Pricing Conventions, Discounting, and Arbitrage

After completing this reading, you should be able to:

• Define discount factor and use a discount function to compute present and future values.
• Define the “law of one price,” explain it using an arbitrage argument, and describe how it can be applied to bond pricing.
• Identify arbitrage opportunities for fixed income securities with certain cash flows.
• Identify the components of a US Treasury coupon bond, and compare and contrast the structure to Treasury STRIPS, including the difference between P-STRIPS and C-STRIPS.
• Construct a replicating portfolio using multiple fixed income securities to match the cash flows of a given fixed-income security.
• Differentiate between “clean” and “dirty” bond pricing and explain the implications of accrued interest with respect to bond pricing.
• Describe the common day-count conventions used in bond pricing.

Discount Factors and Their Use to Compute Present and Future Values

A discount factor for a particular term gives the value today of one unit of currency due at the end of that term. It’s essentially a discount rate. The discount factor for \(t\) years is denoted as d(t). For example, if d(1) =0.85, then the present value of, say, $1 to be received a year from today is given by d(t)×$1=$0.85. Discount factors can easily be extracted from Treasury bond prices. The discount factor d(t) is the factor which, when multiplied by the total amount of money to be received (principal + interest), gives the price (present value) of the bond. However, when performing these calculations, it’s important to note that cash flows with different timings have different discount factors, in line with the time value of money. For example, the discount factor that applies to interest due in six months will be different from the discount factor for interest due in a year, i.e., d(0.5) ≠ d(1), and d(1) < d(0.5).

We can obtain discount factors from coupon-bearing bonds.

To illustrate this, let’s look at an example.

Example: Discount Factors

Suppose that three bonds with semi-annual coupon payments have the following cashflows:

$$\begin{array}{lccc} & \textbf{Maturity} & \textbf{Coupon Rate} &{ \textbf{Dirty Price}}\\
\textbf{Bond 1} & 1/1/2019 & 2\% & 100.2535\\ \textbf{Bond 2} & 1/7/2019 & 3\% & 100.3240 \\
\textbf{Bond 3} & 1/1/2020 & 4\% & 100.1020 \end{array}$$

Assuming $100 face value, what is the value of d(1.5)?

Solution

The cashflows provided by the bonds are as follows:

$$\begin{array}{lccc} & \textbf{1/1/2019} & \textbf{1/7/2019} & \textbf{1/1/2020}\\
\textbf{Bond 1} & 101 &- & -\\ \textbf{Bond 2} & 1.5 & 101.5&-\\ \textbf{Bond 3} & 2 & 2 & 102 \end{array}$$

The price of each bond can be found by discounting the cashflows of that particular bond,

Thus,

$$\begin{align*} 101*d(0.5)&=100.2535\\ \Rightarrow d(0.5)&=\frac{100.2535}{101}=0.992609\end{align*}$$

Similarly,

$$\begin{align*} &1.5*d(0.5)+101.5*d(1)=100.3240\\ \Rightarrow d(1)&=\frac{100.3240-1.5(0.992609)} {101.5}\\&=0.973745\end{align*}$$

$$\begin{align*} &2*d(0.5)+2*d(1)+102*d(1.5)=100.1020 \\ \Rightarrow d(1.5)&=\frac{100.1020-2(0.992609)-2(0.973745} {102}\\&=0.942836\end{align*}$$

Bond Price Quotations

A bond quote is the last price at which a bond is traded, expressed as a percentage of par value (100). Those bonds sold at a discount are priced at less than 100, and another group, although fewer, are sold at a premium and are priced at more than 100.

US T-bonds are quoted in dollars and fractions of a dollar – paving the way for the so-called “32nds” convention. And as the wording suggests, 32 portions of a dollar are considered. For example, if we have a T-bond quoted at 98–16, this means 98 “full” dollars plus 16/32 of a dollar, i.e., 0.5 dollars. Hence, the quote represents a price of $98.50.

Corporate or municipal bonds, on the other hand, use dollars and eight fractions of a dollar.

A “+” sign at the end of a quote represents half a tick. For example,

$$ 98-16+\text{implies } 98+\cfrac { 16.5 }{ 32 } $$

Treasury Bills

Treasury bills are short term debt obligation issued by the government, which usually last for one year or less and do not pay coupons. They are usually quoted at a discount to the face value of 100. The cash price is the face value minus the quoted discount rate.

Let Q be the quoted price of a T-bill and C be the cash price. If the cash price is 97 and there are 90 days until the maturity of the Treasury bill, then:

$$ \begin{align*} \text{Q}& =\cfrac { 360 }{ \text n } \left( 100-\text{C} \right) \\ & =\cfrac { 360 }{ 90 } \left( 100-97 \right) \\ & =12 \end{align*}$$

This means that we would pay 97 today to get the face value of 100 in 90 days. Our discount is 12 for every 100 of face value, which means our annual discount rate is around \(\frac {12}{100} = 12\%\).

Treasury Bonds

Treasury bonds last for more than one year and usually pay coupons. The accrued interest is the amount of coupon payment accrued between two coupon dates. When we are talking of treasury bonds,

$$ \text{Cash price}= \text{Quoted price}+ \text{Accrued interest} $$

The Law of One Price and How it Applies to Bond Pricing

The Law of one price states that the price of a security, commodity, or asset should be the same in two different markets, say, A and B. In other words, if two securities have the same cash flows, they must have the same price. Otherwise, a trader can generate a risk-free profit by buying on market A and selling on market B in one risk-free move. Such a possibility is called an arbitrage opportunity.

The Law of one price describes security price quite well because, in case of an arbitrage opportunity, traders rush en masse to take advantage of it. Within no time, market forces of supply and demand adjust the price to eliminate any deviations.

Let’s look at an example.

Consider a 1-year maturity bond with a face value of $100, a coupon rate of 10%, paying coupons semi-annually. Assume that the borrowing (bank) interest rate is 5% per annum.

The present value of the cash flows from this bond is:

$$ \text{PV}=\cfrac { 5 }{ 1.025 } +\cfrac { 105 }{ { 1.025 }^{ 2 } } =$104.82 $$

If the bond has a price of $100, an investor can borrow $100 from a bank and buy the bond. After six months, they will be able to repay $5 after receiving the first coupon. At this point, the debt outstanding will be equal to:

$$ \text{debt}=$100+$100\times2.5\%-$5=$97.5 $$

At the end of the year, the investor will pocket the principal ($100) as well as the second coupon of $5, making a total of $105.

$$ \text{Debt at this point} =$97.5(1.025)=$99.94 $$

Thus, after fully repaying the debt, they will be left with \($105-$99.94=$5.06\), which would effectively be a risk-free profit.

To exploit this situation, eagle-eyed investors in an efficient market would attempt to buy this bond by borrowing funds from banks. Increased demand would drive the price up so that at the end of the day, there would be no arbitrage opportunity.

Liquidity

Liquidity refers to how easily an asset converts to cash. It affects the price of a bond since it determines how easily the bond could be sold in the future.

Liquidity issues have, at times, causing a violation of the Law of one price, and therefore we can not always conclude that arbitrage opportunities do not exist at all.

Components of US Treasury Coupon Bond and the Structure of Treasury Strips

For instance, suppose we have a 10-year bond with a $100,000 face value and a 10% annual interest rate. Assuming it initially pays coupons semi-annually, 21 zero-coupon bonds can be created. That’s the 20 C-STRIPS plus the principal strip (P-STRIP). Each of the C-STRIPS has a $5,000 face value, which is the amount of each coupon. The P-STRIP to be received at maturity has a face value of $100,000.

Why are STRIPS so popular?

• They have a very high credit quality because US Treasury securities back them.
• Because they are sold at a discount, investors do not need a large stash of money to purchase them
• The payout is known in advance as long as the investor holds them to maturity
• They offer a range of maturity dates and can, therefore, be used to match liabilities due at specific points in the future.
• STRIPS are eligible for inclusion in tax-deferred retirement plans and non-taxable accounts such as pension funds, in which their value would grow tax-free until your retirement.

Disadvantages of STRIPS

• Shorter-term STRIPS tend to trade rich while longer-term STRIPS tend to trade cheap
• Sometimes they can be quite illiquid
• They typically trade very close to the fair value, thus potential profits are small

How to Construct a Replicating Portfolio Using Multiple Fixed Income Securities to Match the Cash Flows of a Given Fixed Income Security

Here’s an example of how a replicating portfolio can be created from multiple fixed income securities:

Assume we have a 2-year fixed-income security with $100 face value and a 20% coupon rate, paid on a semi-annual basis. Assume further that the security has a yield to maturity of 5%.

The present value of the security would be:

$$ { \text{PB} }_{ { \text{B} }_{ 1 } }=\cfrac { 10 }{ { 1.025 }^{ 1 } } +\cfrac { 10 }{ { 1.025 }^{ 2 } } +\cfrac { 10 }{ { 1.025 }^{ 3 } } +\cfrac { 110 }{ { 1.025 }^{ 4 } } =$128.21 $$

If this bond is determined to be trading cheap, a trader can carry out an arbitrage trade by purchasing the undervalued bond and shorting a portfolio that mimics (replicates) the bond’s cash flows. Assume that in addition to our bond above, which we shall call bond 1, we have four fixed income securities with the following characteristics:

$$ \begin{array}{c|c} \textbf{Bond} & \textbf{Coupon} & \textbf{PV} & \textbf{FV} & {\textbf{Time to} \\ \textbf{maturity}}\\ \hline \text{Bond 2} & {14\%} & $\text{106.35} & $\text{100} & \text{Six months} \\ \hline \text{Bond 3} & {24\%} & $\text{122.58} & $\text{100} & \text{12 months} \\ \hline \text{Bond 4} & {10\%} & $\text{113.07} & $\text{100} & \text{18 months} \\ \hline \text{Bond 5} & {12\%} & $\text{120.94} & $\text{100} & \text{24 months} \\ \end{array} $$

Note that these bonds also pay semi-annual coupons.

Using the above multiple fixed-income securities, we can create a replicating portfolio. However, we must first determine the percentage face amounts of each bond to purchase,\(\text F_{\text i}\), where i=1,2,3,4, which match bond 1 cash flows in every semi-annual period.

$$ \text{Bond1}\quad { \text{CF} }_{ \text{t} }={ \text{F} }_{ 2 }\times \cfrac { 14\% }{ 2 } +{ \text{F} }_{ 3 }\times \cfrac { 24\% }{ 2 } +{ \text{F} }_{ 4 }\times \cfrac { 10\% }{ 2 } +{ \text{F} }_{ 5 }\times \cfrac { 12\% }{ 2 }$$

In these types of calculations, the most straightforward approach to obtaining the values of \({\text{F}}_{\text{i}}\) involves starting from the end and then working backward. The logic here is simple. At 24 months, only bond 5 makes a payment. Hence at this point, all other values are equal to zero.

$$ $110={ \text{F} }_{ 2 }\times 0+{ \text{F} }_{ 3 }\times 0+{ \text{F} }_{ 4 }\times0 +{ \text{F} }_{ 5 }\times \left( 100+\cfrac { 12 }{ 2 } \right) \% $$

Solving this gives

$$ { \text{F} }_{ 5 }=\cfrac { 110 }{ 106\% } =103.77\%$$

Thus, we have to purchase 103.77%×100=$103.77 face value of bond 5

At 18 months, only bonds 4 and 5 make a payment. We can, therefore, obtain the value of \( {\text{F}}_{4} \) as follows:

$$ $10={ \text{F} }_{ 2 }\times 0+{ \text{F} }_{ 3 }\times 0+{ \text{F} }_{ 4 }\times \left( 100+\cfrac { 10 }{ 2 } \right) \%+103.77\times \cfrac { 12\% }{ 2 } $$

Solving this gives

$$ { \text{F} }_{ 4 }=\cfrac { 10-103.77\times0.06 }{ 1.05 } =3.59\%$$

Thus, we have to purchase $3.59 face value of bond 4

To solve for \({\text{F}}_{3}\),

$$ \begin{align*} $10 & ={ \text{F} }_{ 2 }\times 0+{ \text{F} }_{ 3 }\times \left( 100+\cfrac { 24 }{ 2 } \right) \%+3.59\times \cfrac { 10\% }{ 2 } +103.77\times \cfrac { 12\% }{ 2 } \\ { \text{F} }_{ 3 }& =\cfrac { 10-0.18-6.23 }{ 1.12 } =3.21\%\end{align*} $$

Similarly,

$$ \begin{align*} $10 & ={ \text{F} }_{ 2 }\times \left( 100+\cfrac { 14 }{ 2 } \right) \%+3.21\times \cfrac { 24\% }{ 2 } +3.59\times \cfrac { 10\% }{ 2 }+103.77\times \cfrac { 12\% }{ 2 } \\ { \text{F} }_{ 2 } & =\cfrac { 10-0.39-0.18-6.23 }{ 1.07 } =2.99915\approx 3.00 \%\end{align*} $$

We can create Cash flows from the replicating portfolio as the product of each bond’s initial cash flows and the face amount percentage. For example, the cash flow from bond five at 12 months is equal to:

\( \cfrac { \left( 12\% \right) }{ 2 } \times $100\times 103.77=$6.23 \)

Similarly, the cash flow from bond two at six months

$$ \cfrac { \left( 14\% \right) }{ 2 } \times $100\times 3.00\%=$0.21 $$

$$\small{\begin{array}{l|c|c|c|c|c|c|} \textbf{Bond} & \textbf{Coupon} & \textbf{Face Amount} & \textbf{CF(t=6)} & \textbf{CF(t=12)} & \textbf{CF(t=18)} & \textbf{CF(t=24)} \\ \hline\text{Bond 2} & 14\% & 2.99\% & 3.2 &  &  &  \\ \hline\text{Bond 3} & 24\% & 3.21\% & 0.39 & 3.59 &  &  \\ \hline\text{Bond 4} & 10\% & 3.59\% & 0.18 & 0.18 & 3.77 & \\ \hline\text{Bond 5} & 12\% & 103.77\% & 6.23 & 6.23 & 6.23 &  \\ \hline\text{Total cashflows} &  &  & 10 & 10 & 10 & 110 \\ \hline\text{Bond 1 cashflows} &  &  & 10 & 10 & 10 & 110\end{array}}$$

Where t= time in months

As can be seen above, the cash flows from the four bonds replicate bond one cash flows.

Clean vs. Dirty Bond Pricing

The dirty price of a bond is a bond pricing quote that’s equal to the present value of all future cash flows, including interest accruing on the next coupon payment date. Bonds do trade in the secondary-market before paying any coupon, or after clearing several coupons. In other words, the day a trader buys or sells the bond could be in between coupon payment dates.

In line with the principle of the time value of money, it’s only fair to compensate the seller of a bond for the number of days they have held the bond between coupon payment dates. We call this compensation of the accrued interest – the interest earned in between any two coupon dates.

$$ \text{Accrued interest}=\text{c}\left( \cfrac { (\text{Number of days that have elapsed since the last coupon was paid} }{ \text{Number of days in the coupon period}) } \right) $$

For example, suppose a $1,000 par value bond pays semi-annual coupons at a rate of 20%, and we’ve had 120 days since the last coupon was paid. Assuming that there are 30 days in a month,

$$ \text{Accrued interest}= \cfrac { 120 }{ 180 } \times$100=$66.70 $$

The seller would be compensated to the tune of $67.70, while the buyer would see out the coupon period and receive the remaining $33.30.

The clean price of a bond is the price that doesn’t include any coupon payments.

The dirty and clean prices are also known as the full and quoted prices, respectively.

Day-Count Conventions

When computing the accrued interest, we use one of several day-count conventions. These include:

• Actual/actual
• Actual/360
• Actual/365
• 30/360
• 30E/360 (E stands for Europe)

Interpretation of these conventions is relatively straightforward. For example, the actual/actual convention considers the actual number of days between two coupon dates. The 30/360 convention assumes there are 30 days in any given month and 360 days in a year.

For purposes of the exam, note the following:

1. We use the actual/actual convention for US government bonds
2. US corporate and municipal bonds use the 30/360 convention
3. The actual/360 convention is common in money markets

Exam tips:

• If coupons are paid semi-annually, the denominator should be 180 in both actual/360 and 30/360 conventions. Similarly, the denominator would be 90 for quarterly coupons.
• Almost all US Treasury trades settle T + 1, which means that the exchange of bonds for cash happens one business day after the trade date.
• Clean price = dirty price – accrued interest

Dirty Price Formula

$$ \text{Price}=\cfrac { \text{C} }{ { \left( 1+\text{y} \right) }^{ \text{k} } } +\cfrac { \text{C} }{ { \left( 1+\text{y} \right) }^{ \text{k}+1 } } +\cfrac { \text{C} }{ { \left( 1+\text{y} \right) }^{ \text{k}+2 } } +\dots \cfrac { \text{C}+\text{F} }{ { \left( 1+\text{y} \right) }^{ \left( \text{k}+\text{n}-1 \right) } } $$

Where:

P = price

C = semi-annual coupon

k = number of days until the next coupon payment divided by the number of days in the coupon period, determined as per the relevant day-count convention.

y = periodic required yield

n = number of periods remaining, including the present one.

F = face value (par value) of the bond

Question 1

A $1,000 par value US corporate bond pays coupons semi-annually on January 1 and July 1 at the rate of 20% per year. Mike Brian, FRM, purchases the bond on March 1, 2018, intending to keep it until maturity. The bond is scheduled to mature on July 1, 2021. Compute the dirty price of the bond, given that the required annual yield is 10%.

A. $1,310.25

B. $502.50

C. $400.25

D. $1,100

The correct answer is A.

As a US corporate issue, this bond is valued based on the 30/360 day-count convention. Under this convention, the number of days between the settlement date (March 1, 2018) and the next coupon date (July 1, 2018) is 120 (= 4 months at 30 days per month).

Each coupon payment is valued at \( \cfrac {20\%} {2}\times$1,000=$100 \)

$$ \text{Price}=\cfrac { \text C }{ { \left( 1+ \text y \right) }^{ \text k } } +\cfrac { \text C }{ { \left( 1+ \text y \right) }^{ \text k+1 } } +\cfrac { \text C }{ { \left( 1+ \text y \right) }^{ \text k+2 } } +\dots \cfrac { \text C+ \text F }{ { \left( 1+ \text y \right) }^{ \left( \text k+ \text n-1 \right) } } $$

Where:

P = price

C = semi-annual coupon

k = number of days until the next coupon payment divided by the number of days in the coupon period, determined as per the relevant day-count convention.

y = periodic required yield

n = number of periods remaining, including the present one.

F = face value (par value) of the bond

In this case, n=7

$$ \begin{align*} \text{Price} & =\cfrac { 100 }{ { \left( 1.05 \right) }^{ 0.67 } } +\cfrac { 100 }{ { \left( 1.05 \right) }^{ 1.67 } } +\cfrac { 100 }{ { \left( 1.05 \right) }^{ 2.67 } } +\cfrac { 100 }{ { \left( 1.05 \right) }^{ 3.67 } } +\cfrac { 100 }{ { \left( 1.05 \right) }^{ 4.67 } } +\cfrac { 100 }{ { \left( 1.05 \right) }^{ 5.67 } } + \cfrac { 100+1000 }{ { \left( 1.05 \right) }^{ 6.67 } } \\ & =96.78+92.18+87.79+83.61+79.62+75.83+794.44=1,310.25 \end{align*} $$

Question 2

An analyst has been asked to check for arbitrage opportunities in the Treasury bond market by comparing the cash flows of selected bonds with the cash flows of combinations of other bonds. If a 1-year zero-coupon bond is priced at USD 97.25 and a 1-year bond paying a 20% coupon semi-annually, is priced at USD 114.50, what should be the price of a 1-year Treasury bond that pays a coupon of 10% semi-annually?

A. $105.88

B. $100

C. $103.35

D. $105

The correct answer is A.

The secret here is to replicate the 1-year 10% bond using the other two treasury bonds whose price we already know. To do this, you could solve a system of equations to determine the weight factors, F1 and F2, which correspond to the proportion of the zero and the 20% bond to be held, respectively.

At every coupon date, the cash flow from the 10% bond should match cash flows from the zero-bond and the 20% bond.

At t=1, the 10% coupon bond pays 105, and both the zero-bond and the 20% also have got payouts of 100 and 110, respectively

$$ 105=\text{F}1\times \left(100\right) +\text{F}2\times110…………\text{equation1} $$

At t=0.5, the 10% coupon bond pays 5, the zero bond pays 0, and the 20% bond pays 10

$$ 5=\text{F}1\times0+\text{F}2\times10…………\text{equation2} $$

Solving equation 2,

$$ \text{F}2=\frac{5}{10}=0.5 $$

Solving equation 1,

$$\begin{align*} 105&=100\text{F}1+0.5\times110 \\ 50&=100\text{F}1 \\ \text{F}1&=0.5 \end{align*} $$

Thus, the price of the

$$ \begin{align*}10\% \text{coupon bond}&=0.5\times \text{price of zero bond}+0.5\times \text{price of 20% bond } \\ &=0.5\times97.25+0.5\times114.5 &=$105.88 \end{align*}$$

Note: You should assume the prices are given as per $100 face value