Fixed Income

When does a bond's expected return equal its yield (YTM)?

Only when all three conditions hold:

1. Held to maturity

2. All payments made on time and in full (implies option-free, no default risk)

3. All coupons reinvested at the original YTM

📌 Key insight: Assumption #3 is the least realistic — if the yield curve isn't flat, coupons can't actually be reinvested at the YTM, so expected return ≠ yield.

Realized return = the actual return earned over the holding period, based on the actual reinvestment rates experienced (not assumed ones).

What is a par rate?

Back:
The coupon rate a bond must have so that it prices at par (100) — i.e., the bond's price equals its face value.

• Given a maturity, the par rate = the yield on a hypothetical bond of that maturity issued today at par.

• The par curve is just the collection of these rates across maturities (1-year, 2-year, 3-year, etc.).

• Par rates are typically what's directly observable/given in the market (e.g., from newly issued bonds).

📌 Key distinction: Par rates are a blended yield across a bond's whole life — they mix together the effect of all the different spot rates. That's why we need to bootstrap them into spot rates, which isolate the discount rate for a single cash flow at a single point in time.

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