Running yield, redemption yield and index-linked bonds

Introduction: Navigating the World of Bond Yields

Welcome! In our journey through the "Equation of Value," we’ve learned how to find the price of a bond. But in the real world, investors often ask the question the other way around: "If I buy this bond at its current market price, what rate of return am I actually making?"

This chapter focuses on the different ways we measure that return—specifically through running yields and redemption yields. We will also explore index-linked bonds, which are clever financial instruments designed to protect investors from the "invisible thief" known as inflation. Whether you are aiming for a career in investment or just trying to pass your CM1 exam, understanding these yields is essential for comparing different financial "deals" on an equal footing.

1. Running Yield: The "Instant" Return

The running yield (also known as the interest yield or flat yield) is the simplest way to look at a bond's return. It tells you how much annual income you get relative to the price you paid for the bond.

The Concept: Imagine you buy a house for \$200,000 and it earns \$10,000 a year in rent. Your "running yield" is simply the rent divided by the price. It ignores whether the house value goes up or down in the future; it only cares about the cash currently "running" into your pocket.

The Formula

For a bond with an annual coupon \(D\) and a price \(P\), the running yield is:

\( \text{Running Yield} = \frac{D}{P} \)

If the bond pays coupons \(p\) times per year, where each payment is \(C\), then the total annual coupon is \(D = p \cdot C\). Note that the running yield is usually expressed as a nominal annual rate.

Key Limitation

The running yield is incomplete. It totally ignores the capital gain or loss you make when the bond is eventually redeemed. Because of this, it’s only a good measure for bonds that have no fixed redemption date (like "Undated" bonds) or for investors who only care about immediate income.

2. Redemption Yield: The "Total" Return

The redemption yield (often called the Yield to Maturity or YTM) is the "holy grail" of bond measures. It is the internal rate of return (IRR) an investor earns if they buy the bond at price \(P\) and hold it until it is redeemed at time \(n\).

Gross vs. Net Redemption Yield

• Gross Redemption Yield (GRY): The yield calculated before any taxes are taken out.
• Net Redemption Yield (NRY): The yield calculated after allowing for income tax on coupons and capital gains tax (CGT) on the profit made at redemption.

The Equation of Value

To find the redemption yield \(i\), we set up an equation of value where the Present Value (PV) of all future cash flows equals the current Price \(P\):

\( P = D \cdot a_{\overline{n}|}^{(p)} + R \cdot v^n \)

Where:
\(P\) = Current Price
\(D\) = Annual Coupon
\(R\) = Redemption Value
\(n\) = Time to redemption
\(v^n = (1+i)^{-n}\)
\(a_{\overline{n}|}^{(p)}\) = The annuity factor for coupons paid \(p\) times per year.

Pro-Tip: Estimating the Yield

Don't worry if you can't solve for \(i\) directly—you usually can't! In the exam, you will use linear interpolation.
1. Try a sensible interest rate (e.g., the coupon rate).
2. If the PV is higher than the price, try a higher interest rate.
3. Use the two results to "zero in" on the answer.

3. Index-Linked Bonds: Fighting Inflation

Standard bonds pay fixed coupons (e.g., \$5 every year). However, if inflation rises, that \$5 will buy fewer groceries in ten years than it does today. Index-linked bonds solve this by adjusting both the coupons and the redemption value in line with an inflation index (like the RPI).

How the Adjustment Works

The payments are scaled by the ratio of the inflation index at the time of payment to the index at the time the bond was issued.

If \(Q(t)\) is the value of the index at time \(t\), a coupon \(C\) due at time \(t\) becomes:

\( C \cdot \frac{Q(t)}{Q(0)} \)

The Time Lag (Conceptual)

In practice, inflation indices are published with a delay. Actuarial models often include a "time lag" (e.g., 3 months or 8 months). This means the payment at time \(t\) might actually be adjusted by the index value from several months prior: \( \frac{Q(t-lag)}{Q(0-lag)} \).

Nominal vs. Real Yields

• Nominal Yield: The return in terms of actual cash (dollars/pounds).
• Real Yield: The return in terms of "purchasing power" after removing the effect of inflation.

If we assume inflation is constant at rate \(e\), the relationship between the nominal yield \(i\) and the real yield \(i'\) is:

\( (1+i) = (1+i')(1+e) \)

Common Mistake: Students often forget to adjust the redemption value. In an index-linked bond, both the coupons and the final capital payment are increased by inflation.

4. Summary of Yield Relationships

Understanding how these yields behave relative to one another helps you spot errors in your calculations:

• If Price < Redemption (Discount): The Redemption Yield > Running Yield > Coupon Rate.
• If Price > Redemption (Premium): The Redemption Yield < Running Yield < Coupon Rate.
• If Price = Redemption (Par): Redemption Yield = Running Yield = Coupon Rate.

Key Takeaways

1. Running Yield = (Annual Coupon) / (Price). It's a "snapshot" of income.
2. Redemption Yield is the IRR of the bond. It takes into account coupons, price, and the final redemption amount.
3. Equation of Value is the tool used to solve for yields (usually via interpolation).
4. Index-linked bonds adjust all cash flows by an inflation index ratio, providing a real rate of return.
5. Taxes (Income tax and CGT) must be subtracted from cash flows to find the Net yield.

Next Step: Now that you've mastered yields, you're ready to look at how these principles apply to project appraisals and more complex loan schedules!