United States
Hong KongBond Valuation and Duration
A fixed-rate bond's price usually moves opposite to market yields.
- Macaulay duration is a weighted average time to receive cash flows; modified duration estimates the percentage price change for a yield change.
- Convexity improves that approximation for larger moves.
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Start with the relationship everything else follows from: price and yield move in opposite directions.
Rates rise, existing bonds paying lower coupons become less attractive, and their prices fall until the yield matches the market. That is the whole mechanism.
That statement assumes a fixed-rate bond and holds other risks steady. A company's credit can deteriorate while interest rates fall, and its bond may still lose value. On a planning case, ask whether the change is a market-rate move, a credit event, or an issuer call before applying the price-yield rule.
The yield measures
| Measure | What it is |
|---|---|
| Coupon rate | The stated rate on par value - fixed, and not a yield |
| Current yield | Annual coupon divided by current price |
| Yield to maturity | Discount rate that equates the price with promised cash flows to maturity; realized return can differ |
| Yield to call | The same, assuming the bond is called at the first call date |
| Taxable equivalent yield | What a taxable bond must yield to match a municipal, given the tax bracket |
For a standard fixed-rate bond trading at a discount and redeemed at par, coupon rate is below current yield, which is below yield to maturity. At a premium, the order reverses. The ranking assumes the issuer pays as promised and that no call changes the cash-flow date.
A current-yield example
Take an illustrative bond with a face value of $1,000, a 5% annual coupon, and a market price of $950. The annual coupon is $50. Current yield is $50 divided by $950, or about 5.26%. The coupon rate remains 5% because the issuer did not rewrite the bond when its market price changed. Yield to maturity would also account for the difference between today's price and the principal repaid at maturity; current yield does not. These are teaching numbers, not a market quote.
| Question in the case | Use this measure | Why |
|---|---|---|
| What is the bond's stated interest rate? | Coupon rate | It is based on face value. |
| What is its annual income relative to today's price? | Current yield | It divides the annual coupon by the price. |
| What rate discounts all scheduled payments to today's price? | Yield to maturity | It includes coupon cash flows and the redemption amount. |
| What if it is redeemed early under a call provision? | Yield to call | The assumed redemption date and price change. |
Duration
Two related measures are often shortened to 'duration.' Macaulay duration is the weighted average time until a bond's promised cash flows arrive. Modified duration converts that measure into an approximate price sensitivity. A modified duration of seven implies roughly a seven percent price change for a one-percentage-point yield move, in the opposite direction, if other factors stay fixed. The estimate is best for a small yield move.
- Longer maturity means higher duration.
- Lower coupon means higher duration.
- A zero-coupon bond's Macaulay duration equals its maturity because all cash arrives at the end.
- Higher yield means slightly lower duration.
The zero-coupon point is a useful check. Everything else pays cash along the way, which pulls the weighted average time of cash flows forward. Do not say that a zero-coupon bond's modified duration exactly equals its maturity; that confuses the two duration measures.
Use duration in a client case
A client who needs cash for a near-term purchase faces a different risk from a client funding a distant liability. A long-duration bond can swing sharply in price before the near-term withdrawal. If the client plans to hold to maturity and the issuer pays, interim price swings matter less, but inflation, reinvestment and credit risks still deserve attention. Duration answers one sensitivity question; it is not a complete measure of safety.
Matching asset duration to the timing of a liability can reduce the first-order effect of a small yield change: price risk and reinvestment risk move in opposing directions. Real immunization also requires attention to cash-flow timing, yield-curve changes and rebalancing. A multiple-choice stem that omits these complications is asking for the basic duration match, not a guarantee.
Convexity
Duration assumes a straight-line relationship. The real one is curved, so duration under-predicts price rises and over-predicts price falls.
Convexity is the correction. Positive convexity is desirable - it means gains are larger and losses smaller than duration alone suggests. Callable bonds can display negative convexity, because the call caps the upside.
Credit and structure
Investment grade above the threshold, high yield below. Callable bonds favor the issuer and therefore pay more. Putable bonds favor the holder and pay less. Convertibles carry equity upside and pay less again.
Municipal-bond interest is generally exempt from federal income tax; state treatment depends on the bond and the investor. The taxable-equivalent-yield calculation compares after-tax income, but it does not settle credit risk, liquidity or maturity. Read the facts of the case before assuming the highest tax-adjusted yield is the best recommendation.
When a question names both a municipal and taxable bond, separate the arithmetic from the recommendation. Calculate the comparable yield using the client's stated tax assumptions, then check whether the bonds carry similar credit quality, call features and term. A municipal bond's tax treatment does not make it automatically safer or more suitable.
Common questions
Why do bond prices fall when rates rise?
Existing bonds paying lower coupons become less attractive, so their prices fall until the yield they offer matches the market. Price and yield move inversely.
What does duration measure?
Macaulay duration measures weighted average cash-flow time; modified duration estimates price sensitivity. A modified duration of seven suggests about a seven percent inverse price move for a one-percentage-point yield change, before convexity.
What increases duration?
Longer maturity and lower coupon generally increase duration. A zero-coupon bond's Macaulay duration equals its maturity because it has no interim coupons.
What is convexity?
The correction to duration's straight-line assumption. Positive convexity means gains are larger and losses smaller than duration predicts; callable bonds can show negative convexity.
How do you compare a municipal bond to a taxable one?
Using the taxable equivalent yield - what a taxable bond must yield to match the municipal after tax at the client's bracket.