Spot Rate (Zero-Coupon Bond Yield)

Definition

Spot Rate, also known as the Zero Coupon Bond Yield, is a financial concept used to estimate the yield an investor would earn if they bought a zero-coupon bond and held it until maturity. It represents the difference between the bond’s purchase price and the face value received at maturity.

Origin of the Name

The term "Spot Rate" in the bond field comes from "spot" transactions in the financial market, i.e., immediate transactions. In a spot transaction, the buyer and seller agree to exchange currency or goods immediately. Similarly, the "Spot Rate" of a bond refers to the rate at which you can trade immediately, without considering future interest rate changes.

Aliases

• Zero Coupon Yield
• Zero Rate

The origin of these names: the "Spot Rate" is commonly used to calculate the yield of zero-coupon bonds (bonds that do not pay interest periodically)

Pros and Cons

Pros

1. Simplicity: The calculation of the zero-coupon bond yield is relatively simple because it only involves the face value paid at maturity and the purchase price of the bond, and does not consider interest payments in the meantime.

2. Lower Risk: Since zero-coupon bonds do not pay interest, investors do not need to worry about reinvestment risk, i.e., the interest income reinvestment rate may be lower than the original bond rate.

3. Certain Returns: Zero-coupon bonds provide certain future returns because investors know how much they will receive at maturity.

Cons

1. Liquidity Issues: Since zero-coupon bonds do not pay interest during their term, investors may have difficulty getting cash inflows before maturity.

2. Inflation Risk: If the inflation rate is higher than the yield of the zero-coupon bond, the real return to the investor may be eroded.

3. Credit Risk: If the bond issuer defaults, investors may lose their entire investment because they can only get a return when the bond matures.

Application

Type	Description	Applicability of Spot Rate
Zero-coupon bonds	These bonds do not pay interest during the term, only paying the face value at maturity.	Yes, the Spot Rate is primarily used to calculate the yield of zero-coupon bonds as it only requires the bond’s face value and purchase price.
Fixed-rate bonds	These bonds pay a fixed interest periodically throughout the term and return the principal at maturity.	Possible, even though the Spot Rate is typically used for zero-coupon bonds, it can also be used to construct the yield curve of fixed-rate bonds, thus evaluating the true value of the bond.
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.	Possible, but more complex. The Spot Rate can be used to construct a yield curve to evaluate the value of floating-rate bonds, but changes in the benchmark rate need to be considered.
Convertible bonds	These bonds allow the holder to convert them into a specific number of common shares under certain conditions.	Possible, but very complex. The Spot Rate can be used to estimate the value of the bond portion, but the evaluation of convertible bonds also needs to consider the value of the conversion rights, which usually requires more complex pricing models.

Formula Derivation

Derivation

Starting from the basic concept of discounted cash flows, the formula for Zero-Coupon Bond Yield (Spot Rate) is gradually derived.

The formula for the present value (PV) of future cash flows is:

$$
PV = \frac{FV}{(1+r)^n}
$$

Where:

• $PV$ is the present value, which is the current value of future cash flows.
• $FV$ is the future value, which is the cash flow you will receive in the future.
• $r$ is the discount rate, which is your investment return rate, i.e., the rate of return your money can get.
• $n$ is the number of periods, i.e., the time from the future cash flow to the present.

Now, let’s consider a special type of bond – a zero-coupon bond. Zero-coupon bonds do not pay interest during their term, but only pay their face value upon maturity. Therefore, for a zero-coupon bond, we only have one future cash flow, which is the face value at maturity.

Suppose you bought a zero-coupon bond with a face value of $FV$ and a maturity of $n$ periods, and you paid $P$ for this bond. We can consider this price as the present value of the future cash flow (i.e., the face value of the bond). Therefore, we can rewrite the present value formula as:

$$
\text{Spot Rate} = \left( \frac{FV}{P} \right)^{1/n} – 1
$$

Where:

• $FV$ is the face value or par value of the bond, which will be repaid at the end of the $T$ period.
• $P$ is the current price of the bond.
• $n$ is the total number of periods until maturity.
• Spot Rate is the zero-coupon bond yield we are looking for.

Spot Rate and Price

The Spot Rate shows a negative correlation with the bond price.

If the bond price rises (you need to spend more money to buy the same bond), the return you get from this investment (relative to your investment cost) will decrease i.e., the Spot Rate will decrease.

Conversely, if the bond price falls (i.e., you need to spend less money to buy the same bond), the return you get from this investment will increase, i.e., the Spot Rate will rise.

Example

Suppose we have the following details about a zero-coupon bond:

• The bond price is 800 units.
• The bond will mature in 2 years.
• The face value of the bond is 1000 units.

We need to find a Spot Rate that makes the price equal to the discounted cash flow. In other words, we need to find a Spot Rate that satisfies the following equation:

$$
P = \frac{FV}{(1+\text{Spot Rate})^T} = 800
$$

Solving this equation, we get:

$$
\text{Spot Rate} = \left( \frac{FV}{P} \right)^{1/T} – 1 = \left( \frac{1000}{800} \right)^{1/2} – 1 = 0.118
$$

So, the yield of this zero-coupon bond is 11.8%.

Spot Rate vs Yield to Maturity (YTM)

Similarities

English	Spot Rate	Yield to Maturity (YTM)
Market-based	It is derived from the market price of zero-coupon bonds.	It is calculated based on the market price of the bond.
Time Value of Money	Takes into account the time value of money.	Takes into account the time value of money.
Risk Indicator	Represents the risk associated with a specific time frame.	Represents the overall risk of the bond until its maturity.
Discount Rate	Used as a discount rate in pricing securities.	Used as a discount rate in pricing bonds.

Differences

	Spot Rate	Yield to Maturity (YTM)
Interest Payments	Does not pay interest, only pays face value at maturity	Pays interest during the period, pays face value at maturity
Calculation Complexity	The calculation is relatively simple, only involving the face value at maturity and the bond price	The calculation is relatively complex, needs to consider interest payments during the period and the face value at maturity
Reinvestment Risk	No reinvestment risk because there is no interest payment	There is reinvestment risk because it assumes that interest can be reinvested at the YTM rate
Applicable Types	Applicable to zero-coupon bonds	Applicable to bonds that pay interest periodically