Financial markets v • d • e

## 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 n times (n is usually 2) every year
• the par or face value F, which is payable at maturity of the bond after T years. In Finance, a bond is a Debt security, in which the authorized issuer owes the holders a debt and is obliged to repay the principal and Interest Options are financial instruments that convey the right but not the obligation to engage in a future transaction on some Underlying security, or in a Futures Callable bond is a type of bond that allows the issuer of the bond to retain the privilege of redeeming the bond at some point before the bond reaches the date of maturity (NB final year payments will include the par value plus the coupon payments for the year)
• Discount rate: the required (annually compounded) yield or rate of return r
• r is the market interest rate for bonds with similar terms and risk ratings
• m is the number of coupons to be paid over the remaining lifetime of the bond, ie n times T. (It is assumed that the previous coupon has just been paid. )
• u is (1+r)^(1/n) ie an interest accumulation factor over one coupon period
Bond Price = $P_0 = \sum_{t=1}^m\frac{C}{u^t} + \frac{F}{(1+r)^T}.$

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). The coupon or coupon rate of a bond is the amount of interest paid per year expressed as a percentage of the face value of the bond

Coupon yield = C / F

Coupon yield is also called nominal yield. Nominal yield is the coupon rate of a Fixed income security, which is a fixed percentage of the par value

### Current yield

The current yield is simply the coupon payment (C) as a percentage of the bond price (P). The current yield, interest yield, income yield, flat yield or running yield is a financial term used in reference to bonds

Current yield = C / P0.

### Yield to Maturity

The yield to maturity (YTM) is the discount rate which returns the market price of the bond. The Yield to maturity ( YTM) or redemption yield is the yield promised to the bondholder on the assumption that the bond or other fixed-interest Market price is an economic concept with commonplace familiarity it is the price that a good or service is offered at or will fetch in the marketplace it is of interest mainly in the It is thus the internal rate of return of an investment in the bond made at the observed price. The internal rate of return (IRR is a Capital budgeting metric used by firms to decide whether they should make Investments It is an indicator of the efficiency YTM can also be used to price a bond, where it is used as the required return on the bond.

In other words, it is identical to r in the above equation.

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

• buy the bond at price P0,
• 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 yield to maturity on the bond is determined based on the bond's rating relative to a government security with similar maturity or duration. In Finance, the duration of a financial asset measures the sensitivity of the asset's price to Interest rate movements expressed as a number of years 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 to obtain the price.

### 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). In Economics and Finance, arbitrage is the practice of taking advantage of a price differential between two or more Markets striking a combination of matching Here, each cash flow is priced separately and is discounted at the same rate as the corresponding government issue Zero coupon bond. A Zero coupon bond (also called a discount bond or deep discount bond) is a bond bought at a price lower than its Face value, with the face value (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. The risk-free interest rate is the Interest rate that it is assumed can be obtained by investing in Financial instruments with no Default risk e. the corresponding government security. Were this not the case, arbitrage would be possible - see rational pricing. Rational pricing is the assumption in Financial economics that asset prices (and hence asset pricing models will reflect the Arbitrage-free price of the asset as