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# Debt

## Debt Valuation

The value of a bond is the present value of the agreed cash flows on the bond, discounted at an interest rate that reflects the default risk in these cash flows. Because the cash flows on a straight bond are set when issued, the value of a bond is inversely linked to the interest rate that investors require for that bond. The interest rate on a bond depends upon both the general degree of interest rates and the default premium specific to the entity issuing the prescribed bond. The overall level of interest rates incorporates expected inflation and a level of real return, and mirrors the term structure, with bonds of different maturities carrying various interest rates. The default premiums change over time, due mainly on the strength of the economy and investors' risk tolerance.

Bonds often have unique term features built in them that have to be factored in to the value. Many of these characteristic include options - for the bondholder to convert into stock (convertible bonds), for the bond issuer to call the bond back if rates of interest go down (callable bonds), for the bond company to call the actual bond back in case interest rates go down (callable bonds), and for the bondholder to put the bond back to the issuer at a set price under specific circumstances (putable bonds). Alternative bond features, including interest rate caps and floors, possess option features. A number of these options reside with the issuer of the bond, some with the buyer of the bond; nonetheless, they all need to be priced. Option pricing models can be used to value this group of unique features, as well as price complex fixed income securities. A few special features with bonds including sinking funds, debt subordination, and the type of collateral used can affect the prices associated with bonds as well.

## Bond Prices and Interest Rates

The value of a straight bond is affected by the level of and movement in interest rates. As interest rates increase, the price of the bond will decline, and vice versa. This inverse relationship between bond prices and interest rates occur directly from the present value relationship that governs bond prices.

## The Present Value Relationship

The value of a bond is the present value of the promised cash flows on the bond, discounted at an interest rate that reflects the default risk associated with the cash flows. There are two things that differentiate bonds from equity investments. First, the promised cash flows on the bond (i.e., the coupon payments and the face value of the bond) are generally set at issue and do not change over the life of the bond. Even when they do change, as in floating rate bonds, the adjustments are typically linked to shifts in interest rates. Second, bonds customarily have a fixed life, unlike equities, since most fixed income securities establish a defined maturity date. Subsequently, the present value of a straight bond with fixed coupons and defined maturity is established entirely by shifts in the discount rate, that combine both the overall level of interest rates and the appropriate default risk of the bond being valued.

The present value of the bond, expected to mature in N time, with coupons every period can be written as:

PV of Bond = ∑ Coupon/(1+r)ᵗ + Face Value/(1+r)ᴺ

where

Coupon = Coupon expected in period t

Face value = Face value of the bond

r = Discount rate for the cash flows

where

Coupon = Coupon expected in period t

Face value = Face value of the bond

r = Discount rate for the cash flows

The discount rate needed to compute the present value of the bond will change from bond to bond, based on default risk, as well as higher rates used for riskier bonds and lower rates for safer ones.

When the bond is traded, and if a market price can be computed, the internal rate of return can be computed for the bond (i.e., the discount rate at which the present value of the coupon and the bond's face value is equal to the market price. This internal rate of return is called the yield to maturity on the bond.

There are specifics tied to both maturity and time of cash flows that can impact the value of a bond as well as its yield to maturity. First, the coupon payment on a bond can be semiannual, where the discounting must allow for semiannual cash flows. (The first coupon will be discounted back half a year, the second 12 months, the third a year and a half, and so on.) Second, once a bond has been issued, it accrues coupon interest between coupon payments, and this accrued interest must be included onto the price of the bond when valuing the bond.

## Interest Rate Risk in Bonds

Once the cash flows on a bond is determined at issue together with the present value relationship affecting bond prices, there is a clear rationale for why interest changes influence bond prices so directly. Any change in interest rates, either at the economy-wide level or because of an increase in the default risk of the firm issuing the bond, will reduce the present value of the flow of expected cash flows and therefore the value of the bond. A decrease in interest rates could have the opposite impact.

The impact of interest rate changes on bond prices will change from bond to bond and can depend on a number of features of the bond:

- Maturity of the bond. Holding coupon prices and default risk constant, increasing the maturity of a straight bond will increase the sensitivity to interest rate changes. The present value of cash flows is higher for cash flows later in the future, as rates of interest shift, than for cash flows that are sooner.

Long-term bonds are far more sensitive to interest rate shifts than the shorter-term bonds. For instance, an increase in interest rates from 8 percent to 10 percent results in the drop in value of 7.71 percent for the 5-year bond and 19.83 percent for any 50-year bond.

- Coupon rate of the bond. Holding maturity and default risk constant, increasing the coupon rate of a straight bond will decrease its sensitivity to interest rate changes. Since higher coupon result in higher cash flows earlier within the bond's life, the present value will change a great deal less as interest rates shift. At the extreme, if the bond is a zero coupon bond, the only real cash flow is the face value at maturity, and the present value is likely to shift much more as a function of changing interest rates.

Even though the maturity and the coupon price are the critical factors of how sensitive the price of a bond is to interest rate changes, a number of other causes affect this sensitivity. Any unique features that the bond has, such as convertibility and callability, make the maturity of the bond less certain and will impact the bond price sensitivity to interest rate changes. If there is any connection between the level of rates of interest and the default premium on bonds, the default risk of the bond can impact the price sensitivity.

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