Bond valuation

Overview

Bond valuation is the process of estimating the fair value of a bond. In the prevalent present-value approach, the value equals the sum of expected cash flows discounted at appropriate rates. In practice, under relative pricing, this discount rate is often inferred by reference to similar, more liquid instruments; arbitrage-free pricing is also common. Several related yield measures can be computed for a given price. If the market price of a bond is below par value, it trades at a discount; if it is above par, it trades at a premium. If a bond has embedded options, valuation combines option pricing with discounting. Depending on the option type, the option value is added to or subtracted from the value of the option-free bond to obtain the total price. See under Bond option § Embedded options.

Bond valuation

The fair price of a “straight” bond (no embedded options - see bond features) is the present value of its expected cash flows discounted at appropriate rates. In practice, prices are often inferred relative to more liquid instruments. Two approaches are common: relative pricing and arbitrage-free pricing. When valuation must reflect uncertainty in future rates, for example when valuing a bond option, analysts use interest-rate models.

Present value approach

A basic calculation discounts each cash flow at a single market rate for all periods. A more realistic variant discounts each cash flow at its own rate along the curve. The formula below assumes a coupon has just been paid. See clean and dirty price for other dates.

P

=

∑

n = 1

N

C

( 1 + i

)

n

+

M

( 1 + i

)

N

=

C

1 − ( 1 + i

)

− N

Relative price approach

Under this approach the bond is priced relative to a benchmark, usually a government bond yield curve. Set the bond's yield to maturity as the benchmark yield plus a credit spread appropriate to its credit rating and maturity or duration. Use this required return in the present-value formula above by replacing

i

{\displaystyle i}

with the bond's YTM.

Arbitrage-free pricing approach

Under this approach each promised cash flow is valued at its own discount rate. View the bond as a package of cash flows and discount each one at the rate implied by a matching zero-coupon of the same maturity and credit quality. Let

C

F

n

{\displaystyle CF_{n}}

be the cash flow at time

t

n

{\displaystyle t_{n}}

and

Z ( 0 ,

t

n

)

{\displaystyle Z(0,t_{n})}

the discount factor for that date. Then

P

=

∑

n = 1

Stochastic calculus approach

When pricing a bond option or other interest rate derivative, future short rates are random, so a single fixed discount rate is not enough. In this setting one uses a one-factor short-rate model and risk-neutral valuation. Under such a model, the price

P ( t , r ; T )

{\displaystyle P(t,r;T)}

of a zero-coupon bond maturing at

T

{\displaystyle T}

satisfies the risk-neutral bond PDE

∂ P

∂ t

+

a

Q

( r , t )

∂ P

∂ r

+

1 2

σ ( r , t

)

2

Clean and dirty price

When a bond is valued between coupon dates the price includes accrued interest for the time since the previous coupon date. The price including accrued interest is the dirty price (also called full price, all-in price, or cash price). The clean price excludes accrued interest. Clean prices are more stable through time than dirty prices. The dirty price rises deterministically between coupons as interest accrues, then drops by roughly the coupon amount when the coupon is paid.

P

dirty

=

P

clean

+

AI

.

{\displaystyle P_{\text{dirty}}\;=\;P_{\text{clean}}\;+\;{\text{AI}}.}

Here

AI

{\displaystyle {\text{AI}}}

is accrued interest for the current coupon period. Under the market day count convention, a common calculation is

AI

=

C × α ,

{\displaystyle {\text{AI}}\;=\;C\times \alpha ,}

where

C

Yield and price relationships

Once the price is known, several yields can be calculated that relate the price to the bond's cash flows.

Yield to maturity

The yield to maturity (YTM) is the discount rate that equates the present value of all promised cash flows to the observed market price for an option-free bond. It is the internal rate of return on the cash flows if they are received as scheduled and reinvested at the YTM. Because YTM can be used in pricing, bonds are often quoted by their YTM. To realise a return equal to the quoted YTM the investor would need to:

buy the bond at the quoted price

P

{\displaystyle P}

, receive all coupons and principal as scheduled with no default, and reinvest each coupon at the YTM until maturity.

Coupon rate

The coupon rate is the stated annual coupon as a percentage of the face value

F

{\displaystyle F}

. If coupons are paid

m

{\displaystyle m}

times per year and the per-period coupon is

C

{\displaystyle C}

, then the annual coupon is

m C

{\displaystyle mC}

and

Coupon rate

=

m C

F

.

{\displaystyle {\text{Coupon rate}}\;=\;{\frac {mC}{F}}.}

The coupon rate is sometimes called the nominal coupon.

Current yield

The current yield is the annual coupon divided by the (clean) price

P

{\displaystyle P}

at the valuation date:

Current yield

=

m C

P

.

{\displaystyle {\text{Current yield}}\;=\;{\frac {mC}{P}}.}

Relationship

The concept of current yield is closely related to other bond concepts, including yield to maturity, and coupon yield. The relationship between yield to maturity and the coupon rate is as follows:

As price falls below par the yield to maturity rises, and it rises more than the current yield because it also reflects the capital gain realised at redemption.