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# Bond valuation

Bond valuation is the process of determining the fair price of a bond. As with any security or capital investment, 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

### The present value relationship

The fair price of a straight bond (a bond with no embedded option; see Callable bond) 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.(NB final year payment will include the par value plus the coupon payment for the year)
• 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 = $P_0 = \sum_{t=1}^n\frac{C}{(1+r)^t} + \frac{F}{(1+r)^T}.$

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 (C) as a percentage of the face value (F).

Coupon yield = C / F

Coupon yield is also called nominal yield.

### Current yield

The current yield is simply the coupon payment (C) as a percentage of the bond price (P).

Current yield = C / P0.

### 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 = $\sum_{t=1}^T\frac{C}{(1+YTM)^t} + \frac{F}{(1+YTM)^T}.$

To achieve a return equal to YTM, the bond owner must:

• reinvest each coupon received at this rate,
• hold the bond until maturity, and
• redeem the bond at par.

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 coupon rate is as follows:

• When a bond sells at a discount, YTM > current yield > coupon yield.
• When a bond sells at a premium, coupon yield > current yield > YTM.
• When a bond sells at par, YTM = current yield = coupon yield.

## Bond pricing

### Relative price approach

Here the bond will be priced relative to a benchmark, usually a government security. The discount rate value of the bond is determined based on the bond's rating relative to a government security with similar maturity or duration. 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 (arbitrage=practice of taking advantage of a state of imbalance between two or more markets). 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, rt, must match that of the corresponding zero coupon bond's rate.

Bond Price = $P_0 = \sum_{t=1}^T\frac{C}{(1+r_t)^t} + \frac{F}{(1+r_T)^T}.$