Trace promised bond payments and see how the discount rate changes their value.
Read Macro SignalsIntermediate4 min
Three annual payments
Plain English
Price is attached to a payment schedule
A conventional fixed-rate bond promises payments on specified dates: coupons during its life and principal at maturity. Its price today depends on what those future payments are worth under a chosen discount rate, as well as whether they are expected to arrive. Reading the schedule first makes the connection between price, yield and maturity easier to follow.
The coupon rate is defined relative to face value. It does not change merely because the bond’s market price changes. Yield to maturity is different: it is the rate that equates the bond’s price with the discounted value of its promised remaining payments under a stated convention. Quoting both as percentages does not make them interchangeable.
For fixed promised payments, raising the discount rate lowers their present value. Lowering it raises present value. This inverse relationship is an arithmetic property of discounting. It is not a promise about the price of every bond on every day, because credit expectations, contractual features and trading conditions can change alongside market interest rates.
Worked Example
Three payments, valued on a coupon date
Consider a hypothetical bond with £100 face value, three years remaining and £5 coupons paid annually. Value it immediately after a coupon date, so no accrued coupon interest is included. The remaining payments are £5 at the end of year one, £5 at the end of year two and £105 at the end of year three. Assume payment in full and no option to repay early.
At a 5% annual yield, discount those payments by 1.05, 1.05 squared and 1.05 cubed respectively. Their sum is £100. At a 6% annual yield, use 1.06 for the same schedule. The sum becomes approximately £97.33. Retain full precision in the calculation and round only the displayed price to pennies.
Nothing in this comparison changes the £5 coupon or £100 contractual principal. The lower price at 6% allows the same promised payments to correspond to a higher yield. The £2.67 displayed price difference is a comparison between two valuation assumptions at the same time, rather than an observed loss over a holding period.
Formula
Price equals five divided by one plus y, plus five divided by one plus y squared, plus one hundred and five divided by one plus y cubed
y is the annual yield as a decimal; all payments occur at year end.
Year 1 payment
£5
Year 2 payment
£5
Year 3 payment
£105
Price at 5% annual yield
£100.00
Price at 6% annual yield
£97.33
Reading the Result
Maturity and realised return are different ideas
Payments further away are generally more sensitive to a discount-rate change than otherwise comparable earlier payments. Duration summarises aspects of this sensitivity, taking account of payment timing rather than just the final maturity date. Bonds with the same maturity can therefore have different rate sensitivity because their coupons and contractual features differ.
Yield to maturity is not an unconditional forecast of the holder’s realised compound return. Payments must occur as assumed, and achieving a particular compound outcome also depends on how coupons are reinvested. Selling before maturity introduces the future sale price. Transaction costs and taxes change the amount retained. The example isolates a price calculation rather than modelling these additional decisions.
Limits and Assumptions
Interest rates are one source of uncertainty
Default risk concerns whether promised payments are made. Liquidity risk concerns the ability to trade at a reasonable price when needed. Neither disappears because a model correctly discounts the contractual schedule. A lower market price can reflect greater concern about repayment as well as a higher general level of rates.
Actual bond quotations can use semi-annual coupons, different day-count rules, accrued interest and clean or dirty prices. Callable bonds add uncertainty over payment timing. The FINRA material below describes common bond conventions and risks, but the example deliberately uses annual payments and a coupon-date valuation. Comparing a quoted yield with this model requires aligning those conventions first.
Common Mistake
Calling the coupon the return
A £5 annual coupon on £100 face value is a 5% coupon rate even if the bond trades below £100. Dividing £5 by market price gives current yield, which still omits the change towards principal repayment. Neither figure alone is the same as yield to maturity or the eventual realised return.
Self-check
Check your understanding
Why is the last payment £105?
It combines the final £5 coupon with the £100 principal repayment.
Why does a 6% yield produce a lower price?
The same promised future payments are discounted more heavily; their present values sum to about £97.33.
Does the calculation remove default risk?
No. It assumes full payment to isolate discounting. Actual repayment and trading conditions remain uncertain.
Continue learning
Connect the ideas
Follow the related articles below to examine these assumptions in another setting.
Disclaimer
Educational Use Only
This article is for informational and educational purposes only. It does not provide personalised investment advice.
