From its past behavior, John knows that the value of a stock has a cyclical comp

Question

From its past behavior, John knows that the value of a stock has a cyclical component that increases for the first three months of each year, falls for the next 6. and rises again for the last three. In additino, inflation adds a linear component to the stock's price. John is seeking a model of the form f(t) = mt+b+Asin(pi . t/6) where t is the number of months since January 1, 2000. He has the following data: Date: Jan 1 Apr 1 Jul 1 Oct 1 Jan 1 Price: $20.00 $37.20 $33.20 $29.20 $46.40 Find values of m, b, and A so that f fits the data. During which months(s) does this stock appreciate the most? The stock appreciates the most between During which period each year is this stock actually losing value? The stock loses value between

Explanation / Answer

Solution - Here b is the base price ie price of Jan 1 and it is $20 (Given )

After working on the equation it seems that the given equation is not correct and it is found that the equation should have had Sin(? X t)/6 instead of Sin(? - t)/6 .Below working is on the corrected equation which is as below

F(t) = mt+b+Asin(?t/6 )

Putting the value of b as 20 and putting the t = 3 for April 1 we have below equation

F(3) = m3+20+Asin(?3/6 )

37.2 = 3m+20+A X sin(3.14* 3/6 ) ................ value of ? = 3.14

17.2= 3m+1A ----------------equation (1)

Similarly Putting the value of b as 20 and putting t = 6 for July 1 we have below equation

F(6) = m6+20+Asin(?6/6 )

33.2 = 6m+20+A X sin(3.14* 6/6 ) ................ value of ? = 3.14

17.2= 6m+0      ----------------equation (2)

Solving equation 1 & 2 Algebraically

we have A = 10.60 & m = 2.19

Ans a) m = 2.19 , b = 20 , A = 10.60

Now putting the derived values above in the given equation we can find the price for all the twelve months by subtituting t as per the month. below is the calculation

b ) Stock appreciated most from jan to feb $7.50 & 37.50%

c) Stok loosing value most during july & August however it never looses value caompare to Jan which is the base price as it is always higher than Jan price throughout the year

Month t mt b(base price) A S= Sin((? X t)/6) F(t) = mt+b+(AXS) Variation from last month Variation from last month (%) Jan 0 0 20 10.60 0 20.00 Feb 1 2.20 20 10.60 0.500 27.50 7.50 37.50% Mar 2 4.40 20 10.60 0.866 33.58 6.08 22.11% Apr 3 6.60 20 10.60 1.000 37.20 3.62 10.78% May 4 8.80 20 10.60 0.866 37.98 0.78 2.10% Jun 5 11.00 20 10.60 0.500 36.30 -1.68 -4.42% Jul 6 13.20 20 10.60 0.000 33.20 -3.10 -8.54% Aug 7 15.40 20 10.60 -0.500 30.10 -3.10 -9.34% Sep 8 17.60 20 10.60 -0.866 28.42 -1.68 -5.58% Oct 9 19.80 20 10.60 -1.000 29.20 0.78 2.74% Nov 10 22.00 20 10.60 -0.866 32.82 3.62 12.40% Dec 11 24.20 20 10.60 -0.500 38.90 6.08 18.52% Jan 12 26.40 20 10.60 0.000 46.40 7.50 19.28%

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