them back at 1.867, our
gross profit would be ($801,562.50 - $1,050,188.00) = -$248,625.50.

Thus the net profit from our hedge would be $45,187.00. A 'perfect' hedge
would have a net profit of zero. 

6. If the OHR is calculated using only 1989 data, we calculate:

. regress dSpot dFut if Yr==89

  Source |       SS       df       MS                  Number of obs =     12
---------+------------------------------               F(  1,    10) = 612.98
   Model |  .032203141     1  .032203141               Prob > F      = 0.0000
Residual |  .000525352    10  .000052535               R-squared     = 0.9839
---------+------------------------------               Adj R-squared = 0.9823
   Total |  .032728492    11  .002975317               Root MSE      = .00725

------------------------------------------------------------------------------
   dSpot |      Coef.   Std. Err.       t     P>|t|       [95% Conf.Interval]
---------+--------------------------------------------------------------------
    dFut |   .8967242   .0362189     24.758   0.000       .8160235   .9774248
   _cons |   -.000193    .002292     -0.084   0.935      -.0052999    .004914
-----------------------------------------------------------------------------

The OHR is now 0.8967 (which still rounds to 9 contracts) and the R-squared
is even higher than it was for the full sample; thus this estimate is very
reliable.