# Bond Pricing

How bonds are priced

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## What is Bond Pricing?

Bond pricing is an empirical matter in the field of financial instruments. The price of a bond depends on several characteristics inherent in every bond issued. These characteristics are:

• Coupon, or lack thereof
• Principal/par value
• Yield to maturity
• Periods to maturity

Alternatively, if the bond price and all but one of the characteristics are known, the last missing characteristic can be solved for.

### Bond Pricing: Coupons

A bond may or may not come with attached coupons. A coupon is stated as a nominal percentage of the par value (principal amount) of the bond. Each coupon is redeemable per period for that percentage. For example, a 10% coupon on a \$1000 par bond is redeemable each period.

A bond may also come with no coupon. In this case, the bond is known as a zero-coupon bond. Zero-coupon bonds are typically priced lower than bonds with coupons.

### Bond Pricing: Principal/Par Value

Each bond must come with a par value that is repaid at maturity. Without the principal value, a bond would have no use. The principal value is to be repaid to the lender (the bond purchaser) by the borrower (the bond issuer). A zero-coupon bond pays no coupons but will guarantee the principal at maturity. Purchasers of zero-coupon bonds earn interest by the bond being sold at a discount to its par value.

A coupon-bearing bond pays coupons each period, and a coupon plus principal at maturity. The price of a bond comprises all these payments discounted at the yield to maturity.

### Bond Pricing: Yield to Maturity

Bonds are priced to yield a certain return to investors. A bond that sells at a premium (where price is above par value) will have a yield to maturity that is lower than the coupon rate. Alternatively, the causality of the relationship between yield to maturity and price may be reversed. A bond could be sold at a higher price if the intended yield (market interest rate) is lower than the coupon rate. This is because the bondholder will receive coupon payments that are higher than the market interest rate, and will, therefore, pay a premium for the difference.

### Bond Pricing: Periods to Maturity

Bonds will have a number of periods to maturity. These are typically annual periods, but may also be semi-annual or quarterly. The number of periods will equal the number of coupon payments.

### The Time Value of Money

Bonds are priced based on the time value of money. Each payment is discounted to the current time based on the yield to maturity (market interest rate). The price of a bond is usually found by:

##### P(T0) = [PMT(T1) / (1 + r)^1] + [PMT(T2) / (1 + r)^2] … [(PMT(Tn) + FV) / (1 + r)^n]

Where:

• P(T0) = Price at Time 0
• PMT(Tn) = Coupon Payment at Time N
• FV = Future Value, Par Value, Principal Value
• R = Yield to Maturity, Market Interest Rates
• N = Number of Periods

### Bond Pricing: Main Characteristics

Ceteris paribus, all else held equal:

• A bond with a higher coupon rate will be priced higher
• A bond with a higher par value will be priced higher
• A bond with a higher number of periods to maturity will be priced higher
• A bond with a higher yield to maturity or market rates will be priced lower

An easier way to remember this is that bonds will be priced higher for all characteristics, except for yield to maturity. A higher yield to maturity results in lower bond pricing.

### Bond Pricing: Other “Soft” Characteristics

The empirical characteristics outlined above affect bond issues, especially in the primary market. There are other, however, bond characteristics that can affect bond pricing, especially in the secondary markets. These are:

• Creditworthiness of issuing firm
• Time to next payment

### Firm Creditworthiness

Bonds are rated based on the creditworthiness of the issuing firm. These ratings range from AAA to D. Bonds rated higher than A are typically known as investment-grade bonds, whereas anything lower is colloquially known as junk bonds.

Junk bonds will require a higher yield to maturity to compensate for their higher credit risk. Because of this, junk bonds trade at a lower price than investment-grade bonds.

### Bond Liquidity

Bonds that are more widely traded will be more valuable than bonds that are sparsely traded. Intuitively, an investor will be wary of purchasing a bond that would be harder to sell afterward. This drives prices of illiquid bonds down.

### Time To Payment

Finally, time to the next coupon payment affects the “actual” price of a bond. This is a more complex bond pricing theory, known as ‘dirty’ pricing. Dirty pricing takes into account the interest that accrues between coupon payments. As the payments get closer, a bondholder has to wait less time before receiving his next payment. This drives prices steadily higher before it drops again right after coupon payment.