How does backward induction work for pricing bonds on a binomial tree?

Backward induction prices a bond by starting at the final nodes (maturity) and working backward to the root, discounting expected future values at each node's rate.

The Rule at Each Node:

Value at node = [0.5 x (Value_up + Coupon) + 0.5 x (Value_down + Coupon)] / (1 + r_node)

You discount by the rate at the current node, not the next node. The coupon is added at each future node because it is received at the end of the period.

Complete Example — Redmond Corp 6% Bond, 2 Years (fictional):

Given binomial tree:

• Year 0: r0 = 5.00%
• Year 1: r1u = 6.50%, r1d = 4.80%

Par = $100, Annual coupon =$6

Step 1: Values at Maturity (Year 2) At all terminal nodes, the bond pays par + final coupon = $100 +$6 = $106.

But we actually work backward from Year 2. At Year 1, the bond will be worth $100 at Year 2 (par) plus the$6 coupon received at Year 2.

Step 2: Values at Year 1

At r1u = 6.50%:

V1u = (0.5 x 106 + 0.5 x 106) / 1.065 = 106 / 1.065 = $99.53

At r1d = 4.80%:

V1d = (0.5 x 106 + 0.5 x 106) / 1.048 = 106 / 1.048 = $101.15

Step 3: Value at Year 0 The bond pays a $6 coupon at Year 1, so the cash flows at Year 1 nodes are:

• Up: $99.53 +$6 = $105.53
• Down: $101.15 +$6 = $107.15

V0 = (0.5 x 105.53 + 0.5 x 107.15) / 1.05 V0 = 106.34 / 1.05 = $101.28

Summary:

Node	Rate	Bond Value
Year 0	5.00%	$101.28
Year 1 Up	6.50%	$99.53
Year 1 Down	4.80%	$101.15
Year 2 (all)	—	$100.00

Common Mistakes:

1. Discounting by the wrong rate (use the rate at the node you're computing, not the child node)
2. Forgetting to add the coupon at each intermediate step
3. Using unequal probabilities (the standard tree uses 0.5/0.5)

Exam Tip: The CFA exam frequently gives you a tree and asks you to price a bond. Be methodical: start at the end, add coupon, discount at the node rate, move one step back, repeat.

Practice backward induction in our CFA Level II question bank.