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# Sinking fund,bond valuation,stock valuation

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### Sinking fund,bond valuation,stock valuation

3. 3. • Public market • Exchange • Securities Bond market • Bond valuation • Corporate bond • Fixed income • Government bond • High-yield debt • Municipal bond Stock market • Common stock • Preferred stock • Registered share • Stock • Stock certificate • Stock exchange • Voting shareDerivatives market • Credit derivative • Futures exchange • Hybrid security • SecuritizationOver-the-counter • Forwards • Options • Spot market • SwapsForeign exchange
4. 4. • Currency • Exchange rate Other markets • Commodity market • Money market • Reinsurance market • Real estate market Practical trading • Clearing house • Financial market participants • Financial regulation Finance series • Banks and banking • Corporate finance • Personal finance • Public finance • v • t • eBond valuation is the determination of the fair price of a bond. As with any security orcapital investment, the theoretical fair value of a bond is the present value of the streamof cash flows it is expected to generate. Hence, the value of a bond is obtained bydiscounting the bonds expected cash flows to the present using an appropriate discountrate. In practice, this discount rate is often determined by reference to similar instruments,provided that such instruments exist. Various related yield-measures are calculated forthe given price.If the bond includes embedded options, the valuation is more difficult and combinesoption pricing with discounting. Depending on the type of option, the option price ascalculated is either added to or subtracted from the price of the "straight" portion. Seefurther under Bond option. This total is then the value of the bond.Contents
5. 5. • 1 Bond valuation o 1.1 Present value approach o 1.2 Relative price approach o 1.3 Arbitrage-free pricing approach o 1.4 Stochastic calculus approach • 2 Clean and dirty price • 3 Yield and price relationships o 3.1 Yield to Maturity o 3.2 Coupon yield o 3.3 Current yield o 3.4 Relationship • 4 Price sensitivity • 5 Accounting treatment • 6 See also • 7 References and external linksBond valuation[1] As above, the fair price of a "straight bond" (a bond with no embedded options; seeBond (finance)# Features) is usually determined by discounting its expected cash flows atthe appropriate discount rate. The formula commonly applied is discussed initially.Although this present value relationship reflects the theoretical approach to determiningthe value of a bond, in practice its price is (usually) determined with reference to other,more liquid instruments. The two main approaches, Relative pricing and Arbitrage-freepricing, are discussed next. Finally, where it is important to recognise that future interestrates are uncertain and that the discount rate is not adequately represented by a singlefixed number - for example when an option is written on the bond in question - stochasticcalculus may be employed.Where the market price of bond is less than its face value (par value), the bond is sellingat a discount. Conversely, if the market price of bond is greater than its face value, thebond is selling at a premium.[2] For this and other relationships relating price and yield,see below.Present value approachBelow is the formula for calculating a bonds price, which uses the basic present value(PV) formula for a given discount rate:[3] (This formula assumes that a coupon paymenthas just been made; see below for adjustments on other dates.)
6. 6. where: F = face value iF = contractual interest rate C = F * iF = coupon payment (periodic interest payment) N = number of payments i = market interest rate, or required yield, or observed / appropriate yield to maturity (see below) M = value at maturity, usually equals face value P = market price of bond.Relative price approachUnder this approach - an extension of the above - the bond will be priced relative to abenchmark, usually a government security; see Relative valuation. Here, the yield tomaturity on the bond is determined based on the bonds Credit rating relative to agovernment security with similar maturity or duration; see Credit spread (bond). Thebetter the quality of the bond, the smaller the spread between its required return and theYTM of the benchmark. This required return is then used to discount the bond cashflows, replacing in the formula above, to obtain the price.Arbitrage-free pricing approach See: Rational pricing: Fixed income securities.As distinct from the two related approaches above, a bond may be thought of as a"package of cash flows" - coupon or face - with each cash flow viewed as a zero-couponinstrument maturing on the date it will be received. Thus, rather than using a singlediscount rate, one should use multiple discount rates, discounting each cash flow at itsown rate.[1] Here, each cash flow is separately discounted at the same rate as a zero-coupon bond corresponding to the coupon date, and of equivalent credit worthiness (ifpossible, from the same issuer as the bond being valued, or if not, with the appropriatecredit spread).Under this approach, the bond price should reflect its "arbitrage-free" price, as anydeviation from this price will be exploited and the bond will then quickly reprice to itscorrect level. Here, we apply the rational pricing logic relating to "Assets with identicalcash flows". In detail: (1) the bonds coupon dates and coupon amounts are known with
7. 7. certainty. Therefore (2) some multiple (or fraction) of zero-coupon bonds, eachcorresponding to the bonds coupon dates, can be specified so as to produce identical cashflows to the bond. Thus (3) the bond price today must be equal to the sum of each of itscash flows discounted at the discount rate implied by the value of the correspondingZCB. Were this not the case, (4) the abitrageur could finance his purchase of whicheverof the bond or the sum of the various ZCBs was cheaper, by short selling the other, andmeeting his cash flow commitments using the coupons or maturing zeroes as appropriate.Then (5) his "risk free", arbitrage profit would be the difference between the two values.Stochastic calculus approachWhen modelling a bond option, or other interest rate derivative (IRD), it is important torecognize that future interest rates are uncertain, and therefore, the discount rate(s)referred to above, under all three cases - i.e. whether for all coupons or for eachindividual coupon - is not adequately represented by a fixed (deterministic) number. Insuch cases, stochastic calculus is employed.The following is a partial differential equation (PDE) in stochastic calculus which issatisfied by any zero-coupon bond.The solution to the PDE - given in [4] - is: where is the expectation with respect to risk-neutral probabilities, and is a random variable representing the discount rate; see also Martingale pricing.To actually determine the bond price, the analyst must choose the specific short ratemodel to be employed. The approaches commonly used are: • the CIR model • the Black-Derman-Toy model • the Hull-White model • the HJM framework • the Chen model.Note that depending on the model selected, a closed-form solution may not be available,and a lattice- or simulation-based implementation of the model in question is thenemployed. See also Jamshidians trick.
8. 8. Clean and dirty priceMain articles: Clean price and Dirty priceWhen the bond is not valued precisely on a coupon date, the calculated price, using themethods above, will incorporate accrued interest: i.e. any interest due to the owner of thebond since the previous coupon date; see day count convention. The price of a bondwhich includes this accrued interest is known as the "dirty price" (or "full price" or "all inprice" or "Cash price"). The "clean price" is the price excluding any interest that hasaccrued. Clean prices are generally more stable over time than dirty prices. This isbecause the dirty price will drop suddenly when the bond goes "ex interest" and thepurchaser is no longer entitled to receive the next coupon payment. In many markets, it ismarket practice to quote bonds on a clean-price basis. When a purchase is settled, theaccrued interest is added to the quoted clean price to arrive at the actual amount to bepaid.Yield and price relationshipsOnce the price or value has been calculated, various yields relating the price of the bondto its coupons can then be determined.Yield to MaturityThe yield to maturity is the discount rate which returns the market price of the bond; it isidentical to (required return) in the above equation. YTM is thus the internal rate ofreturn of an investment in the bond made at the observed price. Since YTM can be usedto price a bond, bond prices are often quoted in terms of YTM.To achieve a return equal to YTM, i.e. where it is the required return on the bond, thebond owner must: • buy the bond at price P0, • hold the bond until maturity, and • redeem the bond at par.Coupon yieldThe coupon yield is simply the coupon payment (C) as a percentage of the face value (F). Coupon yield = C / FCoupon yield is also called nominal yield.Current yield
9. 9. The current yield is simply the coupon payment (C) as a percentage of the (current) bondprice (P). Current yield =RelationshipThe concept of current yield is closely related to other bond concepts, including yield tomaturity, and coupon yield. The relationship between yield to maturity and the couponrate 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 yieldPrice sensitivityMain articles: Bond duration and Bond convexityThe sensitivity of a bonds market price to interest rate (i.e. yield) movements ismeasured by its duration, and, additionally, by its convexity.Duration is a linear measure of how the price of a bond changes in response to interestrate changes. It is approximately equal to the percentage change in price for a givenchange in yield, and may be thought of as the elasticity of the bonds price with respect todiscount rates. For example, for small interest rate changes, the duration is theapproximate percentage by which the value of the bond will fall for a 1% per annumincrease in market interest rate. So the market price of a 17-year bond with a duration of7 would fall about 7% if the market interest rate (or more precisely the correspondingforce of interest) increased by 1% per annum.Convexity is a measure of the "curvature" of price changes. It is needed because the priceis not a linear function of the discount rate, but rather a convex function of the discountrate. Specifically, duration can be formulated as the first derivative of the price withrespect to the interest rate, and convexity as the second derivative (see: Bond durationclosed-form formula; Bond convexity closed-form formula). Continuing the aboveexample, for a more accurate estimate of sensitivity, the convexity score would bemultiplied by the square of the change in interest rate, and the result added to the valuederived by the above linear formula.Accounting treatmentIn accounting for liabilities, any bond discount or premium must be amortized over thelife of bond. A number of methods may be used for this depending on applicable
10. 10. accounting rules. One possibility is that amortization amount in each period is calculatedfrom the following formula: = amortization amount in period number "n+1"Bond Discount or Bond Premium = =Bond Discount or Bond Premium = The 4 Primary Types of BondsThere are millions of different bond issues, but there are only a few types of bonds. Infact, the large majority of bonds fall into one of the 4 categories outlined below. As youcan see from the links in each section, we have lots of information on all the differenttypes of bonds here at Learn Bonds, so enjoy!1. US Government Bonds (Treasuries)When people talk about the US debit being over \$16 trillion, what they are really sayingis the US Government has over \$16 trillion worth of outstanding debt. Much of this