What you will learn
- Distinguish yield quotations from realized outcomes.
- Apply modified duration only as a local approximation.
Yield to Maturity (YTM)
YTM has no simple closed-form solution for most bonds, so it is found by trial and error or a financial calculator. Price and YTM move in opposite directions. A bond priced below face value has a YTM above its coupon rate, at par the two are equal, and above par the YTM is lower.
The figure assumes the bond is held to maturity, every payment is made on time, and coupons are reinvested at the same yield. If the bond is sold early, defaults, is called, or coupons are reinvested at different rates, the realized return will differ.
Duration
A zero-coupon bond's Macaulay duration equals its maturity, while coupon payments return cash earlier and make a coupon bond's duration shorter than its maturity. Longer maturity, lower coupon and lower yield all lengthen duration, meaning greater price sensitivity. Modified duration is Macaulay duration divided by one plus the yield per period.
For a small, parallel shift in yield, the price change is approximately negative modified duration times the yield change. Portfolio managers use it to compare interest-rate sensitivity across bonds and to match assets with liabilities.
Volatility
Historical (realized) volatility is computed from past returns, while implied volatility is inferred from option prices and reflects market expectations. To annualize a daily figure, multiply by √252, the conventional count of trading days in a year; weekly data uses √52, and markets that trade every day, such as many crypto markets, often use 365.
This square-root-of-time rule assumes returns are independent and identically distributed with a stable variance. Volatility tends to cluster in calm and turbulent spells, so a figure from a quiet year can badly understate the next stressful one.
Worked example: A small parallel yield move
Illustrative: a fictional bond portfolio worth HK$100,000 has modified duration 5. A 0.20 percentage-point parallel rise in yields implies an approximate price change of −5 × 0.002 = −1%, or −HK$1,000. The estimate ignores convexity and changes in credit spreads or cash flows; it is not an exact forecast for a large or nonparallel move.
Check your understanding
Using modified duration 4, what is the approximate effect of a 0.50 percentage-point decline in yields?
Show answer and explanation
−4 × (−0.005) = +2%. The sign reverses because falling yields usually raise the value of fixed cash flows. The approximation still depends on its small-move assumptions.