Explain valuation by backward induction, and use it to value a fixed-income instrument from its cash flow at every node

Backward induction starts at maturity, where the cash flow is known, and works back one date at a time. At each node, the value is V = [½ × (value up + cash flow up) + ½ × (value down + cash flow down)] ÷ (1 + that node's rate). The value found at today's node is the instrument's value. For a coupon bond, add the coupon due at each date to the values of the next nodes before averaging and discounting. For bonds with options, compare the node value with the call or put price at each exercise date before moving back. The two standard mistakes are omitting the coupons at intermediate nodes and forgetting the final discounting step to today.