IRD Determination G3: Yield to Maturity Method for Bonds (worked example)

[Snoopy]

So putting the above numbers into the YTM formula we get:

0.162308 = [$140k + (F-$1,012.5)/1.6795] / [(F+$1,012.5)/2]

Now we need to guess a value of F that will make the above equation true. How about we guess a bond face value of $1,059.82?
Put that number in for 'F' and we get the YTM number we are seeking! What an incredible stroke of luck to guess the number right first time ;-) !

For those that have not figured this out, there was no luck involved at all in my guess, because this is an exercise in algebra. We are given the equation below to solve for F

Y = [C + (F-P)/n] / [(F+P)/2]

<=> [(F+P)/2]Y = [C + (F-P)/n]
<=> (F+P)Y = [2C + 2(F-P)/n]
<=> n(F+P)Y = 2nC + 2(F-P)
<=> nFY+nPY = 2nC + 2F-2P
<=> 2P+nPY = 2nC + 2F-nFY
<=> F(2-nY) = P(2+nY) - 2nC
<=> F = [P(2+nY) - 2nC]/(2-nY)

Stick in the numbers you know of the RHS of the equation and you get 'F', the 'capital value' of the bond.

SNOOPY

[Snoopy]

F = [P(2+nY) - 2nC]/(2-n)

Stick in the numbers you know of the RHS of the equation and you get 'F', the 'capital value' of the bond.

I think it is worth going through this equation again, this time using a six month investment period for reference.

C is the Coupon = $70k.
F is the Face Value of the bond (to be determined)
P is the current market price of the bond: $1,012.5k
n is the 'investment periods' to maturity. = 3,3507 sets of six months (post 21)
Y, the yield to maturity we are told is 16.2308% for the year, implying a multiplication factor of 1.162308. For six months the multiplication factor must be the square root of that figure.: 1.078104. This is equivalent to a yield to maturity of 7.8104% compounding for every six months.

F = [P(2+nY) - 2nC]/(2-nY)
= [ $1,012.5k(2 + 3.3507x0.078104) - 2 x 3.3507 x $70k ] / (2-3.3507 x 0.078104)
= [ $2,290.0k - $469.1k ] / 1.7383
= $1,047.52k

This is slightly different result to the annual figure calculation that I did before (post 24), because in this calculation the coupon is generated more frequently (every six months, not summed over a year). That in turn means you need slightly less discount on your bond purchase price to create the same coupon income stream per unit of investment time.

SNOOPY