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Premium experiment • discounted cash-flow valuation

ECONOMICS • FINANCE

Bond Prices, Yields & Duration

Discount coupon and principal payments one period at a time to see why bond prices move inversely with market yields and how duration summarizes interest-rate sensitivity.

● CASH-FLOW DISCOUNTINGfuture cash flows → present value
Bond price
Market yield
Duration
Premium / discount

Reasoning inspector

live model state
Bond price—
Yield—
Coupon payment—
Macaulay duration—
Modified duration—
Price minus face value—

Experiment with the mechanism

A bond is the present value of promised future cash flows. Raising the discount rate lowers the present value of each payment, while duration summarizes how far into the future the value is concentrated.

LEARN WITH THE SIMULATION

Understand Bond Prices, Yields & Duration by experimenting.

START HERE

A conventional fixed-rate bond is valued as the present value of its future coupon payments plus the repayment of face value at maturity. The market yield is the discount rate applied to those cash flows, which is why a higher yield lowers today’s price. Duration summarizes the timing of the discounted cash flows and provides a useful first-order measure of interest-rate sensitivity.

BUILD THE IDEA

What you should understand first

  • Build a bond price by discounting each promised cash flow.
  • Explain the inverse relationship between bond prices and market yields.
  • Interpret Macaulay and modified duration as measures related to timing and price sensitivity.

TRY IT YOURSELF

Which bond is more sensitive to a yield change: a short maturity or a long maturity?

  1. Record price and duration for a short-maturity bond.
  2. Increase maturity while keeping coupon rate and yield fixed.
  3. Move the market yield by the same amount in each case.
  4. Compare both the price changes and modified duration.
Read the visual:

Longer-maturity bonds generally have greater duration and therefore greater price sensitivity to a given yield movement, all else equal.

WHY IT WORKS 01

Why price and yield move in opposite directions

The promised cash flows of a fixed-rate bond are set by contract. When the market yield rises, each future payment is divided by a larger compounding factor, reducing its present value. Because the bond price is the sum of those present values, the market price falls. A lower yield does the reverse, making the same fixed payments more valuable today.

WHY IT WORKS 02

Duration links timing to rate sensitivity

Cash flows received far in the future are more exposed to changes in the discount rate than near-term cash flows. Macaulay duration summarizes the present-value-weighted timing of the bond’s payments, while modified duration rescales that timing to approximate how strongly price responds to a small change in yield. Higher duration therefore usually means more interest-rate risk.

DO NOT MIX THESE UP

Common misconception

A bond’s coupon rate is not the same as its market yield. Yield changes do not alter the promised coupon cash flow on a fixed-rate bond; they change the discount rate and therefore the market price. Duration is an approximation to sensitivity, not a guarantee of the exact price change for a large yield move.

PRACTICE CHALLENGE

Can you make a bond trade at a premium and then turn it into a discount bond without changing its coupon?

  • Start with a market yield below the coupon rate and record the price.
  • Raise the market yield above the coupon rate.
  • Replay the cash-flow discounting sequence.
  • Explain why the same promised coupon stream moves from above face value to below face value.
Reveal the reasoning

Yes. When the market yield is below the coupon rate, the bond’s promised payments are attractive relative to newly required returns, so their discounted value can exceed face value. When yield rises above the coupon rate, those same payments are less attractive and the bond price falls below face value.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The bond has fixed annual coupon payments and repays face value at maturity.
  • A single constant annual market yield discounts every cash flow.
  • There is no default risk, tax, liquidity premium, call option or reinvestment uncertainty.

Model limitations

  • Real bonds can have term-structure effects, credit risk, embedded options and nonannual payment conventions.
  • Modified duration is a local approximation and becomes less accurate for large yield changes because price-yield relationships are convex.

CONNECT IT

Where this matters

  • Fixed-income valuation
  • Interest-rate risk
  • Bond portfolio education
  • Present-value analysis

WORDS THAT MATTER

Key vocabulary

Coupon
The periodic interest payment promised by a fixed-rate bond.
Yield
The market discount rate used here to value the bond’s future cash flows.
Macaulay duration
The present-value-weighted average time until the bond’s cash flows are received.
Modified duration
A duration measure commonly used to approximate percentage price sensitivity to a small yield change.