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

[Snoopy]

I have found a web reference here that might be useful
https://www.wallstreetmojo.com/yield...y-ytm-formula/

Yield to Maturity Formula

YTM considers the effective yield of the bond, which is based on compounding. The below formula focuses on calculating the approximate yield to maturity, whereas calculating the actual YTM will require trial and error by considering different rates in the current value of the bond until the price matches the actual market price of the bond. Nowadays, computer applications facilitate the easy calculation YTM of the bond.

Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the years to maturity.

I think there is a problem with the above formula. It looks 'dimensionally inconsistent' to me. Specifically, what I mean here is that the units measured on each side of any equals sign must be the same. The 'yield to maturity' answer we seek, on the left side of the equation, is a 'percentage figure' which is a dimensionless term.

By contrast on the right hand side of the equation we have 'C', a 'coupon rate' which is also a percentage figure and a dimensionless term (so far so good). But '[F-P]' is measured in 'dollars' and n is measured in 'years'. So the result of (F-P)/n is measured in units of 'dollars per year'. Now I guess you could argue that if you put some money in an interest bearing account at a bank (say $100) at an interest rate of say 5%, then you would earn 5 'dollars per year'. So at a stretch you could say that 'C' and '(F-P)/N' are 'kind of compatible'.

Yet if you accept my above (doubtful) conclusion, you then have to divide a 'percentage figure' by an amount in dollars ( [F+P]/2 ).

So you end up with a percentage number on the LHS of the equation equalling a percentage number divided by a dollar amount on the RHS of the equation. This doesn't make sense. What am I missing?

SNOOPY

[Snow Leopard]

...What am I missing?

SNOOPY

Snoopy.

C is the Coupon (an amount of dollars) not the Coupon Rate ( a fraction or percentage or...)

See example 1 on the page you reference.

The formula is also not precise but approximate and you should refer to this masterful post

[Snoopy]

Snoopy.

C is the Coupon (an amount of dollars) not the Coupon Rate ( a fraction or percentage or...)

See example 1 on the page you reference.

Thanks

The formula is also not precise but approximate and you should refer to this masterful post

An iterative solution is the way to go when you have an equation that cannot be solved algebraically, but has the variable you wish to find on both the RHS and LHS of the equation. But that doesn't seem to apply here. Because the variable you want to find is on the LHS of the equation and all the other variables are known and on the RHS , as far as I can tell.

SNOOPY

[Snow Leopard]

Iterative methods and a spreadsheet

PM me an email address and I will send this Excel Spreadsheet:

Xcel-1.png

Xcel-2.png

Xcel-3.png

Molehill >>> Mountain.

[Snoopy]

PM me an email address and I will send this Excel Spreadsheet:

Thanks for the offer Snow Leopard, but those 3 screen shots you gave allowed me to create the spreadsheet myself. Breaking down the spreadsheet into an underlying form, what we have here is a series of four sequential equations covering the four sequential investment time periods. The job of the spreadsheet is to select an unknown 'yield to maturity' figure to make the whole sequence of equations sum to zero. The four sequential equations from the spreadsheet being:

$989,683.38 (1 + Y/2) - $1,070,000 = 0
$980,141.15 (1 + Y/2) - $70,000 = $989,683.38
$971,315.18 (1 + Y/2) - $70,000 = $980,141.15
$1,012,500.00 (1 + 64Y/365) -$70,000 = $971,315.18

It strikes me that those intermediate principal figures always connect one investment period with a subsequent one. Their only purpose is a connective function . That means you don't need to know the exact figures they are. And the above equations can equally well be represented as below:

$A (1 + Y/2) - $1,070k = 0
$B (1 + Y/2) - $70k = $A
$C (1 + Y/2) - $70k = $B
$1,012.5k (1 + 64Y/365) -$70k = $C

Looking at the $A amount in the first equation, we can eliminate $A by substituting from the second equation.
($B (1 + Y/2) - $70k) (1 + Y/2) - $1,070k = 0

Likewise we can eliminate $B by using the third equation
(($C (1 + Y/2) - $70k) (1 + Y/2) - $70k) (1 + Y/2) - $1,070k = 0

Lastly we can eliminate $C by using the fourth equation
[{($1,012.5k (1 + 64Y/365) -$70k) (1 + Y/2) - $70k} (1 + Y/2) - $70k] (1 + Y/2) - $1,070k = 0

The above equation looks complicated. But taking out the dollar values in the calculation and looking at the above equation representing 5 'sequential events' (numbered sequentially 1,2,3,4,5, the fifth being the final repayment of the bond), the equation takes on the simpler form below.
[{(1)2}3](4) - (5) = 0

That might make it easier to see what is going on for some.


Observations

i/ The above 'combined equation' represents four periods of investment 'sequentially cascaded together' over four distinct time periods.
ii/ The return on the first period (innermost bracket (1) ) gets multiplied by the returns in the subsequent period (2), and so on.
iii/ The negative numbers in the above equation represent the money being 'pulled out' by our investor. Because this money is removed, it does not get 'multiplied up' to produce downstream earnings in subsequent periods, which is all as you might expect.
iv/ The 'total of the money put in' is less than the 'total of the money pulled out'. We can explain this by considering the original investment ($1,012.5k) as being multiplied by a multiplier containing a 'constant yield factor Y' (scaled when any constituent time period is shortened of course), with the retained earnings of such an investment subsequently 'multiplied forwards'. This yield factor Y is unknown and must be 'solved' to complete the equation.

Personally I find the above 'combined equation', that I have highlighted in bold, better communicates what is going on in this valuation process than the spreadsheet. But I guess other peoples' mileage may vary?

SNOOPY