Bond Price-Yield Relationship

The bond price-yield relationship is one of the most heavily tested concepts on the exam, focusing on how bond prices move inversely to yields and how this relationship is affected by bond characteristics like maturity and coupon rate. You must understand why this relationship exists, how to apply it to different scenarios, and which bonds are most sensitive to interest rate changes.

Core Concepts

Inverse Price-Yield Relationship

Bond prices and yields move in opposite directions-when yields rise, prices fall, and when yields fall, prices rise. This happens because a bond's coupon rate is fixed when issued, so if market interest rates increase, new bonds offer higher yields, making existing bonds with lower coupons less valuable. Conversely, when market rates decrease, existing bonds with higher coupons become more valuable.

For example, if you own a bond paying 4% and new bonds start issuing at 5%, investors will only buy your bond at a discount (below par) to compensate for the lower coupon. If new bonds issue at 3%, investors will pay a premium (above par) for your 4% bond.

• This relationship is always inverse-no exceptions
• The bond's coupon rate stays fixed; market price adjusts
• Par value = $1,000 for most corporate and municipal bonds
• Premium bonds trade above par; discount bonds trade below par
• When a bond trades at par, current yield equals coupon rate

When to Use This

• When asked what happens to bond prices if the Federal Reserve raises interest rates (prices fall)
• When determining whether a bond trading at 98 is at a discount (yes-below par of 100)
• When comparing which investor benefits from rate changes-bondholders lose when rates rise
• When identifying risk factors for fixed-income securities on the exam

Duration and Interest Rate Sensitivity

Duration measures how sensitive a bond's price is to interest rate changes. The longer the duration, the more volatile the bond's price when interest rates move. Duration is expressed in years and indicates the approximate percentage price change for a 1% change in yield.

A bond with a duration of 6 years will decrease in price by approximately 6% if interest rates rise by 1%, or increase by approximately 6% if rates fall by 1%. Duration is not the same as maturity, though longer maturity bonds generally have higher duration.

• Higher duration = greater price volatility
• Lower duration = less price volatility
• Duration increases with longer maturity
• Duration decreases with higher coupon rates
• Zero-coupon bonds have duration equal to their maturity
• Duration is always less than maturity for coupon-paying bonds

When to Use This

• When asked which bond has the most interest rate risk (highest duration)
• When comparing a 20-year bond versus a 5-year bond for price stability (5-year has lower duration)
• When determining which bond is most appropriate for an investor worried about rising rates (shortest duration)
• When identifying which bond will gain the most value if rates fall (longest duration)

Maturity Impact on Price Volatility

Longer-maturity bonds are more sensitive to interest rate changes than shorter-maturity bonds. This is because investors are locked into the bond's coupon rate for a longer period, creating greater uncertainty and risk.

If rates rise 1%, a 30-year bond will fall more in price than a 5-year bond with the same coupon rate. The 30-year bondholder must wait much longer to reinvest at higher rates, making the bond less attractive and forcing a steeper price discount.

• Long-term bonds have higher interest rate risk
• Short-term bonds have lower interest rate risk
• Maturity affects duration directly
• The relationship is not linear-the first few years of maturity matter most

When to Use This

• When ranking bonds by interest rate sensitivity (longest maturity = highest sensitivity)
• When a client expects rising rates and wants to minimize loss (recommend shorter maturities)
• When comparing a Treasury note (10 years or less) to a Treasury bond (more than 10 years) for volatility

Coupon Rate Impact on Price Volatility

Lower coupon bonds are more sensitive to interest rate changes than higher coupon bonds with the same maturity. This is because lower coupon bonds return less cash to investors earlier, making the final principal payment (which is further in the future) relatively more important to the bond's value.

A zero-coupon bond, which pays no interest until maturity, has the highest sensitivity to rate changes for any given maturity. A 6% coupon bond will be less volatile than a 2% coupon bond of the same maturity because the 6% bond returns more cash sooner.

• Lower coupon = higher price volatility
• Higher coupon = lower price volatility
• Zero-coupon bonds have maximum volatility for their maturity
• High coupon bonds cushion price drops with regular income

When to Use This

• When comparing two bonds with the same maturity but different coupons for interest rate risk (lower coupon = higher risk)
• When asked which bond type is most volatile (zero-coupon bonds)
• When determining appropriate bonds for conservative investors in rising rate environments (higher coupons)

Premium, Par, and Discount Bonds

A bond trades at a premium when its coupon rate is higher than current market yields for similar bonds. It trades at a discount when its coupon rate is lower than current market yields. It trades at par when the coupon rate equals the current market yield.

If a bond was issued at 5% and market rates drop to 3%, the bond trades at a premium because investors will pay more than $1,000 to lock in that 5% coupon. If market rates rise to 7%, the bond trades at a discount because investors demand a price reduction to compensate for the lower 5% coupon.

• Premium bond: market price above $1,000 (or above 100)
• Discount bond: market price below $1,000 (or below 100)
• Par bond: market price equals $1,000 (or equals 100)
• Bond prices are quoted as a percentage of par (98 = $980, 102 = $1,020)
• Premium bonds have current yield lower than coupon rate
• Discount bonds have current yield higher than coupon rate

When to Use This

• When asked to identify the relationship between a bond's price and its coupon versus market rates
• When calculating current yield and comparing it to coupon rate to determine premium/discount status
• When determining which bonds will experience capital gains (discount bonds held to maturity) or losses (premium bonds held to maturity)

Yield-to-Maturity (YTM) vs. Current Yield

Current yield is the annual coupon payment divided by the current market price. Yield-to-maturity (YTM) accounts for both the coupon payments and any capital gain or loss if the bond is held to maturity, providing a complete measure of return.

For a premium bond, YTM is lower than current yield because the investor will lose money when the bond matures at par (below the purchase price). For a discount bond, YTM is higher than current yield because the investor will gain money when the bond matures at par (above the purchase price).

• Current yield = Annual coupon ÷ Current market price
• YTM includes both income and capital gain/loss
• YTM is the most comprehensive yield measure
• For premium bonds: YTM < Current Yield < Coupon Rate
• For discount bonds: Coupon Rate < Current Yield < YTM
• For par bonds: YTM = Current Yield = Coupon Rate

When to Use This

• When asked which yield is highest or lowest for a premium or discount bond
• When determining total return for a bond held to maturity (use YTM)
• When comparing bonds with different prices and coupons (YTM provides best comparison)
• When identifying the relationship between different yield measures on the exam

Commonly Tested Scenarios / Pitfalls

1. Scenario: A question asks which bond will experience the greatest percentage price decline if interest rates increase by 1%. The choices include bonds with different maturities and coupon rates.

Correct Approach: Select the bond with the longest maturity and lowest coupon rate. This bond has the highest duration and therefore the greatest interest rate sensitivity.

Check first: Compare both maturity and coupon rate together-not just one factor. A 30-year, 2% bond is more sensitive than a 5-year, 8% bond.

Do NOT do first: Do not pick based solely on maturity without considering coupon rate, or assume all bonds with the same maturity have equal sensitivity.

Why other options are wrong: Shorter maturity bonds and higher coupon bonds have lower duration, making them less sensitive to rate changes, even if one factor (like maturity) is high but the other (like coupon) is also high.

2. Scenario: The exam shows a bond trading at 104 and asks whether the current yield is higher or lower than the coupon rate.

Correct Approach: The bond is trading at a premium (above par), so the current yield must be lower than the coupon rate. Current yield = Annual coupon ÷ Market price; dividing by a price above par lowers the yield.

Check first: Determine if the bond is at a premium (above 100), discount (below 100), or par (exactly 100). This immediately tells you the yield relationship.

Do NOT do first: Do not attempt to calculate the actual yields unless specifically asked. The relationship is what matters, and it's determined by premium/discount status.

Why other options are wrong: If the bond is at a premium, current yield cannot be higher than the coupon rate-this only happens with discount bonds. Par bonds have equal current yield and coupon rate.

3. Scenario: A client expects interest rates to rise over the next year and asks which bonds to hold to minimize loss. The choices include long-term bonds, short-term bonds, zero-coupon bonds, and high-yield bonds.

Correct Approach: Recommend short-term bonds. They have the lowest duration and least sensitivity to interest rate increases, minimizing price declines.

Check first: Identify which bonds have the shortest duration or maturity. Duration is the key measure of interest rate risk.

Do NOT do first: Do not recommend long-term or zero-coupon bonds, which have the highest duration and will suffer the largest price drops if rates rise.

Why other options are wrong: Long-term bonds and zero-coupon bonds have maximum sensitivity to rising rates. High-yield bonds carry credit risk but don't specifically address interest rate risk; they can still fall significantly if rates rise.

4. Scenario: The exam asks which bond will have the highest yield-to-maturity: a bond trading at 95, 100, or 105, all with the same coupon rate and maturity.

Correct Approach: The bond trading at 95 (discount) has the highest YTM. Discount bonds have YTM higher than current yield and coupon rate because the investor gains capital appreciation to par at maturity.

Check first: Identify which bond is at the deepest discount. The lower the price, the higher the YTM for bonds with the same coupon and maturity.

Do NOT do first: Do not assume the premium bond (105) has the highest yield-it has the lowest YTM because the investor loses money at maturity when the bond is redeemed at par.

Why other options are wrong: The par bond (100) has YTM equal to the coupon rate. The premium bond (105) has YTM lower than both current yield and coupon rate due to the capital loss at maturity.

5. Scenario: A question presents two bonds with the same maturity: Bond A has a 6% coupon and Bond B has a 2% coupon. It asks which bond has greater interest rate risk.

Correct Approach: Bond B with the 2% coupon has greater interest rate risk. Lower coupon bonds have higher duration and are more sensitive to rate changes because less cash is returned to investors early.

Check first: Compare the coupon rates when maturity is the same. The lower coupon always indicates higher interest rate sensitivity.

Do NOT do first: Do not assume higher coupon bonds are riskier. Higher coupons reduce duration and actually provide a cushion against price volatility.

Why other options are wrong: Bond A's higher coupon reduces its duration and interest rate sensitivity, making it less risky in a rising rate environment despite having the same maturity.

Step-by-Step Procedures or Methods

Task: Determine whether a bond is trading at a premium, discount, or par, and identify the yield relationships.

1. Compare the bond's market price to par ($1,000 or 100).
   - Above par → Premium
   - Below par → Discount
   - At par → Par
2. Compare the bond's coupon rate to current market yields for similar bonds.
   - Coupon higher than market yield → Premium
   - Coupon lower than market yield → Discount
   - Coupon equal to market yield → Par
3. Rank the yields based on premium/discount status.
   - Premium: YTM < Current Yield < Coupon
   - Discount: Coupon < Current Yield < YTM
   - Par: YTM = Current Yield = Coupon
4. If asked about capital gain/loss at maturity:
   - Premium bond → Capital loss (redeemed at par below purchase price)
   - Discount bond → Capital gain (redeemed at par above purchase price)
   - Par bond → No gain or loss

Task: Identify which bond has the greatest interest rate risk among several options.

1. Identify the maturity of each bond. Longer maturity = higher interest rate risk.
2. Identify the coupon rate of each bond. Lower coupon = higher interest rate risk.
3. Combine both factors to determine duration:
   - Longest maturity + Lowest coupon = Highest duration and greatest interest rate risk
4. If a zero-coupon bond is an option, it will have the highest duration for its maturity and is the most sensitive to rate changes.
5. Select the bond with the highest overall duration based on the combination of maturity and coupon.

Practice Questions

Q1: If interest rates rise, what happens to the price of outstanding bonds?
(a) Prices rise
(b) Prices fall
(c) Prices remain unchanged
(d) Prices are unaffected by interest rates

Ans: (b)
Bond prices and yields have an inverse relationship. When interest rates rise, existing bonds with lower coupons become less attractive, causing their prices to fall. (a) is incorrect because prices move opposite to rates. (c) and (d) are incorrect because bond prices are directly affected by interest rate changes.

Q2: A bond is trading at 96. Which statement is TRUE?
(a) The bond's current yield is lower than its coupon rate
(b) The bond's current yield is higher than its coupon rate
(c) The bond's current yield equals its coupon rate
(d) The bond is trading at a premium

Ans: (b)
A bond trading at 96 is at a discount (below par of 100). For discount bonds, current yield is higher than the coupon rate because the denominator (market price) is lower than par, increasing the yield calculation. (a) describes premium bonds. (c) describes par bonds. (d) is incorrect because premiums are above 100.

Q3: Which bond will experience the greatest percentage price increase if interest rates fall by 1%?
(a) 5-year bond with a 6% coupon
(b) 5-year bond with a 2% coupon
(c) 20-year bond with a 6% coupon
(d) 20-year bond with a 2% coupon

Ans: (d)
The 20-year bond with a 2% coupon has the highest duration (longest maturity + lowest coupon), making it the most sensitive to interest rate changes. It will experience the greatest price increase when rates fall. (a) and (b) have shorter maturities and lower duration. (c) has a higher coupon, reducing its duration compared to (d).

Q4: A bond is trading at 102 with a 5% coupon. What is the relationship between its yield-to-maturity (YTM) and current yield?
(a) YTM is higher than current yield
(b) YTM is lower than current yield
(c) YTM equals current yield
(d) Cannot be determined from the information given

Ans: (b)
The bond is at a premium (102 is above par). For premium bonds, YTM is the lowest yield because it accounts for the capital loss when the bond matures at par. The hierarchy is YTM < Current Yield < Coupon. (a) describes discount bonds. (c) describes par bonds. (d) is incorrect because premium status determines the yield relationship.

Q5: An investor is concerned about rising interest rates and wants to minimize price volatility in their bond portfolio. Which strategy is MOST appropriate?
(a) Purchase long-term, low-coupon bonds
(b) Purchase short-term, high-coupon bonds
(c) Purchase zero-coupon bonds
(d) Purchase bonds with the highest yield-to-maturity

Ans: (b)
Short-term, high-coupon bonds have the lowest duration and least sensitivity to interest rate changes, minimizing price volatility. (a) and (c) represent bonds with the highest duration and greatest volatility. (d) does not address interest rate risk; high YTM may indicate high credit risk or discount bonds, which can still be volatile.

Q6: Which bond has the highest interest rate risk?
(a) 10-year Treasury bond
(b) 30-year Treasury bond
(c) 2-year Treasury note
(d) 6-month Treasury bill

Ans: (b)
The 30-year Treasury bond has the longest maturity and therefore the highest duration and interest rate risk. (a) has lower duration than (b). (c) and (d) are short-term securities with minimal interest rate risk.

Quick Review

• Bond prices and yields always move in opposite directions-when yields rise, prices fall, and vice versa.
• Duration measures interest rate sensitivity: higher duration = greater price volatility when rates change.
• Longer maturity bonds have higher duration and more interest rate risk than shorter maturity bonds.
• Lower coupon bonds have higher duration and more interest rate risk than higher coupon bonds with the same maturity.
• Zero-coupon bonds have the highest duration for any given maturity and are the most sensitive to rate changes.
• Premium bonds trade above par (above 100) when the coupon rate exceeds current market yields; for premium bonds, YTM < Current Yield < Coupon Rate.
• Discount bonds trade below par (below 100) when the coupon rate is less than current market yields; for discount bonds, Coupon Rate < Current Yield < YTM.
• Par bonds trade at exactly $1,000 (or 100) when the coupon rate equals current market yields; for par bonds, YTM = Current Yield = Coupon Rate.
• Investors expecting rising rates should hold short-term, high-coupon bonds to minimize price declines.
• Investors expecting falling rates should hold long-term, low-coupon bonds to maximize price gains.