Calculate time-varying monthly variance of return on the S&P 500 Index using the Exponentially Weighted Moving Average (EWMA) method with a decay factor () of 0.85. Using historical data. Calculate summary statistics of monthly returns on INSUR using historical data from January 2016 to December 2020

Details on INSUR

INSUR is an exotic insurance product that aims to help investors manage its exposure to market risk. This product requires policyholders (i.e., buyers of INSUR) to pay insurance premium (i.e., purchasing price of INSUR) at the beginning of the calendar month. INSUR matures at the end of the month and makes payment (if any) to policyholders according to the performance of the S&P 500 Index in that month. If (1) the return on the S&P 500 Index is negative and (2) the monthly standard deviation of return on S&P 500 exceeds 0.025, INSUR will reimburse the insurance premium paid at the beginning of the month and provide additional compensation to its policyholders. The additional compensation amount is the product of the following two components: (1) 150 times the insurance premium, and (2) the monthly standard deviation on the S&P 500 return in excess of 0.025. If the return on S&P 500 is positive or if the monthly standard deviation does not exceed 0.025, INSUR does not make any payment at the end of the month and the policyholder loses all her investment. That is, the monthly return of holding INSUR is

{ 100% 0 0.025

( 0.025) 15000% 0.025 ,

where rt is the return on the S&P 500 Index for month t, is the monthly (non-annualized) standard deviation of return on S&P 500. 2 is estimated using the Exponentially Weighted Moving Average (EWMA) method with a decay factor of 0.85 and historical monthly return prior to month t (i.e., rt is not used to estimate ). As an example, suppose an investor holds INSUR in January 2022. Also assume that the monthly return on S&P 500 for the month is -0.6%, and the monthly standard deviation of return on S&P 500 is 0.0415, calculated using historical data until December 2021. In this case, this investors monthly return for holding INSUR is 15,000% (0.0415 0.025) = 247.5%.