An investment strategist models the daily value of a complex financial derivative, XXX, as
X=3V−4W X = 3V - 4W X=3V−4Wwhere V V\,V and W W\,W represent the daily returns of two independent indices. It is known that V∼N(45,62)V \sim N(45, 6^2)V∼N(45,62) and W∼N(20,32)W \sim N(20, 3^2)W∼N(20,32).
Determine P(X>75)P(X > 75)P(X>75).
The returns of five consecutive independent days V1,V2,V3,V4,V5 V_1, V_2, V_3, V_4, V_5\,V1,V2,V3,V4,V5 are each distributed as VVV. A composite variable S S\,S is defined as the sum of these returns, S=∑i=15ViS = \sum_{i=1}^{5} V_iS=∑i=15Vi.
A benchmark index T T\,T follows the distribution T∼N(240,σ2)T \sim N(240, \sigma^2)T∼N(240,σ2).
Given that P(S−T<−10)=0.6P(S - T < -10) = 0.6P(S−T<−10)=0.6 and that S S\,S and T T\,T are independent,
calculate the variance of TTT, giving your answer to one decimal place.