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Pls explain bond terminology

 
 
joeu2004
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      02-08-2012, 01:42 PM
Someone asked for help with a financial problem. Generally, I know how to
solve the problem. But some of the terminology does not jibe with my (weak)
understanding of bond terminology.

The facts of the user's problem as stated: "A corporate bond has a face
value of $1000 and an annual coupon interest rate of 6%. Interest is paid
annually. 12 years of the life of the bond remain. The current market
price of the bond is $1027, and it will mature at $1100."

What does not make sense to me is "face value of $1000" v. "will mature at
$1100".

The wiki/Face_value page [1] says: "The face value of bonds usually
represents the [...] redemption value. [....] As bonds approach maturity,
actual value approaches face value".

I thought "redemption value" is the amount to be paid at maturity (plus any
accrued interest).

But the wiki/Redemption_value[2] cites a www.12manage.com page [3], which
says: "The Redemption Value is [...] the price at which a bond [...] can be
called by the issuing company. [....] The call price before its maturity
date typically exceeds the Face Value of the bond".

That would suggest that the redemption value is __not__ the face value. (?!)

(Then again, I thought call price and redemption value are two
separately-stated terms of a bond.)

Moreover, the www.12manage.com page defining face value [4] says:
"Corporate Bonds are usually issued with $1000 face values [...]. [....]
Interest on a bond is calculated on this value; for example a $1000, 7%
corporate bond, will pay an annual interest of $70."

So for the bond terms in the user's problem above, would the YTM be the IRR
of the following annual cash flows: -1027, 60 {11 times}, 60+1100?

(I thought the interest coupon would be $66: 6% of $1100.)

Compounding my confusion is the fact that the www.12manage.com "face value"
page [2] also says: "Although the price of bonds fluctuates from the date
they are issued until redemption, they are redeemed at their Maturity date
at their Face Value, unless the issuer defaults".

For user's problem above, that would suggest that $1000, not $1100, is the
amount that the bond will "mature at", contrary to the facts of the problem
above. (?!)

Can anyone clarify the terms "face value", "redemption value" and "value at
maturity" so they are consistent with the cited definition as well as the
user's problem above?

Or is one (or both) misusing the terminology?

The bottom line is: how to compute the YTM based on the facts of the user's
problem?

I know that the YTM is the IRR of "some" cash flows. But the individual
cash flows are unclear to me because I am confused about the terminology.


-----
[1] http://en.wikipedia.org/wiki/Face_value
[2] http://en.wikipedia.org/wiki/Redemption_value
[3] http://www.12manage.com/description_...ion_value.html
[4] http://www.12manage.com/description_face_value.html

 
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      02-25-2012, 10:54 AM
There may be confusion on the wording that is used in between the two articles you cited. Bonds are usually redeemed at par value yet if there is a call price on bond then the bond issuer buys back the bond at call price.

YTM calculation is not possible with a closed form mathematical formula. The reason for not being able to calculate the YTM or YTC with a closed form formula has to do with the fact that the interest rate is trapped in the bond price equation in a way that it is not possible to solve for it

See the following two TVM equation that is used as bond price equation.

FV(1+RATE)^-NPER + PMT[1- {(1+RATE)^-NPER} ]/RATE + PV = 0

or the equivalent form as follows
PV(1+RATE)^NPER + PMT[ {(1+RATE)^NPER} -1 ]/RATE + FV = 0

Where FV is the call price or redemption price of the bond
PV is the market price of the bond
NPER is the years to maturity or years to call (number of periods)
PMT is the periodic coupon paid
RATE is the periodic YTM or periodic YTC

As you can see it is not possible to solve for RATE in the above two TVM equations. Thus one has to resort to using numerical methods such as Newton Raphson, Secant, Bisection, Muller's methods amongst others methods of finding roots of a polynomial

There exist many online calculators that find YTM and YTC such as this one

finance.thinkanddone.com/online-ytc-calculation.html

used to perform the following YTC Calculation for your stated problem

f(x) = 1100 + -1027 * (1+x)^12 + 60 [(1+x)^12 - 1]/x

f'(x) = 12 * -1027 * (1+x)^11 + 60 * (12 x (1 + x)^11 - (1 + x)^12 + 1) / (x^2)

x = 0.1
f(x) = -840.1089
f'(x) = -27449.9403
x1 = 0.1 - -840.1089/-27449.9403 = 0.0693948726727
Error Bound = 0.0693948726727 - 0.1 = 0.030605 > 0.000001

x1 = 0.0693948726727
f(x1) = -127.8602
f'(x1) = -19487.7473
x2 = 0.0693948726727 - -127.8602/-19487.7473 = 0.0628338187866
Error Bound = 0.0628338187866 - 0.0693948726727 = 0.006561 > 0.000001

x2 = 0.0628338187866
f(x2) = -4.7029
f'(x2) = -18069.5261
x3 = 0.0628338187866 - -4.7029/-18069.5261 = 0.0625735499956
Error Bound = 0.0625735499956 - 0.0628338187866 = 0.00026 > 0.000001

x3 = 0.0625735499956
f(x3) = -0.0071
f'(x3) = -18015.1492
x4 = 0.0625735499956 - -0.0071/-18015.1492 = 0.0625731570286
Error Bound = 0.0625731570286 - 0.0625735499956 = 0 < 0.000001
YTC = 6.26%
Annual YTC = 6.26%
 
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