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How to Price BOnds: a Mathematical Approach

RomWharf1

New member
Mar
3
3
Bonds are a useful tool for companies to raise money or refinance some existing debt. A bond has three main components:

  • Coupon: Fixed payments the corporation will pay at the agreed periods (quarterly, semi-annual, annual payments or even PIK)
  • Face value: Also called the Principal, this is the gross amount of money the company will have to repay at maturity
  • Maturity: Date when the bond will have to be repaid
The coupon is a crucial element in bond pricing. When issuing a bond, a company will have to compare with other similar companies that have issued bonds. This is required to know at what coupon level the company would be able to attract investors, without under or over pricing it. Having a rating from a major rating agency also helps significantly: It will be easier for companies to find comparable corporations that recently issued a bond, and see at what level the coupon priced at, and for what discount – they would just need to compare ratings and operations. Having a rating is also often a requirement for many investors - most mainstream asset managers will not invest in a bond that has not at least two ratings from the main rating agencies (S&P, Moody’s or Fitch). Usually, bonds issued by corporations will have a higher coupon than government bonds given the higher risk of default. This will of course depend on the company and the country.

Another important element when issuing the bond is the discount at issuance. The price of a bond is between 0 and 100: this represents the % of the Face Value. Par means the price is 100% of the bond’s principal value. The issuing company will then receive 100% of the amount raised. For example, a company issuing EUR 200 million in bonds will effectively receive EUR 200 million. However, if the bond is, for example, issued at 95, the company will only receive 95% of 200 million = EUR 190 million. The investor will however have a bond which value is 100%: when the company repays the bond, the company will have to repay EUR 200 million even if they only received EUR 190 million at issuance.

Further to the coupon, investors will look at the yield of the bond. Yield is different from the coupon because it takes into account the current trading price of the bond. Indeed, the price might be higher or lower than par (i.e., trading at a premium or a discount). An investor can purchase a bond on the market at 90% and hold the bond until maturity and receive 100% of the principal then. The investor will have received the coupon from the bond, but also get the benefit of being repaid an amount higher than what was invested.

Bonds can also be issued in different denominations – for example, one bond denomination can be EUR 1,000 or EUR 100,000. This will be determined by the issuing company and will impact the investor’s decision as well. Indeed, some institutional investors might not be interested in investing in a bond with EUR 1,000 denominations. Such a low denomination means that retail investors will be able to purchase and sell the bond easily and the bond will be at risk of higher volatility: a retail investor holding a small portion of the bond wanting to sell the bond quickly might be willing to sell at a discounted price. This will impact the trading price of the bond, which will impact the fund’s value of asset managers also holding that same bond. However, it might be the company’s decision to use EUR 1,000 denominations to include retail investors for a broader outreach.

Cash flows received by bond investors are made of the regular coupons, and the principal repayment at maturity. Similar to equity valuation, bond valuation will look to bring forward all the future cash flows to the date of the investment by discounting these future cash flows. The formula is:

View attachment 149

Where:

B= Bond price

CFn= Cash flow at period n

r = Discount rate (market rate)

T = Maturity date

In more details, cash flows are made of the bond coupons, and the Face Value (or principal) of the bond:

View attachment 150

Where:

Cn is the coupon in period n

F is the Face Value of the bond, received at maturity

Similar to equity investments, calculating the bond value mathematically will give you a theoretical price. Practically, investors will compare the current trading level of the bond to the theoretical value and use that information (amongst many other things) to make a decision to buy or sell. However, bond prices will also be affected by the current market environment, and how the company is performing. For example during the COVID crisis, many (if not most) bond prices went down significantly even though the cash flows had not changed, theoretically. This is because investors reacted negatively to the pessimistic COVID news, and decided to sell their positions, even at a discounted value compared to what it was initially invested at, potentially resulting in a loss for the investor.

Bond prices will also be impacted by negative news from the company itself. For example if, when approaching maturity, investors feel the company will not be in a position to repay the face value of the bond (i.e. because the company was unable to refinance the bond or does not have enough cash on balance sheet), bond holders might prefer to sell at a discount the bond now, and avoid being involved in a lengthy process further down the line because the company defaulted on its payment. Having bonds trading at a low price can also be negative for the public relations of the company, making it more difficult to maintain investors’ trust and hence to refinance later on should this be required.

Some bond investors will have different strategies. Amongst other strategies, the main ones are:

  • Trade the bonds, holding them for a short period of time;
  • Hold the bond to maturity;
  • Opportunistic investors might look for companies in trouble that are likely to default or have defaulted on their bond. This is a way for investors to get hold of the bond and ultimately arrive at a point where the company would have to give the bond holders their shares to make up for the repayment default.
 
Great summary! I just want to elaborate on one point that you mention - " discount at issuance". There is a specific type of bond called zero-coupon. The issuers of zero-coupon bonds do not have to make periodic payments and must repay the principal at the end of the period. These bonds are priced below par value when they are issued and their prices increase over time (until maturity). For example, the price of a one-year zero-coupon bond with a 10% interest rate will be 90.91$ at the time of issuance and is expected to reach 100$ after 1 year. In this case, the company issuing the bond will receive 90.91$ (not taking into account the fees charged by investment banks and others) and will be expected to repay the principal of 100$ in 1 year. It should be also mentioned that there is a slight difference in portfolio duration of investing in traditional vs zero-coupon bonds.
 
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