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Time Value of Money in Relation to Bond Valuation.

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Meaning-

• The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time.

• The method also allows the valuation of a likely stream of income in the future, in such a way that the annual incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream.

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Factors

• Present value of an annuity An annuity is a series of equal payments or receipts

that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due. 

• Present value of a perpetuity It is an infinite and constant stream of identical cash

flows.

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• Future value

It is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. 

• Future value of an annuity

(FVA) is the future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.

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Bond Pricing

• It is important for prospective bond buyers to know how to determine the price of a bond because it will indicate theyield received should the bond be purchased. In this section, we will run through some bond price calculations for various types of bond instruments.

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•

C = coupon payment 

n = number of payments 

I = interest rate, or required yield 

M = value at maturity, or par value 

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Bond valuation Process

• Bond valuation is the process of determining the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is determined by discounting the bond's expected cash flows to the present using the appropriate discount rate. Determining this rate in practice - i.e. "pricing" the bond - is done with reference to other instruments. Once the price or value has been calculated, the sensitivity of the price can then be estimated; the various yields, which relate the price of the bond to its coupons, can also be determined.

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Present value relationship

• 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. (note: final year payments will include the par value plus the coupon payments for the year). In some of the bonds, their Maturity Redemption Price might be more than par value, in this case the F is actually the Redemption Price.

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• 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 =

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Clean and dirty price

• When the bond is not valued precisely on a coupon date, the present value relationship as above, will incorporate accrued interest: i.e. any interest due to the owner of the bond since the previous coupon date; see day count convention. The price of a bond which includes this accrued interest is known as the "dirty price"; the "clean price" is the price excluding any interest that has accrued. The value returned by the above formula is thus the dirty price.

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Bond Pricing

• Relative price approach Under this approach, the bond will be priced relative to a benchmark,

usually a government security; see Relative valuation. Here, the yield to maturity on the bond is determined based on the bond's Credit rating relative to a government security with similar maturity or duration; see Credit spread (bond).

• Arbitrage-free pricing approach Under this approach, the bond price will reflect its arbitrage-free price.

Here, each cash flow (coupon or face) is separately discounted at the same rate as a zero-coupon bond corresponding to the coupon date, and of equivalent credit worthiness (if possible, from the same issuer as the bond being valued).

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• Bond Issuers 

• Priority 

• Coupon Rate 

• Redemption Features 

• Unlimited Types of Bonds 

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Yield and price relationships

• Yield to Maturity

The yield to maturity (YTM) is the discount rate which returns the market price of the bond; it is identical to r (required return) in the above equation. YTM is thus the internal rate of return of an investment in the bond made at the observed price. Since YTM can be used to price a bond, bond prices are often quoted in terms of YTM.

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• 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

(current) bond price (P).

Current yield =

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Yield Curve

• Normal Yield Curve

• Flat Yield Curve: 

• Inverted Yield Curve

• The Theoretical Spot Rate Curve 

• The Credit Spread