Premium and Discount Bonds and Pull to Par

Buy a bond above face value and hold it to maturity, and you're guaranteed to get back less than you paid — the issuer only ever repays par. Buy one below face value, and you're guaranteed to get back more. Neither outcome is a surprise or a loss in disguise: it's baked into the price from day one, and the bond's price converges toward par steadily as that final repayment gets closer. This convergence is called pull to par.

Why premium and discount happen at all

A bond trades at a premium when its coupon rate is higher than the market's required yield for bonds like it, and at a discount when the coupon is lower. Investors pay more than par to lock in an above-market coupon, and less than par to compensate for a below-market one:

$$P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}$$

In words: price is the sum of every discounted coupon $C$ plus the discounted face value $F$ at yield $y$. If $C$ is large relative to $y$, that sum comes out above $F$ (premium); if $C$ is small relative to $y$, it comes out below $F$ (discount).

Why price is pulled to par

Hold the yield $y$ fixed and just let time pass. Each year, one coupon disappears from the sum, and the remaining coupons and the face value get discounted over fewer periods. Both effects shrink the gap between price and par: fewer above-market coupons left to inflate the premium, and less discounting suppressing the (still fixed) face value below par. At the final coupon date, the sum collapses to a single cash flow — the last coupon plus face value — and immediately after that payment, price equals exactly par, because there's nothing left to own but the redemption.

Worked example

A 5-year bond has a 6% annual coupon and trades to yield 4%, above the market rate, so it's a premium bond.

1. Price today (5 years to maturity): $P = \sum_{t=1}^{5}\frac{60}{(1.04)^t} + \frac{1000}{(1.04)^5} \approx 1089$.
2. Price after 2 years pass (3 years left, yield still 4%): $P = \sum_{t=1}^{3}\frac{60}{(1.04)^t} + \frac{1000}{(1.04)^3} \approx 1055$.
3. Price after 4 years pass (1 year left, yield still 4%): $P = \frac{60}{1.04} + \frac{1000}{1.04} \approx 1019$.
4. At maturity: price is exactly $1,000 (par), the moment before the final coupon and principal are paid.

Even though the yield never moved, the price fell steadily from 1089 toward 1000 — that decline is pull to par, not a market reassessment.

What this means in practice

Pull to par matters most for total return accounting: a premium bond's price decline over time is not a capital loss to worry about — it's the flip side of collecting an above-market coupon, and the two roughly offset in total return terms (see bond amortization). It also matters for anyone marking positions to market near maturity: short-dated premium and discount bonds show price convergence that can look like a trading signal but is actually just the calendar doing its job.