Скачать или смотреть Mastering Bonds: From Price & Yield to Duration & Convexity | A Complete Beginner’s Guide

What is a Bond?
Fundamentally, a bond represents a loan from an investor to an issuer, such as a government or a corporation. It is a "certificate of indebtedness" where the issuer promises to pay the investor a guaranteed return over a specified period.
To understand how a bond functions, one must grasp its three primary components:
• Face Value (Par Value): The amount the bond issuer promises to pay back to the investor when the bond matures.
• Maturity: The specific future date on which the face value is repaid. This can range from short-term (such as Treasury bills maturing in under a year) to long-term (such as Treasury bonds maturing in over 10 years).
• Coupon: The interest payment made to the bondholder, usually paid semi-annually or annually. The "coupon rate" is the annual interest payment expressed as a percentage of the face value.
The Price-Yield Relationship
A critical concept in fixed-income investing is the inverse relationship between market interest rates and bond prices. They move in opposite directions, acting like a see-saw.
• When Interest Rates Rise: The prices of existing fixed-rate bonds fall. This occurs because new bonds are issued with higher coupon rates, making existing bonds with lower coupons less attractive. To sell an existing bond, the holder must lower the price (sell at a discount) to match the market’s new higher yield.
• When Interest Rates Fall: The prices of existing bonds rise. Existing bonds with higher coupon rates become more valuable compared to new, lower-yielding issues, allowing them to trade at a premium.
Yield to Maturity (YTM) The YTM is the internal rate of return on a bond’s cash flows if held until maturity. It accounts for the current market price, the coupon payments, and the repayment of the principal. If a bond sells at par, the YTM equals the coupon rate; if it sells at a discount, the YTM exceeds the coupon rate.
Valuation of Bonds
Bond valuation determines the fair price of a bond. This is calculated using the Present Value (PV) approach. The value of a bond is the sum of the present values of all expected future cash flows (coupons and principal repayment), discounted at the appropriate market rate.
Clean vs. Dirty Price When buying a bond between coupon payment dates, the valuation must account for interest that has accumulated since the last payment.
• Dirty Price (Full Price): The price including accrued interest. This is the actual amount paid by the buyer.
• Clean Price: The quoted price of the bond excluding accrued interest. Clean prices are more stable, whereas dirty prices rise daily as interest accrues until the coupon is paid.
Measuring Interest Rate Risk
Investors use two primary metrics to estimate how sensitive a bond's price is to changes in interest rates: Duration and Convexity.
1. Duration
Duration measures the first-order sensitivity of a bond's price to interest rate changes. It is often described as the weighted average time until cash flows are received.
• Macaulay Duration: This calculates the weighted average time (in years) to receive the bond's cash flows. For a zero-coupon bond (which pays no interest until maturity), the Macaulay duration is exactly equal to its time to maturity.
• Modified Duration: This adjusts Macaulay duration to estimate the percentage change in price for a 1% change in yield. For example, if a bond has a duration of 3, a 1% rise in interest rates will cause the bond's price to fall by approximately 3%.
Key Factors Affecting Duration:
• Time to Maturity: The longer the time to maturity, the higher the duration (and risk). Cash flows received further in the future are more sensitive to discount rate changes.
• Coupon Rate: The higher the coupon rate, the lower the duration. Higher coupons return cash to the investor sooner, reducing sensitivity to rates.
2. Convexity
While duration provides a useful linear estimate, it becomes inaccurate for large changes in interest rates because the relationship between price and yield is actually curved (convex).
• The Curvature: Convexity measures the curvature of the price-yield relationship (the second derivative). It shows that bond prices rise more when rates fall than they drop when rates rise.
• Adjustment: To get a precise price estimate, analysts add a convexity adjustment to the duration estimate. This accounts for the non-linear behaviour of bond prices.
• Benefit: Generally, higher convexity is desirable for investors (assuming equal duration and yield) because it implies better price protection against rate rises and greater appreciation when rates fall.
Opportunity Cost
Investing in bonds also involves opportunity cost—the value of the next best alternative forgone. For instance, if an investor locks money into a bond yielding 6%, and market rates subsequently rise to 9%, the opportunity cost is the missed additional return. Conversely, choosing a safe bond might mean foregoing potential (but riskier) gains in the equity market