linkIntroduction

Fixed Income encapsulates securities that pay predictable cash flows at regular intervals. This includes bonds, ...

Bond → A bond is a promise. It's essentially a piece of paper that defines an agreement between two parties, the issuer and the bondholder. At the time of the issue,

Any bond can be described by the following characteristics:

• coupon rate schedule
• frequency / schedule of payment dates
• principal (face value)
• maturity date
• issuer
• risk (e.g. credit risk, interest rate risk, liquidity risk)
• additional covenants (e.g. callability, convertibility, sinking fund provisions)

These characteristics are unique to each bond issue and are listed in a bond indenture which is a legal document that outlines the terms and conditions of the bond. Bonds can therefore be grouped and classified based on these characteristics. For example, a bond with a fixed coupon rate is called a fixed-rate bond. A bond with no coupon (i.e. only one bullet payment at maturity) is called a zero-coupon bond. Short-term bonds (less than one year) issued by the U.S. treasury are called treasury bills or T-bills. Below is a picture of how phyisical bonds looked like in the past. They are now mostly digital but the concept is the same.

linkCommon types of bonds

Straight bond → A bond that pays a fixed coupon rate at regular intervals and returns the principal at maturity.

Zero-coupon bond → A bond that does not pay periodic interest. Instead, it is issued at a discount to its face value and pays the full face value at maturity.

Bullet bonds → A bond that pays periodic interest but does not return the principal until maturity.

Amortizing bonds → A bond that pays periodic interest and also returns a portion of the principal with each payment. Example: residential mortgage loans.

Bonds with embedded options → callable bonds, puttable bonds, extendible bonds, convertible bonds

Organizations issue bonds to raise capital. Investors lend to organizations to earn income. Speculators and traders buy and sell bonds to earn from active management. When governments and companies issue fresh bonds, they almost always make it available to select investors to purchase. These are large institional investors with massive money. It's a win-win siutation for both issuers and institutional investors. Issuers don't have to negotiate with millions of small, retail investors and institutional investors can also get a slight discount because of the wholesale rate. This is called the primary market. The process makes sense logistically, however I wonder if automated systems will replace the primary market.

For organizations, bonds are one of the tools to raise money. Capital structure, which is the mix of equity and debt sources used to fund operations and investments, is important to balance. It usually varies by industry and geography.

# Pricing and Valuation of Bonds

linkPricing a bond

There are strict rules used across the industry to price bonds. It's a carefully calculated symphony of what you pay today versus what you'll get in return in the future given the current situation and future expectations. Since large issuers dictate the terms of the issue, it is up to the investor to evaluate the worth of the securities being issued. As you'll see, it is not necessary that the price of a bond is equal to it's par value (principal).

Let's take a real example. The current interest rate the U.S. government is offering on its 2-year treasury notes is 4.125% (fixed for the tenor) paid every 6 months. One can lend as much as one wishes but let's say you have $10,000. Your next best option is to park your money in the bank account at 4.2% paid monthly. Here's a comparison of the cash flows.

[Insert here]

Why is the bank offering a higher interest rate?

For banks, this interest rate is the cost of borrowing. Their cost of borrowing is usually higher than the government because people the government has more creadibility than the banks.

The computation below calculates the price of the bond at issuance and assumes that this is the best return you get for the given level of risk.

$$P = \frac{206.25}{1+\frac{4.125\%}{2}} + \frac{206.25}{(1+\frac{4.125\%}{2})^2} + \frac{206.25}{(1+\frac{4.125\%}{2})^3} + \frac{206.25+10,000}{(1+\frac{4.125\%}{2})^4} = 10,000$$

What if your opportunity cost by taking identical risk is different than 4.125%? In such case, the price of the bond would also change and not equate to the par value. We consider two examples.

• $r_d = 4\%$
  
  

$P = \frac{206.25}{1+\frac{4\%}{2}} + \frac{206.25}{(1+\frac{4\%}{2})^2} + \frac{206.25}{(1+\frac{4\%}{2})^3} + \frac{206.25+10,000}{(1+\frac{4\%}{2})^4} = \$10,023.80$

• $r_d = 4.25\%$
  
  

$P = \frac{206.25}{1+\frac{4.25\%}{2}} + \frac{206.25}{(1+\frac{4.25\%}{2})^2} + \frac{206.25}{(1+\frac{4.25\%}{2})^3} + \frac{206.25+10,000}{(1+\frac{4.25\%}{2})^4} = \$9,976.27$

Here's a summary of what the price would be given the opportunity cost.

$r$	$P$	%Δ from par
3.750%	$10,119.72	1.1972%
3.875%	$10,047.67	0.4767%
4.000%	$10,023.80	0.2380%
4.125%	$10,000	NIL
4.250%	$9,976.27	0.2373%
4.375%	$9,952.62	0.4738%
4.500%	$9,929.04	0.7096%

Notice that the

We can also calculate the yield-to-maturity (YTM) from the price of the bond. YTM is the internal rate of return of the bond => the yield on the bond given its price. Many textbooks present YTM as a separate concept but I think of it as another way to represent the price of the bond. So price and YTM are the same thing... just represented differently.

linkValuation of a bond

You may be wondering what's the difference between bond pricing and valuation.

Bond Price Calculator

/calculators/financial/bond-price-calculator

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linkHandwritten Notes

You can download my handwritten notes here.

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