The Central Inverse Relationship
The most important rule in bond mathematics is this: when yields rise, bond prices fall; when yields fall, bond prices rise. This inverse relationship is not a market convention or a quirk — it is a mathematical necessity that follows directly from how a bond's value is calculated.
A bond is a promise to pay a series of fixed future cash flows: periodic coupon payments and a lump sum at maturity. The present value of any future cash flow is calculated by discounting it at the prevailing interest rate. If that rate rises, the discounted value of each fixed payment falls. Since the bond's cash flows are fixed, the only thing that can adjust is the bond's price.
A Worked Example
Consider a bond with a 5% coupon on £1,000 face value, maturing in 10 years, currently priced at par (£1,000). It pays £50 per year.
Now suppose market interest rates rise to 6%. Newly issued bonds now pay £60 per year on a £1,000 investment. No rational investor will pay £1,000 for our bond when they can get £60 a year from a new one — our bond must fall in price until it offers a comparable return. The price must drop to approximately £926, at which point the £50 annual coupon plus the £74 gain at maturity (from £926 back to £1,000) equates to roughly a 6% annual return.
Conversely, if rates fall to 4%, our 5% bond becomes attractive — investors will bid its price above par until the yield is competed down to 4%.
Yield to Maturity
Yield to maturity (YTM) is the single discount rate that, when applied to all of a bond's future cash flows, exactly equals the bond's current market price. It is the most complete single-figure summary of a bond's return if held to maturity.
YTM accounts for:
- The coupon payments received over the bond's life
- The difference between the purchase price and the face value at maturity (a gain if bought below par, a loss if bought above par)
- The timing of all cash flows
When investors or the financial press quote a bond's "yield," they almost always mean YTM.
Duration as a Measure of Price Sensitivity
While the inverse relationship tells us the direction of price change in response to yield moves, duration tells us the magnitude. Duration measures how sensitive a bond's price is to a change in yield — a longer duration means a larger price move for a given yield change.
A bond's duration depends on its maturity and coupon rate. Long-dated bonds with low coupons have the highest duration and are therefore most sensitive to interest rate moves. Short-dated or high-coupon bonds have lower duration and react less to rate changes. Understanding this relationship is essential for managing bond portfolios, particularly in environments where rates are moving — as they did sharply upward in 2022, causing significant mark-to-market losses in long-duration bond holdings. Duration is explored in depth in the Duration and Interest Rate Risk tile.