Daily Economics, Business & Finance Study

October 18, 2026 (KST) · Day 28
Today’s Topic: Why Bond Prices and Yields Move in Opposite Directions — Reading Duration, Convexity, and Valuation Gains or Losses

1. Review of the Previous Lesson

Day 27 explained that a long-term yield consists of the average expected path of short-term rates plus a term premium, and that the slope of the yield curve jointly reflects the economy, inflation, monetary policy, and bond supply and demand. Today, we will explain why and by how much the price and valuation gain or loss of an already-issued bond change when those yields move.

2. Today’s Core Concept

A bond is an asset whose price equals the present value of the coupons and principal to be received in the future. If the promised cash flows stay the same, a higher market yield raises the discount rate and lowers the current price, while a lower market yield raises the price.

Today’s learning objectives are to:

1. Distinguish the coupon rate, market yield, and bond price.
2. Explain the inverse relationship between yields and bond prices using present value.
3. Use duration to approximate the price effect of a small yield change.
4. Use convexity to improve the duration approximation.
5. Connect bond valuation gains and losses to company, bank, and insurer disclosures and investment decisions.

The core relationship is:

\[ Yield\ increase \Rightarrow Higher\ discount\ rate \Rightarrow Lower\ bond\ price \]

3. Concept Explanation

3-1. A Bond’s Price Is the Present Value of Its Future Cash Flows

A typical fixed-rate bond pays a predetermined coupon until maturity and repays the last coupon and principal at maturity. The price of a simple bond that pays coupons once a year is:

\[ P =\sum_{t=1}^{n}\frac{C}{(1+y)^t} +\frac{F}{(1+y)^n} \]

• \(P\): current bond price
• \(C\): annual coupon payment
• \(F\): face value repaid at maturity
• \(y\): yield to maturity required by the market
• \(n\): remaining term

At this point, when the market yield \(y\) rises, the denominator applied to every cash flow increases and the present value falls. When the market yield falls, the fixed coupon on an existing bond becomes relatively more attractive and its price rises. The U.S. SEC’s investor guidance likewise explains that fixed-rate bond prices and market interest rates generally move in opposite directions.

3-2. The Coupon Rate and Market Yield Are Not the Same Number

Coupon rate is the annual interest rate set at issuance. If face value is KRW 1 million and the coupon rate is 5% per year, the annual coupon is normally KRW 50,000. A fixed-rate bond’s coupon does not change when market rates change after issuance.

Market yield, or yield to maturity (YTM), is the internal rate of return implied by the current market price if the investor buys the bond, receives all promised cash flows, and holds it to maturity. This assumes no default and reinvestment of coupons at the same yield.

A higher coupon does not necessarily mean a higher expected return. An existing bond that pays a high coupon may trade at a correspondingly higher price.

3-3. Longer-Maturity Bonds Tend to Be More Sensitive to the Same Yield Change

Cash flows further in the future remain exposed to a change in the discount rate for longer. Therefore, all else equal, a longer-maturity bond is more sensitive to yield changes. For the same maturity, a lower-coupon bond tends to be more price-sensitive because the center of gravity of cash recovery moves further into the future.

However, maturity and duration are different.

• Maturity: the time remaining until the final repayment of principal
• Duration: the present-value-weighted average time to receive cash flows and the basis of interest-rate sensitivity

A zero-coupon bond pays only once at maturity, so its Macaulay duration equals its maturity. A coupon bond makes cash payments before maturity, so its duration is generally shorter than its maturity.

3-4. Macaulay Duration Is the Time-Based Center of Gravity of Cash Flows

Macaulay duration weights the timing of each cash flow by its share of the bond’s total present value.

\[ D_{Mac} =\frac{\sum_{t=1}^{n}t\times PV(CF_t)}{P} \]

• \(D_{Mac}\): Macaulay duration
• \(PV(CF_t)\): present value of the cash flow at time t
• \(P\): bond price

A duration of six years does not mean that maturity is exactly six years. It means that the present-value-weighted average time to receive the cash flows is six years.

3-5. Modified Duration Approximates Price Sensitivity

In practice, the modified duration measure is used to quickly estimate the price effect of a yield change. If coupons are paid \(m\) times a year, it relates to Macaulay duration as follows:

\[ D_{mod} =\frac{D_{Mac}}{1+y/m} \]

The first-order approximation for a small yield change is:

\[ \frac{\Delta P}{P} \approx -D_{mod}\times\Delta y \]

The minus sign represents the opposite directions of yield and price. The yield change must be entered as a decimal.

• 1 bp = 0.01 percentage point = 0.0001
• 50 bps = 0.50 percentage point = 0.005
• 100 bps = 1.00 percentage point = 0.01

For example, a bond with a modified duration of 6 is estimated to lose about 6% in price when its yield rises by 1 percentage point. This is an approximation based on a small parallel shift, not a guaranteed loss.

3-6. The Main Drivers of Duration

All else equal, the following general relationships apply.

The duration of a bond fund or ETF may keep changing as its holdings change. The maturity range in a product’s name is not enough to determine its actual interest-rate sensitivity.

3-7. Convexity Improves the Curved Price–Yield Relationship

The relationship between bond prices and yields is curved, not linear. Duration approximates the price change with a tangent at the current point, so the error grows as the yield change becomes larger. This adjustment is called convexity.

\[ \frac{\Delta P}{P} \approx -D_{mod}\Delta y +\frac{1}{2}\times Convexity\times(\Delta y)^2 \]

A plain fixed-rate bond without embedded options normally has positive convexity. All else equal, the percentage price gain from a given yield decline is slightly larger than the percentage price loss from an equal yield increase.

However, a callable bond may be redeemed early by the issuer, and a mortgage-backed security (MBS) may experience faster prepayment by borrowers when rates fall. The upside in price can then be capped, reducing convexity or making it negative in some ranges. Bonds with embedded options therefore require option-adjusted analysis beyond simple duration and convexity.

3-8. Total Return Includes More Than the Coupon

A bond’s holding-period total return can be simplified as:

\[ Holding\ period\ total\ return =Coupon\ income +Price\ change +Reinvestment\ income \]

Higher yields hurt the price of an existing bond but allow new coupons and maturing principal to be reinvested at higher rates. Lower yields help the existing bond’s price but may reduce reinvestment income. When the investment horizon and duration are similar, price risk and reinvestment risk can partly offset each other, which is the foundation of immunization.

3-9. Holding to Maturity and Valuation Gains or Losses Must Be Distinguished

If a fixed-rate bond does not default and is not sold early, an investor who holds it to maturity can receive the promised coupons and face value. That does not make an interim price decline economically meaningless.

• There is an opportunity cost from forgoing newly issued bonds with higher yields.
• If cash is needed and the bond is sold early, the valuation loss can become realized.
• Falling collateral value or deposit withdrawals can force a sale.
• Credit risk and liquidity risk remain separate from interest-rate risk.
• Depending on accounting classification, unrealized gains and losses may be reported in net income or other comprehensive income.

Under international accounting standards, debt instruments may be classified at amortized cost, fair value through other comprehensive income, or fair value through profit or loss based on factors including the business model and contractual cash-flow characteristics. Under U.S. accounting, the presentation of valuation gains and losses also differs among held-to-maturity, available-for-sale, and trading debt securities. Economic interest-rate risk and accounting presentation must therefore be checked separately.

4. A Simple Numerical Example

The following are hypothetical figures for learning purposes. They do not represent an actual bond or investment recommendation. Taxes, transaction costs, default, and coupon reinvestment are excluded.

① The Same Bond Has a Different Price at a Different Market Yield

Assume a bond with a face value of KRW 1 million, a 5% annual coupon, three years to maturity, and one coupon payment per year. The annual coupon is KRW 50,000.

If the market yield is 5%:

\[ P =\frac{50{,}000}{1.05} +\frac{50{,}000}{1.05^2} +\frac{1{,}050{,}000}{1.05^3} =KRW\ 1{,}000{,}000 \]

If the market yield rises to 6%:

\[ P =\frac{50{,}000}{1.06} +\frac{50{,}000}{1.06^2} +\frac{1{,}050{,}000}{1.06^3} \approx KRW\ 973{,}270 \]

If the market yield falls to 4%:

\[ P =\frac{50{,}000}{1.04} +\frac{50{,}000}{1.04^2} +\frac{1{,}050{,}000}{1.04^3} \approx KRW\ 1{,}027{,}751 \]

The coupon and principal stayed the same. The price moved solely because the yield required by the market changed.

② Approximate a Valuation Gain or Loss With Modified Duration

Assume a bond portfolio has a market value of KRW 10 billion and a modified duration of 6.8. Its market yield rises by 50 bps, or 0.005.

\[ Percentage\ price\ change \approx-6.8\times0.005 =-0.034 =-3.4\% \]

\[ Valuation\ loss \approx KRW\ 10\ billion\times3.4\% =KRW\ 340\ million \]

The estimated market value using duration alone is approximately KRW 9.66 billion.

③ Improve the Approximation With Convexity

Assume the same portfolio has convexity of 55.

\[ Convexity\ adjustment =\frac{1}{2}\times55\times(0.005)^2 =0.0006875 =0.06875\% \]

\[ Percentage\ price\ change \approx-3.4\%+0.06875\% =-3.33125\% \]

After convexity, the estimated valuation loss is about KRW 333.125 million and the estimated market value is about KRW 9.666875 billion. The actual price may differ because of nonparallel yield-curve shifts, credit spreads, options, and transaction costs.

④ The Same Yield Change Produces Different Results at Different Durations

Assume both portfolios have market values of KRW 10 billion and yields rise by 1 percentage point.

Despite the same yield shock, B has a much larger estimated valuation loss because it has more weight in distant cash flows.

5. How to Use This in Stock and Company Analysis

① Compare Yield and Duration Before the Coupon

When comparing bonds or bond products, review at least the following together:

• Current price and yield to maturity
• Modified duration and remaining maturity
• Credit rating and credit spread
• Call, put, conversion, and early-redemption terms
• Liquidity and transaction costs
• Cash flows to maturity and the investor’s actual holding period

A 6% coupon bond is not necessarily more attractive than a 3% coupon bond. If purchase prices, maturities, credit risks, or embedded options differ, expected returns and risks differ as well.

② Distinguish the Benefit of Lower Government Yields From Credit Deterioration

Even if a government bond yield falls by 0.5 percentage point, a 1.0-percentage-point widening in a company’s credit spread can raise the final corporate bond yield.

\[ Corporate\ bond\ yield \approx Government\ bond\ yield+Credit\ spread \]

Consequently, government yields can fall on recession concerns while a low-rated corporate bond loses value. Decompose the total yield change into the risk-free rate, credit spread, and liquidity components.

③ Review the Duration of Both Assets and Liabilities at Financial Companies

Banks and insurers hold large bond portfolios, so they can be highly exposed to yield changes. Looking only at valuation losses on bond assets is incomplete.

• If asset duration exceeds liability duration, rising yields can reduce asset value more than liability value.
• Deposit outflows or collateral calls can force bond sales regardless of the original intention to hold to maturity.
• An insurer’s long-term liabilities are also yield-sensitive, so assets and liabilities must be analyzed together.
• If derivatives are used for hedging, review notional amounts, maturities, hedge effectiveness, and accounting treatment.

The key question is not only the size of the valuation loss, but also the probability that liquidity pressure will force the loss to be realized.

④ Distinguish an Ordinary Company’s Bond Investments From Its Borrowings

Bond assets held by an ordinary company can lose fair value when yields rise. Conversely, the economic value of low fixed-rate debt issued in the past declines when yields rise, but if the liability is measured at amortized cost, that valuation gain may not immediately appear in earnings.

In stock analysis, distinguish:

• Whether investment-asset valuation gains and losses go through net income or other comprehensive income
• Whether borrowings have fixed or floating rates
• What yield will apply at the next refinancing
• The gap between reported carrying value and fair value
• Whether interest-rate derivatives are genuinely reducing risk

⑤ Discount Rates Also Affect Retirement Benefits and Impairment

Higher yields can increase the discount rate used for a defined benefit obligation and reduce the liability’s calculated present value. Lower yields can increase that present value. In value-in-use testing for goodwill and long-lived assets, a higher discount rate can reduce the recoverable amount and raise impairment risk.

The bond-pricing principle therefore extends beyond financial assets to pension liabilities, leases, provisions, long-term projects, and DCF valuation.

⑥ Remember the Limits of Duration

A single duration figure cannot explain every risk.

• It approximates a small, parallel shift in yields.
• A large yield change requires a convexity adjustment.
• If short- and long-term yields move differently, key-rate durations by maturity segment are needed.
• Corporate bonds add credit-spread and liquidity changes.
• Embedded options cause the cash flows themselves to change with yields.
• A bond fund’s holdings and duration may change continually.

Duration is best understood not as a forecasting tool but as a sensitivity tool that translates a yield shock into profit or loss.

⑦ Use Stress Tests to Build a Range of Outcomes

When analyzing a company or portfolio, calculate ranges rather than relying on one forecast.

Even a correct interest-rate forecast can produce a wrong investment result if the credit-spread or liquidity forecast is wrong.

6. What to Find in Disclosures

Central-Bank and Market Data: Check Outside Company Disclosures

A company’s financial statements do not contain every current bond price, yield curve, benchmark government yield, credit spread, or market duration. Obtain values for the same valuation date from exchanges, central banks, government statistics, and reliable bond-data sources.

When comparing market prices, use consistent definitions for clean price versus settlement price including accrued interest, yield compounding and coupon frequency, duration, and observation date.

Korean Companies: DART

Korea’s DART annual, semiannual, and quarterly reports contain the following items to review.

An ordinary company may not disclose duration or convexity directly. In that case, calculate them from bond cash flows, maturities, and yields or obtain them from a reliable market-data provider, while stating the definition and valuation date used.

U.S. Companies: 10-K and 10-Q

SEC EDGAR, review:

• 10-K Item 8 / 10-Q Part I, Item 1 — Financial Statements and Notes
• Debt Securities / Investments: amortized cost, unrealized gains and losses, fair value, maturity schedule
• Fair Value Measurements: Level 1·2·3, valuation techniques and inputs
• Accumulated Other Comprehensive Income: accumulated unrealized valuation gains and losses and period changes
• Credit Losses: allowance, write-offs, credit deterioration
• Debt: fixed/floating rates, maturities, fair value and covenants
• Derivatives and Hedging: swaps, notional amounts, cash-flow or fair-value hedges
• Pension and Postretirement Benefits: discount rate and sensitivity
• 10-K Item 7A / 10-Q Part I, Item 3 — Quantitative and Qualitative Disclosures About Market Risk: sensitivity of value or earnings to yield shocks

Connect the disclosure figures with three questions:

1. Are unrealized gains and losses reported in net income, other comprehensive income, or only in a note outside the carrying amount?
2. Does the company have enough cash and stable funding to withstand the valuation loss until maturity?
3. How much of the result came from government-yield changes versus credit spreads and impairment?

7. Key Takeaways

• A bond’s price is the present value of its promised coupons and principal.
• For fixed cash flows, a higher market yield lowers the price and a lower yield raises the price.
• The coupon rate is the interest rate set at issuance, while YTM is the market return reflected in the current price.
• All else equal, longer maturity and lower coupons tend to produce greater interest-rate sensitivity.
• Macaulay duration is the present-value-weighted average time to receive cash flows.
• Modified duration approximates the percentage price change from a small yield move as -duration × yield change.
• Convexity reflects the curved price–yield relationship and improves the duration approximation.
• A bond’s total return includes the price change and reinvestment income as well as the coupon.
• Even when a bond is intended to be held to maturity, liquidity pressure, opportunity cost, credit risk, and accounting valuation gains or losses cannot be ignored.
• In company analysis, review the classification of bond assets, OCI, fair value, interest-rate sensitivity, asset and liability maturities, and hedging together.

8. Three Review Questions

Question 1

The market yield on a plain bond with fixed face value and coupon rises from 4% to 5%. Which direction does the bond price move, and why?

Question 2

A bond portfolio has a market value of KRW 20 billion and a modified duration of 5.5. Its yield rises by 40 bps. Ignoring convexity, credit spreads, and transaction costs, calculate the percentage price change and valuation gain or loss.

Question 3

Two fixed-rate bonds have the same maturity, credit rating, and market yield. Bond A has a 2% coupon and Bond B has a 6% coupon. All else equal, which bond tends to have the greater duration and interest-rate sensitivity? Explain why.

Answers and Explanations

1

The bond price falls. The promised coupons and principal stay the same, but the market-required discount rate rises from 4% to 5%, reducing the present value of those future cash flows. If credit, liquidity, or embedded-option conditions also change, the actual price movement will reflect those effects too.

2

Forty basis points equals 0.40 percentage point, or 0.004 as a decimal.

\[ Percentage\ price\ change \approx-5.5\times0.004 =-0.022 =-2.2\% \]

\[ Valuation\ gain\ or\ loss \approx KRW\ 20\ billion\times(-2.2\%) =-KRW\ 440\ million \]

The duration approximation therefore indicates a decline of about 2.2% and a valuation loss of about KRW 440 million. Actual results will differ with convexity, nonparallel yield-curve shifts, and credit-spread changes.

3

Bond A, with the lower coupon, generally has greater duration and interest-rate sensitivity. Its interim coupon payments are smaller, so the cash-flow weight is more concentrated in principal at maturity and the present-value-weighted average recovery time moves further into the future.

Next Lesson — Day 29:
“What Is Added to a Government Yield to Produce a Corporate Bond Yield? — Reading Credit Spreads, Ratings, and Default Risk”