Bonds III  and bond yields: a continuation of Part II

If we keep a bond to maturity, and the Maturity Value at that time is $B, then the current bond price, V, is related to c, the coupon rate per coupon period (which may be a year or six months or whatever), and y, the current yield (per period), and n, the number of periods to maturity, according to:

(1)     V = B/(1+y)n + cB/y (1 - (1+y)-n)
where B/(1+y)n = present value of the bond price after n periods = the present value of the Maturity Value, and
cB is the dollar value of each coupon - expressed as a fraction of the Maturity Value of the bond, and
cB/y(1 - (1+y)-n) is the present value of all n coupons, and
n is the number of periods until maturity (when the bond is worth $B).

>And if I sell the bond before maturity ... and it's worth just a buck or two? What is ...?
Okay, we're going to sell the bond after n periods and, at that time, it's value is some fraction of the Maturity Value. Let's say it's worth fB where, for example, f = 0.75 means I sell the bond for 75% of its Maturity Value. The formula above changes only in the first term:

(2)     V = fB/(1+y)n + cB/y (1 - (1+y)-n)
where fB/(1+y)n = present value of the bond price after n periods = the present value of the sale price fB, and
cB is the dollar value of each coupon - expressed as a fraction of the Maturity Value of the bond, and
cB/y(1 - (1+y)-n) is the present value of all n coupons, and
n is now the number of periods until the bond is sold (at the price fB).

Our problem is to determine the return on our bond if we sell it early and get just $fB after n periods.
That means we must find y, from equation (2). To do this we first rearrange Equation (2) so it reads:

(3)     F(y) = (V/B - c/y)(1+y)n + c/y -f = 0
If we buy the bond (with maturity value B and coupon rate c) for a price V and sell it at a price fB, n periods later, then our goal is to determine the yield y so that ...
>I assume you've got a spreadsheet to do this.
Well, yes. It's the very same spreadsheet we met in Part II. A piece of it looks like this:

If you haven't already downloaded the spreadsheet, you can do it now by Right-clicking on this picture and downloading the .ZIPd file (not the picture!).

>The spreadsheet assumes that the first coupon is after one year, right?
Right, but if the first coupon occurs immediately, you can enter a Purchase Price which is smaller by the amount of that first coupon. For example, in the above situation, if the Purchase price is 1271.90 - 60.00 = 1211.90, then the yield would change to 4.39% so that ...

>So the yield would lie between 4.00% and 4.39%, right?
Very good.
>But yields change, from week to week. What if ...?

There's a Monte Carlo spreadsheet (see: Monte Carlo) where you can assume a Mean Return and a Standard Deviation for your bond, and annual re-investments at some Coupon Rate and get an estimate of where your bond portfolio is going over the next umpteen years.

For example, consider a bond whose Coupon Rate is 8.0% (hence an 8% annual re-investment) and an expected Yield of 4.0% but with a Standard Deviation of 6.0% then a possible future evolution of your bond might look like one of these:    

for Part IV