#9: (a) Iterate the defining formula for the SONIA compounded index I(t) on pg. 5 of the lecture to justify the formula S(t+m 1)T(t+m 1) I(t+m) = I(t) (1 + S(t)r(t)) X XX (1+ 365 Hint: Start with small values m = 1,2,3. Note that m = 1 is just the defining formula for I(t). 2 For instance, the first 2 cases of the defining formula are I(t + 1) = I(t) (1 + S(t)r(t) 365 I(t + 2) = I(t + 1) (1 + S(t + Dr(t+1)) Substitution of the first formula into the second yields the identity for the case m = 2. Ultimately, this can be proved by induction by assuming it is true form m and using the defining formula to extend that to the case m +1. (b) Demonstrate that SONIA compounded in arrears over a period is the simple term rate which, it were earned over that period would be the same return earned by investing every day at the daily SONIA rate, compounding daily. (c) Download daily SONIA interest rates from the Bank of England website (www.bankofengland.co.uk/markets/sonia-benchmark) to calculate SONIA compounded in arrears on January 17, 2019 For the 1 week term ending on that date. #9: (a) Iterate the defining formula for the SONIA compounded index I(t) on pg. 5 of the lecture to justify the formula S(t+m 1)T(t+m 1) I(t+m) = I(t) (1 + S(t)r(t)) X XX (1+ 365 Hint: Start with small values m = 1,2,3. Note that m = 1 is just the defining formula for I(t). 2 For instance, the first 2 cases of the defining formula are I(t + 1) = I(t) (1 + S(t)r(t) 365 I(t + 2) = I(t + 1) (1 + S(t + Dr(t+1)) Substitution of the first formula into the second yields the identity for the case m = 2. Ultimately, this can be proved by induction by assuming it is true form m and using the defining formula to extend that to the case m +1. (b) Demonstrate that SONIA compounded in arrears over a period is the simple term rate which, it were earned over that period would be the same return earned by investing every day at the daily SONIA rate, compounding daily. (c) Download daily SONIA interest rates from the Bank of England website (www.bankofengland.co.uk/markets/sonia-benchmark) to calculate SONIA compounded in arrears on January 17, 2019 For the 1 week term ending on that date