Bond Pricing, Yield and Duration
5 minutes
5 Questions
In the Certified Treasury Professional (CTP) body of knowledge, bond pricing, yield, and duration are core concepts for managing long-term capital and understanding how debt fits into a firm's capital structure. Bond Pricing: A bond's price equals the present value of its future cash flows—periodic coupon payments plus the face (par) value repaid at maturity—discounted at the market's required rate of return. If the coupon rate equals the required yield, the bond trades at par. If market rates rise above the coupon rate, the bond trades at a discount; if rates fall below the coupon, it trades at a premium. This inverse relationship between price and interest rates is fundamental: when rates go up, prices go down, and vice versa. Yield: Several yield measures matter to treasury professionals. The current yield is simply the annual coupon divided by the current price. Yield to maturity (YTM) is the internal rate of return that equates the bond's price to the present value of all its cash flows, assuming it is held to maturity and coupons are reinvested at that rate. Yield to call (YTC) applies to callable bonds and measures return if the issuer redeems early. For an issuer, YTM on outstanding bonds approximates the pre-tax cost of debt, a key input into the weighted average cost of capital (WACC). Duration: Duration measures a bond's price sensitivity to interest rate changes. Macaulay duration is the weighted average time to receive cash flows; modified duration estimates the percentage price change for a 1% change in yield. Longer maturities and lower coupons increase duration, meaning greater interest rate risk. Treasury managers use duration to match assets and liabilities, hedge rate exposure, and evaluate refinancing decisions. Convexity refines this estimate for larger rate movements. Together, these concepts help CTPs evaluate debt financing costs, manage interest rate risk, and optimize the firm's long-term capital structure.
Bond Pricing, Yield and Duration: A Complete Guide for the AFP CTP Exam
Why This Topic Matters
Bonds are a core component of long-term capital for corporations, governments and financial institutions. Treasury professionals issue bonds to raise capital, invest in bonds to manage excess cash, and evaluate bond portfolios for interest rate risk. The CTP exam tests this topic within the Capital Structure and Long-Term Capital domain because understanding how bond prices respond to interest rates, how yields are measured, and how duration quantifies interest rate sensitivity is essential for cost-of-capital decisions, investment policy compliance, and risk management. Expect both conceptual questions (relationships and definitions) and calculation questions (present value pricing, yield measures and duration-based price change estimates).
What It Is: Key Definitions
Bond: A debt instrument in which the issuer promises to pay periodic interest (coupons) and repay the face (par) value at maturity.
Par (Face) Value: The principal repaid at maturity, usually $1,000 per bond.
Coupon Rate: The fixed annual interest rate stated on the bond, applied to par value. A 6% coupon on a $1,000 bond pays $60 per year ($30 semiannually).
Market Yield / Required Return: The rate investors currently demand for bonds of similar risk and maturity. This is the discount rate used to price the bond.
Price: The present value of all future cash flows (coupons plus par) discounted at the market yield.
Duration: A measure of a bond's price sensitivity to interest rate changes, expressed in years (Macaulay) or as a percentage change factor (modified).
How Bond Pricing Works
The price of a bond is simply the present value of its cash flows:
Price = Sum of [Coupon / (1 + y)^t] + Par / (1 + y)^n
where y is the yield per period and n is the number of periods. For semiannual bonds, divide the annual coupon and the annual yield by two and double the number of periods.
Worked Example: A 3-year bond with a 5% annual coupon, $1,000 par, and a market yield of 6%.
Year 1: $50 / 1.06 = $47.17
Year 2: $50 / 1.1236 = $44.50
Year 3: $1,050 / 1.1910 = $881.60
Price = $973.27
Because the coupon rate (5%) is below the market yield (6%), the bond sells at a discount to par.
The Three Pricing Rules You Must Memorize
1. Coupon rate = Market yield: Bond trades at par.
2. Coupon rate < Market yield: Bond trades at a discount (below par).
3. Coupon rate > Market yield: Bond trades at a premium (above par).
The Inverse Relationship: Bond prices and yields move in opposite directions. When market rates rise, existing bond prices fall; when rates fall, prices rise. This is the single most tested concept in the topic.
Other Pricing Relationships
- Longer maturity bonds are more sensitive to rate changes than shorter maturity bonds.
- Lower coupon bonds are more sensitive to rate changes than higher coupon bonds.
- As a bond approaches maturity, its price converges toward par (the pull-to-par effect).
- Price changes are not symmetric: a given decrease in yield raises price by more than the same increase in yield lowers it (convexity).
Yield Measures
Nominal (Coupon) Yield: Annual coupon / Par value. Fixed at issuance and ignores market price.
Current Yield: Annual coupon / Current market price. Using the example above: $50 / $973.27 = 5.14%. Current yield ignores capital gain or loss at maturity and the time value of money.
Yield to Maturity (YTM): The single discount rate that equates the present value of all future cash flows to the current price. It is the internal rate of return of the bond if held to maturity, assuming all coupons are reinvested at the YTM. YTM is the most complete measure and is what the market quotes as the bond's yield. For a discount bond, YTM > current yield > coupon rate. For a premium bond, coupon rate > current yield > YTM.
Yield to Call (YTC): Same calculation as YTM but uses the call date and call price instead of maturity and par. Relevant for callable bonds, especially when trading at a premium, since issuers typically call when rates have fallen.
Yield to Worst: The lowest of YTM and all possible YTC values; the conservative measure for callable bonds.
Realized (Holding Period) Yield: Actual return earned, including coupons, reinvestment income and price change, over the period the bond is actually held.
Effective Annual Yield: Converts a semiannual YTM to a true annual compound rate: (1 + y/2)^2 - 1. A bond quoted at 6% semiannual has an effective annual yield of 6.09%.
Duration: How It Works
Macaulay Duration is the weighted average time (in years) until the bond's cash flows are received, where each payment is weighted by its present value as a proportion of the bond's price. It answers the question: on average, how long does it take to get my money back in present value terms?
Modified Duration converts Macaulay duration into a direct measure of price sensitivity:
Modified Duration = Macaulay Duration / (1 + y/k)
where k is the number of coupon periods per year. Modified duration tells you the approximate percentage change in price for a 1 percentage point (100 basis point) change in yield.
Estimating Price Change:
% Change in Price = - Modified Duration x Change in Yield
Worked Example: A bond has a Macaulay duration of 4.0 years, pays annual coupons, and yields 5%. Modified duration = 4.0 / 1.05 = 3.81. If yields rise by 1% (100 bps), the price is expected to fall by approximately 3.81%. On a $1,000 bond, that is a decline of about $38.10 to roughly $961.90. If yields fall by 50 bps, the price rises by approximately 1.9%.
Factors That Affect Duration
- Maturity: Longer maturity means higher duration (more rate sensitivity).
- Coupon rate: Lower coupon means higher duration because more of the value is received at maturity.
- Yield level: Lower market yields mean higher duration because distant cash flows are discounted less heavily.
- Zero-coupon bonds: Duration equals maturity because the only cash flow is at maturity. A zero has the highest duration of any bond with the same maturity.
- Callable bonds: Effective duration is lower than for an otherwise identical non-callable bond, because the call feature caps price appreciation when rates fall.
- Floating rate notes: Very low duration, approximately equal to the time until the next reset date.
Convexity
Duration is a linear approximation of a curved price-yield relationship. For large rate moves, the duration estimate understates price increases when rates fall and overstates price decreases when rates rise. Convexity is the correction term. Positive convexity is desirable for investors. Callable bonds exhibit negative convexity at low yields because the call price acts as a ceiling.
Portfolio Duration and Immunization
Portfolio duration is the market-value-weighted average of the durations of the individual bonds. Treasury professionals use immunization by matching the duration of assets to the duration of liabilities so that interest rate changes affect both sides equally, protecting the net position. This appears in pension and liability management questions.
Why Treasury Professionals Care
- Evaluating the cost of long-term debt and refinancing decisions when rates change.
- Setting investment policy limits on duration for short-term and long-term portfolios.
- Assessing mark-to-market risk on fixed income holdings.
- Understanding how credit rating changes (which shift required yields) affect the value of outstanding debt.
- Comparing fixed versus floating rate financing in the capital structure.
Exam Tips: Answering Questions on Bond Pricing, Yield and Duration
1. Start with the inverse relationship. If a question describes rates rising, prices fall; if rates fall, prices rise. Eliminate any answer choice that violates this before doing anything else.
2. Compare coupon rate to market yield to find premium or discount. Coupon below yield means discount; coupon above yield means premium; equal means par. Many questions can be answered with this comparison alone without any calculation.
3. Check the compounding convention. If the question says semiannual, divide coupon and yield by two and double the periods. Using annual figures on a semiannual bond is the most common calculation error.
4. Know the ranking of yields. For a discount bond: YTM > current yield > coupon rate. For a premium bond: coupon rate > current yield > YTM. Questions frequently test this ordering.
5. Remember what each yield measure ignores. Nominal yield ignores price; current yield ignores capital gain or loss and time value; YTM assumes reinvestment at the YTM and holding to maturity. If a question asks which measure accounts for all cash flows and the time value of money, the answer is YTM.
6. For duration questions, memorize the direction of each factor. Longer maturity, lower coupon and lower yield all increase duration. Higher coupon, shorter maturity and higher yield all decrease duration. A zero-coupon bond's duration equals its maturity.
7. Use the modified duration shortcut for price change. Percentage price change equals negative modified duration times the yield change. Convert basis points to percentage (100 bps = 1%). A bond with modified duration of 6 falls about 6% if rates rise 100 bps, or about 3% if rates rise 50 bps.
8. Do not confuse Macaulay and modified duration. Macaulay is in years and is a weighted average time; modified is Macaulay divided by (1 + y/k) and measures price sensitivity. If a question gives Macaulay duration and asks for a price change, convert to modified first.
9. Watch for callable bond traps. A callable bond trading at a premium should be evaluated on yield to call or yield to worst, not YTM. Callable bonds have lower effective duration and negative convexity when rates are low.
10. Apply convexity qualitatively. If a question asks whether the duration estimate overstates or understates the actual price change, remember that for a plain bond the actual price is always higher than the duration estimate, meaning gains are understated and losses are overstated.
11. Read for what is asked: price, yield or sensitivity. Pricing questions want present value; yield questions want a rate that equates price to cash flows; sensitivity questions want duration or a percentage change. Identifying the category immediately narrows the formula you need.
12. Sanity check your numbers. A price must be below par if the coupon is below the yield. A duration can never exceed the bond's maturity. A percentage price change from a 100 bp move should be roughly equal to the modified duration. If your result fails these checks, recompute.
13. Link to treasury decisions. Scenario questions may ask why a treasurer would issue debt now, refinance, or shorten portfolio duration. Falling rates favor issuing fixed debt and calling existing bonds; rising rate expectations favor shortening portfolio duration and locking in fixed-rate borrowing early.
Quick Reference Summary
- Price = PV of coupons + PV of par, discounted at market yield.
- Price and yield move inversely.
- Coupon vs yield determines premium, par or discount.
- Current yield = annual coupon / price.
- YTM = IRR of the bond held to maturity.
- Macaulay duration = weighted average time to cash flows (years).
- Modified duration = Macaulay / (1 + y/k).
- % price change = - Modified duration x change in yield.
- Duration rises with maturity, falls with coupon and yield; zero-coupon duration = maturity.
- Convexity means actual prices beat the duration estimate for non-callable bonds.
Master these relationships and the handful of formulas above, practice converting between annual and semiannual conventions, and you will be well prepared for every bond pricing, yield and duration question the CTP exam presents.
Bonds are a core component of long-term capital for corporations, governments and financial institutions. Treasury professionals issue bonds to raise capital, invest in bonds to manage excess cash, and evaluate bond portfolios for interest rate risk. The CTP exam tests this topic within the Capital Structure and Long-Term Capital domain because understanding how bond prices respond to interest rates, how yields are measured, and how duration quantifies interest rate sensitivity is essential for cost-of-capital decisions, investment policy compliance, and risk management. Expect both conceptual questions (relationships and definitions) and calculation questions (present value pricing, yield measures and duration-based price change estimates).
What It Is: Key Definitions
Bond: A debt instrument in which the issuer promises to pay periodic interest (coupons) and repay the face (par) value at maturity.
Par (Face) Value: The principal repaid at maturity, usually $1,000 per bond.
Coupon Rate: The fixed annual interest rate stated on the bond, applied to par value. A 6% coupon on a $1,000 bond pays $60 per year ($30 semiannually).
Market Yield / Required Return: The rate investors currently demand for bonds of similar risk and maturity. This is the discount rate used to price the bond.
Price: The present value of all future cash flows (coupons plus par) discounted at the market yield.
Duration: A measure of a bond's price sensitivity to interest rate changes, expressed in years (Macaulay) or as a percentage change factor (modified).
How Bond Pricing Works
The price of a bond is simply the present value of its cash flows:
Price = Sum of [Coupon / (1 + y)^t] + Par / (1 + y)^n
where y is the yield per period and n is the number of periods. For semiannual bonds, divide the annual coupon and the annual yield by two and double the number of periods.
Worked Example: A 3-year bond with a 5% annual coupon, $1,000 par, and a market yield of 6%.
Year 1: $50 / 1.06 = $47.17
Year 2: $50 / 1.1236 = $44.50
Year 3: $1,050 / 1.1910 = $881.60
Price = $973.27
Because the coupon rate (5%) is below the market yield (6%), the bond sells at a discount to par.
The Three Pricing Rules You Must Memorize
1. Coupon rate = Market yield: Bond trades at par.
2. Coupon rate < Market yield: Bond trades at a discount (below par).
3. Coupon rate > Market yield: Bond trades at a premium (above par).
The Inverse Relationship: Bond prices and yields move in opposite directions. When market rates rise, existing bond prices fall; when rates fall, prices rise. This is the single most tested concept in the topic.
Other Pricing Relationships
- Longer maturity bonds are more sensitive to rate changes than shorter maturity bonds.
- Lower coupon bonds are more sensitive to rate changes than higher coupon bonds.
- As a bond approaches maturity, its price converges toward par (the pull-to-par effect).
- Price changes are not symmetric: a given decrease in yield raises price by more than the same increase in yield lowers it (convexity).
Yield Measures
Nominal (Coupon) Yield: Annual coupon / Par value. Fixed at issuance and ignores market price.
Current Yield: Annual coupon / Current market price. Using the example above: $50 / $973.27 = 5.14%. Current yield ignores capital gain or loss at maturity and the time value of money.
Yield to Maturity (YTM): The single discount rate that equates the present value of all future cash flows to the current price. It is the internal rate of return of the bond if held to maturity, assuming all coupons are reinvested at the YTM. YTM is the most complete measure and is what the market quotes as the bond's yield. For a discount bond, YTM > current yield > coupon rate. For a premium bond, coupon rate > current yield > YTM.
Yield to Call (YTC): Same calculation as YTM but uses the call date and call price instead of maturity and par. Relevant for callable bonds, especially when trading at a premium, since issuers typically call when rates have fallen.
Yield to Worst: The lowest of YTM and all possible YTC values; the conservative measure for callable bonds.
Realized (Holding Period) Yield: Actual return earned, including coupons, reinvestment income and price change, over the period the bond is actually held.
Effective Annual Yield: Converts a semiannual YTM to a true annual compound rate: (1 + y/2)^2 - 1. A bond quoted at 6% semiannual has an effective annual yield of 6.09%.
Duration: How It Works
Macaulay Duration is the weighted average time (in years) until the bond's cash flows are received, where each payment is weighted by its present value as a proportion of the bond's price. It answers the question: on average, how long does it take to get my money back in present value terms?
Modified Duration converts Macaulay duration into a direct measure of price sensitivity:
Modified Duration = Macaulay Duration / (1 + y/k)
where k is the number of coupon periods per year. Modified duration tells you the approximate percentage change in price for a 1 percentage point (100 basis point) change in yield.
Estimating Price Change:
% Change in Price = - Modified Duration x Change in Yield
Worked Example: A bond has a Macaulay duration of 4.0 years, pays annual coupons, and yields 5%. Modified duration = 4.0 / 1.05 = 3.81. If yields rise by 1% (100 bps), the price is expected to fall by approximately 3.81%. On a $1,000 bond, that is a decline of about $38.10 to roughly $961.90. If yields fall by 50 bps, the price rises by approximately 1.9%.
Factors That Affect Duration
- Maturity: Longer maturity means higher duration (more rate sensitivity).
- Coupon rate: Lower coupon means higher duration because more of the value is received at maturity.
- Yield level: Lower market yields mean higher duration because distant cash flows are discounted less heavily.
- Zero-coupon bonds: Duration equals maturity because the only cash flow is at maturity. A zero has the highest duration of any bond with the same maturity.
- Callable bonds: Effective duration is lower than for an otherwise identical non-callable bond, because the call feature caps price appreciation when rates fall.
- Floating rate notes: Very low duration, approximately equal to the time until the next reset date.
Convexity
Duration is a linear approximation of a curved price-yield relationship. For large rate moves, the duration estimate understates price increases when rates fall and overstates price decreases when rates rise. Convexity is the correction term. Positive convexity is desirable for investors. Callable bonds exhibit negative convexity at low yields because the call price acts as a ceiling.
Portfolio Duration and Immunization
Portfolio duration is the market-value-weighted average of the durations of the individual bonds. Treasury professionals use immunization by matching the duration of assets to the duration of liabilities so that interest rate changes affect both sides equally, protecting the net position. This appears in pension and liability management questions.
Why Treasury Professionals Care
- Evaluating the cost of long-term debt and refinancing decisions when rates change.
- Setting investment policy limits on duration for short-term and long-term portfolios.
- Assessing mark-to-market risk on fixed income holdings.
- Understanding how credit rating changes (which shift required yields) affect the value of outstanding debt.
- Comparing fixed versus floating rate financing in the capital structure.
Exam Tips: Answering Questions on Bond Pricing, Yield and Duration
1. Start with the inverse relationship. If a question describes rates rising, prices fall; if rates fall, prices rise. Eliminate any answer choice that violates this before doing anything else.
2. Compare coupon rate to market yield to find premium or discount. Coupon below yield means discount; coupon above yield means premium; equal means par. Many questions can be answered with this comparison alone without any calculation.
3. Check the compounding convention. If the question says semiannual, divide coupon and yield by two and double the periods. Using annual figures on a semiannual bond is the most common calculation error.
4. Know the ranking of yields. For a discount bond: YTM > current yield > coupon rate. For a premium bond: coupon rate > current yield > YTM. Questions frequently test this ordering.
5. Remember what each yield measure ignores. Nominal yield ignores price; current yield ignores capital gain or loss and time value; YTM assumes reinvestment at the YTM and holding to maturity. If a question asks which measure accounts for all cash flows and the time value of money, the answer is YTM.
6. For duration questions, memorize the direction of each factor. Longer maturity, lower coupon and lower yield all increase duration. Higher coupon, shorter maturity and higher yield all decrease duration. A zero-coupon bond's duration equals its maturity.
7. Use the modified duration shortcut for price change. Percentage price change equals negative modified duration times the yield change. Convert basis points to percentage (100 bps = 1%). A bond with modified duration of 6 falls about 6% if rates rise 100 bps, or about 3% if rates rise 50 bps.
8. Do not confuse Macaulay and modified duration. Macaulay is in years and is a weighted average time; modified is Macaulay divided by (1 + y/k) and measures price sensitivity. If a question gives Macaulay duration and asks for a price change, convert to modified first.
9. Watch for callable bond traps. A callable bond trading at a premium should be evaluated on yield to call or yield to worst, not YTM. Callable bonds have lower effective duration and negative convexity when rates are low.
10. Apply convexity qualitatively. If a question asks whether the duration estimate overstates or understates the actual price change, remember that for a plain bond the actual price is always higher than the duration estimate, meaning gains are understated and losses are overstated.
11. Read for what is asked: price, yield or sensitivity. Pricing questions want present value; yield questions want a rate that equates price to cash flows; sensitivity questions want duration or a percentage change. Identifying the category immediately narrows the formula you need.
12. Sanity check your numbers. A price must be below par if the coupon is below the yield. A duration can never exceed the bond's maturity. A percentage price change from a 100 bp move should be roughly equal to the modified duration. If your result fails these checks, recompute.
13. Link to treasury decisions. Scenario questions may ask why a treasurer would issue debt now, refinance, or shorten portfolio duration. Falling rates favor issuing fixed debt and calling existing bonds; rising rate expectations favor shortening portfolio duration and locking in fixed-rate borrowing early.
Quick Reference Summary
- Price = PV of coupons + PV of par, discounted at market yield.
- Price and yield move inversely.
- Coupon vs yield determines premium, par or discount.
- Current yield = annual coupon / price.
- YTM = IRR of the bond held to maturity.
- Macaulay duration = weighted average time to cash flows (years).
- Modified duration = Macaulay / (1 + y/k).
- % price change = - Modified duration x change in yield.
- Duration rises with maturity, falls with coupon and yield; zero-coupon duration = maturity.
- Convexity means actual prices beat the duration estimate for non-callable bonds.
Master these relationships and the handful of formulas above, practice converting between annual and semiannual conventions, and you will be well prepared for every bond pricing, yield and duration question the CTP exam presents.
Test mode:
CTP - Bond Pricing, Yield and Duration Example Questions
Test your knowledge of Bond Pricing, Yield and Duration
Question 1
A company's investment portfolio holds a fixed-rate corporate note that was bought at par. The note pays a 5% annual coupon and has seven years left to maturity. Since the purchase, market yields on comparable notes have risen to 6%. The treasury analyst must report the note's current market value to the treasurer. Which statement best describes how the note is now valued?
Question 2
A bond priced at $1,000 has a Macaulay duration of 5.30 years and a yield to maturity of 6.00% with annual compounding. Using modified duration, what is the estimated price change if the yield rises by 100 basis points? Round to the nearest cent.
Question 3
Which bond measure expresses the weighted average time, in years, until a bond's cash flows are received, with each cash flow weighted by its present value?
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