There is minor typo in my message changing the sense: "As a result of this activity asking about the "rate of return" can be answered." Here it should be from the context:

As a result of this activity asking about the "rate of return" can be unanswered.

Best Regards,

Valerii

The definition of "bond yield" Didier refers to is actually the "Current Yield". Generally when people refer to "bond yield" they mean "yield to maturity" and WH's definition is correct when he refers to this being the IRR of the bond's cash flows. The yield in this context is the interest rate which, when used to present-value the remaining cash flows of a bond, makes the sum of those cash flows equal to the current (dirty) price of the bond.

However, the term "yield to maturity" is a misnomer because it suggests that an investor who buys a bond to yield, say, 4%, and holds it to maturity will actually earn 4%. In fact, the investor will only earn the YTM if: a) he/she holds the bond to maturity, and b) he/she can invest all the coupons received at a rate equal to the bond's YTM. This is most unlikely in practice.

That's where the "rate of return" or "total return" comes in. This is the annualized rate of return of the bond's cash flows including the actual interest earned on reinvested coupons. You will only know this when the bond is evetually sold or matures, although it can be calculated in advance if you make assumptions about future reinvestment rates.

There's a lot more and many books on bond math have been written, notably those by Donald Smith "Bond Math: The Theory Behind the Formulas" and several by Frank Fabozzi.

Short Answer:

Yield is only one component of the Bond's return.

Longer Answer:

Foremost, remember that Bond is a debt. When you buy a bond from the issuer, you are giving her/him a debt and have an assurance from him/her that you will receive the principle and interest in a timely manner.

Bond's price at any point of time is the price the market is willing to pay for this promise of future stream of payments by the bond issuer.

Bond's Price fluctuates due to two reasons:

1) The remaining part of the future stream of payments. That's because, every time you redeem a coupon, the remaining stream of payments is reduced by the amount on the coupon, and hence the market should pay you less by that very amount.

2) Depending on the demand and supply of money in the market and the credibility of the bond issuer, the market may not be willing to pay the same price for a similar debt as it did in the past.

THEREFORE, a bond's return is composed of these two components:

1) The Coupon Payments which is characterized by YIELD.

2) Change in the cost of raising capital, which is characterized by DURATION and CONVEXITY.

In the language of Calculus:

The bond return can written as a Taylor series expansion in two variables: time and market interest rates [with respect to the bond issuer in question].

Yield Term is that term of the Taylor Series which has the first order partial derivative with respect to time.

Duration Term is that term in the Taylor Series which has first order partial derivative with respect to interest rate.

Convexity Term is that term in the Taylor Series which has second order partial derivative with respect to interest rate.

Other higher order terms are ignored in the expansion.

Steven Pikowski: "Yield is The income return on an investment."

Valerii Salov: "the "yield to maturity" Y is the value making the following equation valid: P = C/(1 + Y) + C/(1 + Y)^2 + ... + C/(1 + Y)^N + M/(1 + Y)^N"

Lawrence Galitz: "...the investor will only earn the YTM if: a) he/she holds the bond to maturity, and b) he/she can invest all the coupons received at a rate equal to the bond's YTM."

Let us see how these three statements work together. Due to Steven we can say that for our bond the income return R on investment P is equal to

(1) R = (M + N * C - P) / P = (M + N * C) / P - 1

Here M is the face value paid at maturity, C is a coupon, N is the number of coupons and periods, and P is the bond price - our investment. Total money IN = M + N * C minus total money OUT = P we divide by total money OUT. This implies that C are obtained but not reinvested.

If we shall follow to Lawrence and reinvest all C at Y from Valerii's formula, then our income can only increase and our rate of return R' for the same investment P will be only greater than R. For the first coupon C we shall get now C * (1 + Y)^(N - 1). The power is not N because the number of periods left for reinvestment is one less than from the beginning. Accordingly, new R' > R is given by

(2) R' = (M + C * (1 + Y)^(N - 1) + C * (1 + Y)^(N - 2) + ... + C) / P - 1 > R

Let us make all fractions in the formula for P having one denominator (1 + Y)^N. This will give us one fraction

(3) P = (C * (1 + Y)^(N - 1) + C * (1 + Y)^(N - 2) + ... + C + M) / (1 + Y)^N

Now division by P in (2) is getting trivial and yields

(4) R' = (1 + Y)^N - 1 = (P * (1 + Y)^N - P) / P

This is nothing else but the return on P at the rate Y for N periods.

Valerii's (obviously not mine) formula can be written in a compact form. Recollect the sum of the first N terms of a geometric progression

SUM[j=0;j=N -1]{q^j} = q^0 + q^1 + q^2 + ... q^(N - 1)

We have forgotten the formula for the left expression and the right one is good for a computer loop and less suitable for a human being. If both sides are equal, then we can multiply them by non-zero (1 - q) without breaking validity of the statement. Would the latter be zero, then after multiplication both sides would become equal each to other, even, if they were not equal from the beginning. Therefore, we assume that q is not equal to 1 and get

(1 - q) * SUM[j=0;j=N -1]{q^j} = [q^0 + q^1 + q^2 + ... q^(N - 1)] * (1 - q) = q^0 + q^1 + q^2 + ... q^(N - 1) - (q^1 + q^2 + ... q^N) = q^0 - q^N and finally recover the well known formula for the sum dividing both by non-zero (1 - q)

SUM[j=0;j=N -1]{q^j} = (q^0 - q^N) / (1 - q) = (1 - q^N) / (1 - q), q is not equal to 1.

For q = 1 the sum is equal to N. We can rewrite the formula for P (I have taken common multiplier C out and added and subtracted -1 + 1 in brackets)

P = C * (-1 + 1 + 1/(1 + Y) + 1/(1 + Y)^2 + ... + 1/(1 + Y)^N) + M/(1 + Y)^N =

After -1 we get the sum for q = 1/(1 + Y) for N + 1 terms ( because the power is not N - 1 but N).

= C * (-1 + (1 - 1/(1 + Y)^(N + 1) * (1 + Y)) / (1 + Y - 1)) + M/(1 + Y)^N = C * (1 - 1 / (1 + Y)^N) / Y + M/(1 + Y)^N

(5) P = C * (1 - 1 / (1 + Y)^N) / Y + M / (1 + Y)^N

The first term is also known as the "present value of annuity" for N coupons, where the rate for all periods remain equal to Y, "yield to maturity" or "internal rate of return" (see messages of WH Chan and mine earlier for the terms). Every time when one say "annuity" think that a discussion can turn to a sum of the first N members of a geometric progression. The formula for the sum is valid for any N and q != 1 and in addition, it has the limit 1 / (1 - q) for N -> infinity, if |q| < 1.

Best Regards,

Valerii

Thanks All

Its very help full ...