Work out a bond's price from a given discount rate
The idea
A bond's price is the present value of its cash flows: each coupon and the principal, discounted at the required yield. Price and yield move inversely — a rise in required yield lowers the price of the fixed cash flows. A bond prices at par when its coupon equals its required yield, at a premium when the coupon is higher, and at a discount when it is lower. Longer maturity and lower coupon both increase sensitivity to yield changes, which is the intuition duration later makes precise.
Formulas
Price of a bond as the present value of its cash flows
PV = Σ [PMT ÷ (1 + r)ᵗ] + FV ÷ (1 + r)ⁿ
- PMT
- the coupon paid each period, in money
- FV
- the principal repaid at maturity, usually par
- r
- the discount rate per period
- t
- the period a coupon falls in, 1 through n
- n
- the number of periods to maturity
One rate discounting every cash flow, which is what a yield to maturity is. With semiannual coupons r is the annual yield ÷ 2 and n is years × 2, and the coupon is halved with them.
A practice question
If the yield required by investors rises, the price of an existing fixed-coupon bond:
- falls — correct
- is unchangedWhy not: The coupons are fixed, so the only variable left is the price. If it did not move, the bond would still offer the old yield.
- risesWhy not: Backwards, and it is the foundational relationship in fixed income: price and yield move in opposite directions.
The cash flows are fixed, so the only way the bond can offer a higher yield is to cost less. Price and yield move inversely.
A harder one
A bond whose coupon rate exceeds its required yield will trade:
- at a discount to parWhy not: That is the case where the coupon falls short of the required yield.
- at a premium to par — correct
- at parWhy not: Par requires the two rates to match. An above-market coupon is worth paying extra for.
It pays more than the market currently demands, so investors bid its price above par until the yields match.
The hardest tier
Which bond is most sensitive to a given change in yields?
- Short maturity, low couponWhy not: The low coupon lengthens duration but the short maturity caps it. Both factors have to point the same way.
- Short maturity, high couponWhy not: The least sensitive combination of the three — value returns early on both counts.
- Long maturity, low coupon — correct
Long maturity pushes cash flows further out, and a low coupon concentrates value in the distant principal repayment. Both lengthen duration.
In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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