(2) Yield to Maturity (YTM)

Definition

YTM, or Yield to Maturity, is a financial concept often used to estimate the total return an investor would receive if they held a bond or other fixed-interest security until it matures. It is the internal rate of return (IRR) of the cash flows from a bond given its price.

Pros and Cons

Pros

1. Comparing Investments: YTM provides a standard metric to compare different bonds and other fixed-income investments. It reflects the total return a bondholder will receive if the bond is held until maturity, assuming all interest payments are reinvested at the same rate.

2. Price Determination: YTM can be used to calculate the present value of a bond’s future coupon payments and principal repayment, helping to determine its fair price.

3. Investment Decision Making: Investors often use YTM to decide whether to buy a bond. If the YTM of a bond is higher than the investor’s required rate of return, the investor might consider buying the bond.

4. Risk Assessment: YTM can be a useful tool to evaluate the risk of bond investments. Bonds with higher YTMs typically carry higher risks (e.g., default risk), and vice versa.

5. Portfolio Management: YTM is often used in managing fixed-income portfolios. Portfolio managers can use YTM to strategize their investment decisions, such as whether to invest in short-term or long-term bonds based on their yield-to-maturity.

Cons

Yield to Maturity (YTM) is a useful tool for evaluating and comparing the potential investment returns of bonds or other fixed income securities. However, it is not without limitations. Here are some of the key restrictions:

1. Reinvestment Risk: The calculation of YTM assumes that all coupon payments can be reinvested at the same YTM rate. However, in reality, market interest rates may fluctuate, making it impossible to reinvest at the anticipated YTM. This introduces reinvestment risk, which could impact actual returns.

2. Default Risk: The calculation of YTM does not consider the risk that the bond issuer might default. If the issuer defaults before maturity, investors might not be able to receive the expected returns. When assessing the true returns of a bond, the credit risk of the issuer should be taken into account.

3. Expected Life: The calculation of YTM assumes that the bond will be held until maturity. However, many bonds (for instance, callable bonds or convertible bonds) might be redeemed or converted before maturity. In these situations, YTM may not be an accurate measure.

4. Changes in Market Interest Rates: The calculation of YTM is based on the bond’s purchase price and fixed payment stream. However, if market interest rates change, the price of the bond may fluctuate, affecting the actual returns to investors.

5. Computational Complexity: The calculation of YTM typically requires solving a complex mathematical equation known as a "nonlinear equation". This can become quite complex in some situations, particularly for bonds with unconventional cash flows (for example, irregular coupon payments or face values).

Therefore, while YTM is an important tool for evaluating fixed income investments, investors also need to consider these limitations and may need to use other assessment tools in conjunction, to fully understand the risks and returns of potential investments.

Application

Type	Description	Applicability of Yield to Maturity
Zero-coupon bonds	These bonds do not pay interest during the term, only paying the face value at maturity.	Not applicable, YTM is more suitable for bonds that pay interest periodically.
Fixed-rate bonds	These bonds pay a fixed interest periodically throughout the term and return the principal at maturity.	Yes, YTM is primarily used to calculate the yield of fixed-rate bonds.
Floating-rate bonds	The interest payment of these bonds is tied to a benchmark interest rate (like LIBOR), and the rate changes as the benchmark rate changes.	Yes, even though calculating the YTM of floating-rate bonds is more complex, YTM can still be used to evaluate the bond yield.
Convertible bonds	These bonds allow the holder to convert them into a specific number of common shares under certain conditions.	Yes, but very complex. The calculation of the YTM for convertible bonds needs to consider the value of the bond portion and the conversion rights portion, which usually requires more complex pricing models.

Formula Derivation

The formula for YTM is similar to the formula for IRR, and is derived from the time value of money principle.

You might need some background knowledge on the time value of money – link # The Time Value of Money (TVM)

The YTM is the discount rate that makes the following equation hold:

$$
P = \frac{C_1}{(1+YTM)^1} + \frac{C_2}{(1+YTM)^2} + … + \frac{C_T + FV}{(1+YTM)^T}
$$

Where:

• $C_t$ is the coupon payment (interest) in the $t$th period.
• $T$ is the total number of periods till maturity.
• $P$ is the current price of the bond.
• $FV$ is the face value or par value of the bond, which is repaid at the end of period $T$.
• $YTM$ is the Yield to Maturity we are trying to find.

This formula can be simplified to

$$
P = \sum_{t=1}^{T} \frac{C_t}{(1+YTM)^t} + \frac{FV}{(1+YTM)^T}
$$

Approximate Solution of the Equation

Let’s consider we have the following bond details:

• The bond price is 950 units.
• The annual coupon payment is 50 units.
• The bond will mature in 3 years.
• The face value of the bond is 1000 units.

We need to find a YTM that makes the price equal to the discounted cash flows. That is, we need to find a YTM that satisfies the following equation:

$$
P = \frac{50}{(1+YTM)^1} + \frac{50}{(1+YTM)^2} + \frac{50 + 1000}{(1+YTM)^3} = 950
$$

As with the IRR, the solution to equations usually involves methods such as the Bisection Method and Newton’s Method.

Please refer to this chapter for details on the solution methods – link # Methods of Equation Solving: Bisection and Newton’s Approach

YTM (Yield to Maturity) vs IRR (Internal Rate of Return)

Similarities

	Yield to Maturity (YTM)	Internal Rate of Return (IRR)
Discounted Cash Flows	The sum of discounted future cash flows equals the current investment or price	The sum of discounted future cash flows equals the current investment or price
Investment Assessment	Used to evaluate the effectiveness of an investment	Used to evaluate the effectiveness of an investment
Interest Rate Calculation	Requires a numerical method to solve an equation involving an unknown interest rate	Requires a numerical method to solve an equation involving an unknown interest rate
Investment Assumptions	Assumes that bond’s interest and principal will be continually paid until the bond matures	Assumes that future cash flows are determined based on the forecast of the investment project

Differences

	Yield to Maturity (YTM)	Internal Rate of Return (IRR)
Field of Application	Mainly used for investment analysis of bonds and other fixed income securities	Commonly used in areas such as capital budgeting and project investment
Certainty of Cash Flows	Future cash flows (such as interest and principal repayments) are generally known and fixed	Future cash flows (such as project income and costs) may be projected and have a certain degree of uncertainty
Regularity of Cash Flows	Typically considers equal, periodic cash flows	May need to handle unequal, non-periodic cash flows
Reinvestment Assumption	Assumes all interest income will be reinvested at the YTM rate	Assumes all positive cash flows will be reinvested at the IRR rate