Bond valuation is the process of determining the fair price of a bond. As with any security, the fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the price or value of a bond is determined by discounting the bond's expected cash flows to the present using the appropriate discount rate.

General relationships

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The present value relationship

The fair price of a straight bond (a bond with no embedded option) is determined by discounting the expected cash flows:

• Cash flows:
  • the periodic coupon payments C, each of which is made once every period;
  • the par or face value F, which is payable at maturity of the bond after T periods.

• Discount rate: the required (annually compounded) yield or rate of return r.
  • r is the market interest rate for new bond issues with similar risk ratings

Bond Price = $P\_0\; =\; \backslash sum\_\{t=1\}^T\backslash frac\{C\}\{(1+r)^t\}\; +\; \backslash frac\{F\}\{(1+r)^T\}.$

Because the price is the present value of the cash flows, there is an inverse relationship between price and discount rate: the higher the discount rate the lower the value of the bond (and vice versa). A bond trading below its face value is trading at a discount, a bond trading above its face value is at a premium.

Coupon yield

The coupon yield is simply the coupon payment as a percentage of the face value.

Coupon yield = C / F

Coupon yield is also called nominal yield.

Current yield

The current yield is simply the coupon payment as a percentage of the bond price.

Current yield = $C\; /\; P\_0.$

Yield to Maturity

The yield to maturity, YTM, is the discount rate which returns the market price of the bond. It is thus the internal rate of return of an investment in the bond made at the observed price. YTM can also be used to price a bond, where it is used as the required return on the bond.

Solve for YTM where

Market Price = $\backslash sum\_\{t=1\}^T\backslash frac\{C\}\{(1+YTM)^t\}\; +\; \backslash frac\{F\}\{(1+YTM)^T\}.$

To achieve a return equal to YTM, the bond owner must invest each coupon received at this rate.

Bond pricing

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Relative price approach

Here the bond will be priced relative to a benchmark, usually a government security. The discount rate used to value the bond is determined based on the bond's rating relative to a government security with similar maturity. The better the quality of the bond, the smaller the spread between its required return and the YTM of the benchmark. This required return is then used to discount the bond cash flows as above.

Arbitrage free pricing approach

In this approach, the bond price will reflect its arbitrage free price. Here, each cash flow is priced separately and is discounted at the same rate as the corresponding government issue Zero coupon bond. (Some multiple of the bond (or the security) will produce an identical cash flow to the government security (or the bond in question).) Since each bond cash flow is known with certainty, the bond price today must be equal to the sum of each of its cash flows discounted at the corresponding risk free rate - i.e. the corresponding government security. Were this not the case, arbitrage would be possible - see rational pricing.

Here the discount rate per cash flow, $r\_t$, must match that of the corresponding zero coupon bond's rate.

Bond Price = $P\_0\; =\; \backslash sum\_\{t=1\}^T\backslash frac\{C\}\{(1+r\_t)^t\}\; +\; \backslash frac\{F\}\{(1+r\_T)^T\}.$

See also

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• Bond duration
• Bond convexity

External links

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Discussion

• Calculating and comparing nominal yield, current yield, yield to maturity, and yield to call for bonds, thisMatter.com
• Bond Price Volatility Investment Analysts Society of South Africa
• Duration and convexity Investment Analysts Society of South Africa

Fixed income analysis