A fixed-income security, such as a bond, provides three different types of returns, and a yield measure used by the investor should consider all these three sources of return. These are: the coupon payment received, capital gains/losses, and reinvestment income. Let’s look at each of these sources of income in detail.

Coupon Payments

When an investor purchases a regular bond, the issuer of the bond makes coupon payments to the investor periodically based on the bond specifications. For example, a 5-year $100 par 8% semi-annual coupon paying bond will pay $4 coupon to the investor semi-annually for the next 5 years. At maturity the investor will receive the final coupon payment and the par value ($100) from the bond. Unlike a regular bond, a zero-coupon bond does not make coupon payments. Instead the investor purchases the zero-coupon bond at a deep discount and receives the par value at maturity, compensating him for the interest.

Capital Gains/Losses

Bonds also provide capital gains or losses. While purchasing the bond, the price that an investor pays may be above par or below part. However, at maturity the bond always pays par value. So, if a bond has a par value of $100 and the investor purchased the bond for $95 (below par) and held it till maturity, he will have a capital gain of $5. Alternatively, if he purchased the bond at $102 and held it till maturity, then he will have a capital loss of $2. If the investor does not hold the bond till maturity, then the capital gain/loss will be calculated as the difference between the purchase price and the sale price. If a bond is called by the issuer, then the capital gain/loss will be calculated as the difference between the call price and the purchase price.

Reinvestment Income

The periodic income (coupons) provided by the bond can be reinvested in other securities, and income from such reinvestment is called reinvestment income. When we talk about <a title="Yield to Maturity (YTM) of a Bond" href="https://financetrain.com/yield-to-maturity-ytm-of-a-bond/">yield-to-maturity (YTM)</a>, it specifies the yield considering that all your interim cash flows continue to earn the same yield. However, it may not always be possible for the investor to be able to invest these interim funds at the same interest rate. On the other hand, since zero-coupon bonds don’t pay coupon, there is no reinvestment income. All the money remains invested in the zero-coupon bond till its maturity ensuring a fixed yield.

Sources of Return from Investing in a Bond

The current yield of a bond measures the interest income that an investor gets from the bond. It’s represented using the following formula:

Current Yield = Annual Coupon Interest/Bond Price

Where,
<ul>
	<li>Annual coupon interest is the total coupon payment received by the bond annually</li>
	<li>Bond price is the market price of the bond</li>
</ul>
Let’s take a $1,000 par bond, with 8% coupon and 7 years to maturity. The price of the bond is $1,112.96.

The current yield of the bond can be calculated as follows:

Current yield = $80/1112.96 = 7.19% (or 8/111.296)

One of the biggest drawbacks of current yield is that is does not take into account the capital gain or loss from the bond and hence doesn't represent the true yield from the bond. It does not account for the reinvestment risk or the fact that the bond matures at par value. In the above example, Current yield &gt; YTM because YTM accounts for price dropping to par at maturity (capital loss).

If we know YTM and the capital gains from the bond, then the current yield will be = YTM – capital gains yield.

To know the actual yield from the bond, <a href="https://financetrain.com/how-to-calculate-yield-to-maturity/">Yield-to-maturity</a> (YTM) is a better measure. YTM, coupon yield and Current Yield are compared below:
<ul>
	<li>For a par bond, YTM = Current Yield = Coupon Yield</li>
	<li>For a premium bond, YTM &lt; Current Yield &lt; Coupon Yield</li>
	<li>For a discount bond, YTM &gt; Current Yield &gt; Coupon Yield</li>
</ul>
&nbsp;

How to Calculate Current Yield

Yield to Maturity (YTM) is the most commonly used and comprehensive measure of risk. In fact, if someone talks about just ‘Yield’ they are most likely referring to Yield to Maturity.

In simple terms, YTM is the discount rate that makes the present value of the future bond payments (coupons and par) equal to the market price of the bond plus any accrued interest.

Consider the following equation:

<img class="aligncenter size-full wp-image-9895" src="https://financetrain.com/wp-content/uploads/tym.gif" alt="Yield to Maturity" width="522" height="80" />

Where:
<ul>
 	<li>P is the bond price</li>
 	<li>C is the periodic coupon payment</li>
 	<li>N is the number of years to maturity</li>
 	<li>M is the (face value) payment at maturity</li>
 	<li>y is the “risk-adjusted discount rate” (or yield to maturity, or IRR)</li>
</ul>
In the above equation, we solve for y, which is the yield to maturity of the bond.

It’s a trial and error process, and you need a spread sheet or a calculator to calculate YTM.

Let’s take a simple example to understand how YTM is calculated. Consider a $1,000 par bond, with 8% coupon and 7 years to maturity. The price of the bond is $1,112.96. Assume that the bond pays coupon semi-annually.

C = $40

N = 2 x 7 = 14

M (face value) = $1,000

P = $1,112.96

Plugging all these numbers in the above equation, we calculate y = 3.00%

In excel you can use Goal Seek to get the YTM. You can also use a <a href="https://financetrain.com/how-to-compute-bond-yield-ytm-using-texas-instruments-ba-ii/">calculator such as Texas Instrument BA II to calculate YTM</a>.

Since this bond paid semi-annual coupon, the y we calculated is the semi-annual YTM. We use the <a href="https://financetrain.com/bond-equivalent-yield-convention">bond-equivalent yield convention</a> to calculate the annual YTM. So, YTM = 3.00x 2 = 6%

On the other hand, the effective annual yield is (1.03)2 – 1 = 6.09%

Again, note that YTM is a bond equivalent yield (BEY) which is calculated by applying simple interest rules. So, we annualize the rate by multiplying, instead of using raise to the power. For semi-annual yield, we double the semi-annual yield. For quarterly yield, we will quadruple the quarterly yield.

YTM is a preferred yield measure over <a href="https://financetrain.com/how-to-calculate-current-yield/">current yield</a> because it considers both interest income and capital gains for calculating the returns from the bond. It also includes the reinvestment income.

How to Calculate Yield to Maturity

In the bond market the convention is to annualize the semi-annual yield by simply doubling it. So, if the semi-annual yield is 3%, the annual yield is calculated simply as 3% x 2 = 6%. The annual yield so calculated is called the bond-equivalent yield (BEY).

This convention doesn’t follow the time value of money rules where you would compound the semi-annual yield to calculate the effective annual yield. Instead the doubling convention is followed across the market.

A common question asked by students is ‘Why have such a convention and why not instead use effective annual yield?’ The answer to this question is that since it’s a convention everybody uses it and therefore yields are comparable. It doesn’t really affect performance or comparison between bonds because everyone would have used the convention to quote the yield. Conventions are usually made to make things simpler. In this case if someone tells you that the bond-equivalent yield is 6%, you instantly know that semi-annual yield is 3%, which you can use to perform any calculations or to calculate the effective annual yield if you require it.

If the convention was to use effective annual yield, it may have been better, but does it really matter? In fact there are many other limitations of YTM that far outweigh the problem of BEY convention. So, my suggestion to you would be to just follow the convention and don’t fret over it. It is important however, that you use the convention correctly.

Bond Equivalent Yield Convention

<a href="https://financetrain.com/how-to-calculate-yield-to-maturity/">Calculating yield to maturity</a> can sometimes be a tedious process because we need to do trial and error to arrive at the exact yield.

In this article we provide an approximation formula to calculate YTM that you can use for quick calculation. The following formula assumes semi-annual coupon payments.

<img class="aligncenter size-full wp-image-9906" alt="YTM Approximation" src="https://financetrain.com/wp-content/uploads/ytm-approximate.gif" width="351" height="167" />

Where:
<ul>
	<li>F = Face Value = Par Value (Usually $1,000)</li>
	<li>P = Bond Price</li>
	<li>C = the semi-annual coupon interest</li>
	<li>N = number of semi-annual periods left to maturity</li>
</ul>
Let’s take an example to understand how to use the formula. Let us find the yield-to-maturity of a 5 year 6% coupon bond that is currently priced at $850. The calculation of YTM is shown below:

<img class="aligncenter size-full wp-image-9908" alt="ytm-example" src="https://financetrain.com/wp-content/uploads/ytm-example.gif" width="535" height="227" />

Note that the actual YTM in this example is 9.87%. However, our approximation is good enough for exams or for quick comparisons.

Yield to Maturity (YTM) Approximation Formula

<a title="How to Calculate Yield to Maturity" href="https://financetrain.com/how-to-calculate-yield-to-maturity/">Yield to maturity</a> calculates the total return an investor would earn by holding the bond till maturity. This includes both the coupon income and the capital gains from the bond. It also considers reinvestment income, however, with the assumption that coupon payments can be invested at the same rate as YTM. This means that you will actually earn a yield equal to YTM, even if you hold the bond to maturity, only if you are able to reinvest all the coupon payment at the YTM until the maturity of the bond. This is very unlikely and that's why we have reinvestment risk.

This assumption can be misleading because we have a term structure of interest rates, which can result in different yields at different points in time.

Let’s demonstrate this with the help of an example.

Let’s say that an investor has purchased a 15-year 8% semi-annual coupon bond at par. We assume that he purchased the bond at par for simplicity. Since there is no capital gain/loss, the bond’s YTM will also be 8%.

We can calculate the total future value of all cash flows from this bond as follows:

Future dollar value = $100 x (1.04)30 = $324.34

This value is made up of $100 of principal return and $224.34 of the total dollar return from coupon and reinvestment income. In fact if there was no reinvestment income the coupon income would only be $120. Remember that there is no capital gain, so the balance ($224.34 - $120) = $104.35 is earned from reinvestment income. This is the money which is earned by investing the periodic coupon payments at YTM of 8%.

If the coupons are not reinvested at 8%, the investor will not be able to realize the YTM of 8%.

How much reinvestment risk is present in a bond depends on several factors such as coupon rate and bond’s maturity. Here are some observations.
<ul>
	<li>A bond that has high coupon is more dependent on reinvestment income because more money needs to be reinvested at the YTM to maintain the YTM.</li>
	<li>For a bond selling at premium, there is capital loss, which further increases the dependency on reinvestment income.</li>
	<li>For a bond selling at a discount, there is capital gain, which reduces the dependency on reinvestment income.</li>
	<li>The longer the bond’s maturity, the higher is the reinvestment risk.</li>
</ul>

YTM and Reinvestment Risk

As we learned in the previous article, coupon paying bonds have reinvestment risk because the investor is expected to invest the cash flows from the bond at the same rate as <a href="https://financetrain.com/how-to-calculate-yield-to-maturity/">yield-to-maturity (YTM)</a> to be able to realize the YTM if he holds the bond till maturity.

There are two key characteristics of a bond that influence the quantum of reinvestment risk in the bond. They are the maturity of the bond and the coupon rate. Let’s look at these factors.

Bond’s Maturity

Other factors remaining the same, a bond with a higher maturity will have higher reinvestment risk. This is because the interim coupon payments need to be reinvested for a longer period of time to realize the YTM. We can say that it is difficult to rely on YTM for long-term bonds to measure earnings potential especially if the investor wants to hold the bond till maturity.

Bond’s Coupon Rate

Other factors remaining the same, a bond with a higher coupon will have the higher reinvestment risk. This is because higher dollar amount needs to be reinvested to realize the YTM. This again may not always be possible.

What do you think about the reinvestment risk in a zero-coupon bond? A zero-coupon bond actually has no reinvestment risk because there are no interim coupons paid and all the money is in a way reinvested in the bond itself and a final amount is paid. This is true especially if the bond is held till maturity.

Also read: <a title="YTM and Reinvestment Risk" href="https://financetrain.com/ytm-and-reinvestment-risk/">YTM and Reinvestment Risk</a>

Let’s take an example. Consider a $1000 par, 8% coupon 10 year maturity bond, and a $1000 par, 8% coupon and 12 year maturity bond.
<ul>
	<li>Which bond will have more investment risk? The second bond because they both pay the same coupon but the second bond has higher maturity and therefore higher dependency on reinvestment income.</li>
	<li>What if the first bond is sold at a discount while the second bond is sold at par? Then the first bond will have capital loss and its dependency on reinvestment risk will increase. Depending on at how much discount the bond is selling, the first bond may even have higher reinvestment risk.</li>
</ul>

Factors Affecting Reinvestment Risk

In the US, most bonds are generally semi-annual coupon paying bonds, so we calculate the semi-annual yield and then calculate the <a href="https://financetrain.com/bond-equivalent-yield-convention">bond-equivalent yield</a> (annualized) by simply doubling the semi-annual yield. This is done when the bonds have semi-annual coupon payments.

However, not all bonds pay semi-annual coupon, especially there are many non-US bonds that pay coupon annually and hence will have the annual yield that is calculated by the compounding rules.

The bond-equivalent yield of a semi-annual coupon bond and the annual yield from an annual-pay bond cannot be directly compared. To make them comparable, we need to convert the annual yield from the annual-pay bond into bond equivalent yield, which is done using the following formula:

BEY of annual-pay bond = 2 x [(1+ yield on annual-pay bond)0.5 -1]

What the above formula does is first calculates the semi-annual yield and then applies the convention of doubling it to arrive at the bond equivalent yield.

Let’s take an example. Let say that an annual coupon paying bond has a yield of 6.5%. The bond-equivalent yield will be calculated as follows:

BEY = 2*(1.065^0.5 – 1) = 6.398%

Note that BEY will always be lesser than the effective annual yield or the <a title="How to Calculate Yield to Maturity" href="https://financetrain.com/how-to-calculate-yield-to-maturity">YTM</a> of annual pay bond because BEY doesn’t consider the effect of compounding.

Alternatively you could also convert the BEY of a semi-annual pay bond into YTM on annual-pay basis by using the formula:

YTM on annual-pay basis = (1+BEY/2)2 - 1

So, if BEY is 6%, then YTM on annual-pay basis will be:

YTM (annual-pay basis) = (1+6%/2)^2 – 1 = 6.09%

Again note that YTM on annual pay basis is more than Bond-equivalent Yield.

Calculate Bond-Equivalent Yield of Annual-Pay Bonds

For callable bonds that are likely to be called before their maturity, it is more useful to calculate yield to call instead of yield to maturity. The formula and steps to calculate yield to call are exactly the same as how we <a href="https://financetrain.com/how-to-calculate-yield-to-maturity/">calculate yield to maturity</a>, i.e., you calculate the discount rate that makes the present value of the future bond payments (coupons and par) equal to the market price of the bond plus any accrued interest. There are two deviations from the standard formula:
<ol>
	<li>For calculating yield to call the bond price will be taken as the price at which the bond is called back.</li>
	<li>The maturity will be the date with expected call date.</li>
</ol>
Bonds will have a call schedule which will specify how much the issuer will pay on the date the bond is called. Investors can calculate various types of yield to call such as yield to first call or yield to next call.

Let’s take an example:

Consider a $1,000 par 8% coupon, 5 years maturity bond selling at $800. The bond is callable and the first call date is 2 years from now at a call price of $1010.

The cash flows from the bond upto call date are the coupon payments every 6 months, and the call price after two years. So, the yield to call will be the interest rate that will make the present value of these cash flows equal to the bond price of $800.

Assuming semi-annual coupon payments, the yield to call will be calculated as follows:

$800 = 40/(1+y) + 40/(1+y)^2 +  40/(1+y)^2 + 1050/(1+y)^2

Solving for y, we get:

Y = 10.61%

As you can see, yield to call consider all sources of income such as coupon income, capital gains, and reinvestment income upto the call date. This measure assumes that investor is going to hold the bond till the call date and that the bond will be called on the scheduled date. These assumptions are not practical and it’s also not possible to directly compare the yield to call of a callable bond with the yield to maturity of a non-callable bond.

Just like yield to call we can also calculate yield to put, where the bond can be put (sold) back by the investor back to the issuer as a scheduled at and at a specified put price.

How to Calculate Yield to Call of a Bond

For some debt products such as <a title="Introduction to Mortgage-backed Securities – Part 1 of 3" href="https://financetrain.com/introduction-to-mortgage-backed-securities/">mortgage-backed securities</a>, we calculate cash flow yield instead of <a title="How to Calculate Yield to Maturity" href="https://financetrain.com/how-to-calculate-yield-to-maturity/">yield to maturity</a>. This is because investors in mortgage-backed securities receive their cash flow from the cash flow of the underlying collateral pool and this cash flow is not fixed. In the underlying collateral pool, the home owners make the payments for the loans they have taken which acts as the cash flow to mortgage-backed securities. However, the homeowners can prepay their loans based on the changing interest rate conditions. This will affect the principal and interest payments to the MBS. Since the cash flow is uncertain, typically an assumption is made about the <a title="Single Monthly Mortality (SMM) &amp; Conditional Prepayment Rate (CPR)" href="https://financetrain.com/single-monthly-mortality-smm-conditional-prepayment-rate-cpr/">prepayment rate</a>, and based on that cash flows are estimated. This cash flow is then used to calculate the cash flow yield.

The procedure to calculate cash flow yield will be the same as for YTM, i.e., the interest rate that will make the present value of all cash flows equal to the current price of the MBS.
<h3>Cash Flow Yield on a Bond-Equivalent Basis</h3>
Just like for semi-annual bonds, here also we need to convert the cash flow yield to <a title="Bond Equivalent Yield Convention" href="https://financetrain.com/bond-equivalent-yield-convention/">bond-equivalent yield</a>.

The cash flows to MBS are paid monthly. So, the cash flow yield we calculate will also be monthly yield. Bond equivalent yield calculation is a two-step process. Let’s say the monthly yield is 1%.

Step 1: Calculate Effective Semi-annual Yield

The first step is to convert the monthly yield into effective semi-annual yield.

Effective semi-annual yield = (1+ym)^6 – 1

Effective semi-annual yield = (1.01)^6 – 1 = 6.152%

Step 2: Calculate Bond Equivalent Yield (BEY)

We calculate BEY by simply doubling the effective semi-annual yield

BEY = 2*6.152 = 12.304%

As you can see, BEY is calculated as per the convention in the market.

Cash Flow Yield

A spot rate curve, also known as a zero curve refers to the yield curve constructed using the spot rates such as Treasury spot rates instead of the yields.



A spot rate Treasury curve is more suitable to <a href="https://financetrain.com/calculate-clean-and-dirty-price-of-a-bond/">price bonds</a> because most bonds provide multiple cash flows (coupons) to the bond holders at different points in time, and it is better to use the spot rates as the discount rates for different time periods rather than using a single discount rate.



We can build a spot rate curve using on-the-run treasuries, off-the-run treasuries, or both. In this article we will learn how to construct build a spot rate curve using a method called bootstrapping. In the bootstrapping technique one repetitively applies a no-arbitrage implied forward rate equation to yields on the estimated Treasury par yield curve. Given below is the step-by-step process to arrive at the spot curve using the bootstrapping method.



Step 1: Decide on the Instrument for Yield Curve



The spot curve can be obtained by using on-the-run Treasury securities, off-the-run treasury securities, or a combination of both, or Treasury coupon strips. The instrument you select should not have credit risk, liquidity risk, embedded options, or any pricing anomalies. In our example, let’s use on-the-run Treasury yields.



Step 2: Select the Par Yield Curve



Typically, you will not find Treasury securities for only a few maturities such as 3-month, 6-month, 2-year, 5-year, 10-year, and 30 years.



Step 3: Interpolate the Missing Yields



We will have to interpolate the in between yields for different periods, such as for 3-year, 3-year, 6-year, and so on. This can be done using the simple interpolation formula however it will not be very accurate.



Step 4: Calculate Spot Rates Using Treasury Yields



In this step we will apply the bootstrapping method to calculate the spot rates.



To reiterate, the spot curve is made up of spot interest rates for zero coupon bonds of different maturities. For example, a 2-year spot rate tells us for the interest rate is for a <a href="https://financetrain.com/calculate-value-zero-coupon-bond/" class="rank-math-link">zero-coupon bond</a> of two-year maturity. We bootstrap this data from the Treasury yield curve.



Let’s say we have the following data:



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/zero1.gif" alt="zero1" class="wp-image-9942"/></figure></div>



Let’s use the above data to calculate the spot rates. Note that first two securities, i.e., the 6 month and 1 year Treasury securities are T-bills which are discount securities, essentially zero-coupon securities. So, for these the spot rate will be the same as the yield, i.e., 4% and 4.3%.



0.5 year spot rate, z1 = 4%



1 year spot rate, z2 = 4.3%



We can now use this data to calculate the 1.5 year spot rate. Since 1.5 year bond is selling at par, its coupon will be 4.5%. The three cash flows are:



Cash flow at 0.5 year = 100*0.045*0.5 = 2.25



Cash flow at 1 year = 100*0.045*0.5 = 2.25



Cash flow at 1.5 year = 100 + 100*0.045*0.5 = 102.25



We want the 1.5 year zero rate such that the present value of all cash flows discounted by their respective spot rates is equal to the bond’s price i.e., 100.



100 = 2.25/(1+4%/2)^1 + 2.25/(1+4.3%/2)^2 + 102.25/(1+z3/2)^3



Solving for z3, we get z3 = 4.511, which is the 1.5 year spot rate.



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/zero2.gif" alt="zero2" class="wp-image-9943"/></figure></div>



When we plot the spot rates against the maturities, we get the spot rate or the zero curve.



Similarly, we can calculate the 2-year spot rate. I’ll leave for the readers to calculate that. Once&nbsp;you've&nbsp;calculated, please add your answer in the comments.

Bootstrapping Spot Rate Curve (Zero Curve)

The simplest way to calculate the value of a bond is to take the cash flows of the bond till its maturity and then discount them by a single discount rate. The method is quick but not very accurate because the yield curve is not flat and the interest rates are different for different maturities. A better way to price the bonds is to discount each cash flow with the spot rate (zero coupon rate) for its respective maturity.

Example 1

Let’s take an example. Suppose we want to calculate the value of a $1000 par, 5% coupon, 5 year maturity bond. We also have the following spot rates for the next 5 years:

<img class="aligncenter size-full wp-image-9949" alt="price1" src="https://financetrain.com/wp-content/uploads/price1.gif" width="135" height="129" />

Assuming this is an annual pay bond, the bond will have the following cash flows.

Year 1: $50

Year 2: $50

Year 3: $50

Year 4: $50

Year 5: $1000 + $50

The value of the bond can be calculated by discounting these cash flows by their respective spot rate.

Bond Value = 50/(1.04)^1+50/(1.0430)^2+50/(1.0451)^3+50/(1.047)^4+1050/(1.048)^5

Bond Value = $1010.033

Example 2

Let’s take another example. Suppose we have a bond that matures in 2 years, that has a coupon rate of 6%, and pays coupon semi-annually. The spot rates are 3.9% for 6 months, 4% for 1 year, 4.15% for 1.5 years, and 4.3% for 2 years.

The cash flows from this bond are $30, $30, $30, and $1030.

The value of the bond will be calculated as follows:

Bond value = $30/(1+3.9%/2)^1+$30/(1+4%/2)^2+$30/(1+4.15%/2)^3+$1030/(1+4.3%/2)^4

Bond value = $1032.45

You can use the above formula to value any bond with any maturity. All you need is the spot rate for the respective maturity. Since <a title="Bootstrapping Spot Rate Curve (Zero Curve)" href="https://financetrain.com/bootstrapping-spot-rate-curve-zero-curve/">spot rates are bootstrapped from yield curve</a>, we may have to interpolate yields for certain maturities for which Treasury securities are not available.

How to Price a Bond Using Spot Rates (Zero Curve)

Nominal Spread, or the nominal yield spread, is the most simple yield spread for non-Treasury bonds. Nominal spread measures the difference between the yield of a bond and the <a title="How to Calculate Yield to Maturity" href="https://financetrain.com/how-to-calculate-yield-to-maturity/">yield to maturity</a> of a similar maturity Treasury bond. Consider the following two 10-year bonds:
<ul>
	<li>A Treasury bond having a YTM of 6.5%</li>
	<li>A non-Treasury bond having a YTM of 8%</li>
</ul>
Nominal Spread = Yield of non-Treasury Bond – Yield of Treasury Bond

Nominal Spread = 8% - 6.5% = 1.5%

The difference between the YTM for the two bonds is 1.5% (150 bps). This is the nominal spread.

A non-Treasury bond usually provides a higher yield compared to a Treasury bond because of the additional risk involved, especially the credit risk and the liquidity risk. It could also be because of other features such as the risk due to the embedded options.

Nominal spread is a way to price the bonds. Generally a spread is taken over the Treasury yield, and used as a discount factor to value the bond.

Even though it’s simple to calculate, it’s not a very strong measure because it doesn’t consider the spot rates for different maturities and it also ignores the effect of embedded options.

Due to this reason other spread measures such as the <a href="https://financetrain.com/z-spread-definition-and-calculation/" title="Z-Spread: Definition and Calculation">z-spread</a> are more popular.

Nominal Spread

The problem with <a title="Nominal Spread" href="https://financetrain.com/nominal-spread/">nominal spread</a> is that it measures the spread at just one point on the yield curve. The z-spread solves this problem by considering the spot yield curve instead of the standard yield curve.

The z-spread, also known as the zero-volatility spread or the static spread, measures the spread that the investor will receive over the entire <a title="Bootstrapping Spot Rate Curve (Zero Curve)" href="https://financetrain.com/bootstrapping-spot-rate-curve-zero-curve/">Treasury spot rate curve</a>.

For the purpose of calculation, we start with an assumption for the z-spread. One takes the Treasury spot rates for each maturity, adds the z-spread to it, and uses this new rate as a discount rate for each maturity to price the bond. The correct z-spread is the one that makes the present value of cash flows equal to the price of the bond.

P = C1/(1+r1 + z) + C2/(1+r2 + z)2 + C3/(1+r3 + z)3 ... T(1+rn + z)n

Where,
<ul>
	<li>P represents the clean price of the bond plus any accrued interest.</li>
	<li>C represents the coupon payments.</li>
	<li>T represents the total cash flow received at maturity</li>
	<li>r1,r2… represent the zero spot rates for each maturity</li>
	<li>z represents the z-spread.</li>
</ul>
Note that the benchmark for calculating z-spread is the spot rate curve. Unlike nominal spread, the z-spread is spread over the entire Treasury spot rate curve.

The z-spread represents the additional risk the investor is taking in the form of credit risk, liquidity risk, and option risk.

In most cases, such as for the vanilla coupon paying bonds, the z-spread will only slightly diverge from the nominal spread. The difference mainly comes from the shape of the term structure and the bond characteristics. For example, the difference will be high for amortizing bonds for which the principal is repaid over time, bonds with high coupon, and when the yield curve is steep.

Z-Spread: Definition and Calculation

The <a title="Z-Spread: Definition and Calculation" href="https://financetrain.com/z-spread-definition-and-calculation/">Z-spread</a> handles one problem present in <a title="Nominal Spread" href="https://financetrain.com/nominal-spread/">nominal spread</a> effectively, i.e., that is, it measures the spread over the entire spot rate curve instead of only at one point in the curve. However, there is another problem that comes because of the embedded options. Due to the embedded options in the bonds, there is uncertainty about the future cash flows but both nominal spread and z-spread do not account for it. Option-adjusted spread is the third spread measure which takes care of this problem as well.

The option-adjusted spread removes the effect of embedded options on future returns, to reflect non-option risks when comparing a bond to a benchmark. It is the spread over the entire Treasury spot rate curve, but after accounting for the embedded options.

Unlike the other two spread measures, OAS reflects only credit and liquidity risk.

This is the best spread measure for comparing the yields of bonds with embedded options to Treasury yields.

The option-adjusted spread is model dependent and will vary based on the model used to compute it. It also depends on our assumption of the interest rate volatility. The higher the assumption of volatility, the lower will be the spread.

Below are a few points on how OAS compares to z-spread
<ul>
	<li>For bonds without options, OAS = Z-spread</li>
	<li>For a bond with call option, OAS is lesser than Z-spread</li>
	<li>For a bond with put option, OAS is great than Z-spread</li>
	<li>The difference between Z-spread and OAS is called Option cost.</li>
</ul>
When the OAS for a bond is higher than the OAS of comparable bonds relative to the same benchmark, the bond is considered undervalued.

Alternatively, when the OAS for a bond is lower than the OAS of comparable bonds against their relevant benchmark, the bond is considered overvalued.

Option-adjusted Spreads (OAS)

Till now we have looked at the spot rates and we also learned about how to <a title="Bootstrapping Spot Rate Curve (Zero Curve)" href="https://financetrain.com/bootstrapping-spot-rate-curve-zero-curve/">construct spot rate curve</a>. Essentially a spot rate is the borrowing or lending interest rate today for a specified period of time. For example, s1 refers to the one-year spot rate, i.e., the interest rate you will realize in one year. s2 represents the annualized two-year spot rate, i.e., the rate you will realize if you invest/lend/borrow for two years.

A forward rate, on the other hand, is the interest rate for the future. For example, you may want to know what will be the one-year interest rate one year from now. Similarly, you can find out what will be the 2-year forward rate one year from now. Just like spot rates, forward rates can also plotted on a graph to arrive at the forward rate curve.

Forward rates are represented with specific terminology, and once you know it, it will be very easy for you to read any forward rate. The forward rate itself is represented by an f, and there are two subscripts, one before f and the other after f, such as 1f2.

The first subscript represents the time period for which the rate applies, for example, 1-year forward rate. The second subscript represents when the forward rate begins, for example, one year from now or two years from now.

Let’s take a few examples:

1f1 represents 1-year forward rate 1 year from now.

1f2 represents 1-year forward rate 2 year from now.

1f3 represents 1-year forward rate 3 year from now.

2f1 represents 2-year forward rate 1 year from now.

2f2 represents 2-year forward rate 2 year from now.

Note that the above notations assume that each period is for one year. In some cases, you can assume one period equal to 6-months also. In that case 1f2 represents 6-month forward rate 1 year from now. Some other resources may have totally different notations, but this is the standard way of reading forward rates.

We can derive forward rates from the spot rate curve, which we will learn in our next article.

What are Forward Rates?

Once we have the <a title="Bootstrapping Spot Rate Curve (Zero Curve)" href="https://financetrain.com/bootstrapping-spot-rate-curve-zero-curve/">spot rate curve</a>, we can easily use it to derive the forward rates. The key idea is to satisfy the no arbitrage condition – no two investors should be able to earn a return from arbitraging between different interest periods. Let’s take an example of how this works. Let’s say an investor wants to invests his funds for two years. He is faced with two choices:
<ol>
	<li>Directly invest in a 2-year bond</li>
	<li>Invest in a one-year bond, and again invest the proceeds after one year in a one year bond.</li>
</ol>
Assuming the same nature of investments, the returns from both choices should be the same.

Let’s say s1 is the one-year spot rate, s2 is the two-year spot rate and 1f1 is the one year forward rate one year from now.

Assuming $1 as the initial investment, the value of investment in first choice after two years:

= (1+s2)2

The value of investment in second choice after two years:

= (1+s1) (1+1f1)

If there are no arbitrage opportunities, both these values should be the same.

(1+s2)2 = (1+s1) (1+1f1)

If we have the spot rates, we can rearrange the above equation to calculate the one-year forward rate one year from now.

1f1 = (1+s2)2/(1+s1) – 1

Let’s say s1 is 6% and s2 is 6.5%. The forward rate will be:

1f1 = (1.065^2)/(1.06) – 1

1f1 = 7%

Similarly we can calculate a forward rate for any period.

How to Calculate Forward Rates from Spot Rates?

We have seen that a <a title="How to Price a Bond Using Spot Rates (Zero Curve)" href="https://financetrain.com/how-to-price-a-bond-using-spot-rates-zero-curve/">bond can be valued using spot rates</a> by discounting each cash flow by the spot rate for the maturity. We also saw that <a title="How to Calculate Forward Rates from Spot Rates?" href="https://financetrain.com/how-to-calculate-forward-rates-from-spot-rates/">forward rates can be derived from spot rates</a>. If so, we can also value a bond using forward rates instead of spot rates.

Let’s take a specific cash flow in a bond to understand this. Say, a bond is going to pay $100 as coupon after 2 years. s2 is the 2-year spot rate is 6%. The present value of this cash flow will be:

PV of $100 = $100/(1+s2)2

We also know that

(1+s2)2 = (1+s1) (1+1f1)

Replacing this is the PV calculation:

PV of $100 = $100/(1+s1) (1+1f1)

If s1 is 6% and 1f1 is 7%.

Here 1/(1+s1) (1+1f1) is called the forward discount factor.

The resulting valuation using either spot rates or forward rates will be the same.

How to Value a Bond Using Forward Rates

Yield Measures, Spot Rates, and Forward Rates

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Yield Measures, Spot Rates, and Forward Rates

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