Demond owned an annuity for which he paid $$100,000$ in premiums. When he exchanged the annuity in a tax$-$free exchange for a new annuity, the value of the old annuity was $$150,000.$ In the first year after the exchange, Demond paid premiums of $$20,000$ and the value of the new annuity grew to $$200,000.$ What is Demond's cost basis in the new annuity?