Bonds IV and bond ladders:
a continuation of Part III 
Here we consider Bond Ladders where we ...
>Ladders? With rungs and ...?
Rungs, yes. Pay attention. Here's what we do:
 Today is January 1 and we buy three bonds.
 The first matures 1 year from today, the second in 2 years, the third in 3 years.
 The maturity dates are the rungs of the ladder.
 At the first maturity date (one year from now) a bond matures and we immediately
buy a 3year bond
(recognizing that the yield may change for this new bond).
 At the next maturity date we buy another 3year bond.
 etc. etc.
 On January 1 of each year we always have three bonds maturing in 1, 2 and 3 years
(with, usually, different yields).
Remember the formula for the Value of our bond purchases
(For simplicity, we'll assume zero coupon bonds.)
Suppose we spend equal amounts on our three bonds (meaning equal Vvalues).
Then, if the maturity values and yields are (B_{1},R_{1}),
(B_{2},R_{3}) and (B_{3},R_{3}), then we pay, for each bond:
V = B_{1}/(1+R_{1}) = B_{2}/(1+R_{2})^{2} = B_{3}/(1+R_{3})^{3}
Note that the maturity values of nyear bonds, purchased for $V, are given by B_{n} = V(1+R)^{n}.

V = B/(1+R)^{N}
where
N = number of years to maturity
B = value of Bond at maturity,
R = Annual Yield to Maturity


One year from now, we cash in the first bond for $B_{1}, reinvest in a 3year
bond at a yield of R_{4}.
the maturity value is B_{1}(1+R_{4})^{3} =
V(1+R_{1})(1+R_{4})^{3}.
The year after that, we cash in a bond for $B_{2}, reinvest in a 3year
bond at a yield of R_{5}:
the maturity value is B_{2}(1+R_{5})^{3} =
V(1+R_{2})^{2}(1+R_{4})^{3}.
The year after that, we cash in a bond, get $B_{3}, we reinvest in a 3year ...
>Can't you just make a table? I mean ...
Okay, but to make things simpler, let's define the Gain Factors
G_{1} = (1+R_{1}), G_{2} = (1+R_{2}), etc. etc.
Year  Cash Value  Maturity Value 
1  V G_{1}  V G_{1} G_{4}^{3} 
2  V G_{2}^{2}  V G_{2}^{2} G_{5}^{3} 
3  V G_{3}^{3}  V G_{3}^{3} G_{6}^{3} 
4  V G_{1} G_{4}^{3}  V G_{1} G_{4}^{3} G_{7}^{3} 
5  V G_{2}^{2} G_{5}^{3}  V G_{2}^{2} G_{5}^{3} G_{8}^{3} 
6  V G_{3}^{3} G_{6}^{3}  V G_{3}^{3} G_{6}^{3} G_{9}^{3} 
etc.  etc.  etc. 
Table 1
 Do you see how things are progressing?
>No.
The Cash Values keep getting multiplied by a (Gain Factor)^{3}
... with a possibly different yield. That becomes the Maturity Value.
That Maturity Value reappears 3 years later (when it matures)
as a Cash Value, then it gets multiplied by ...
>A (Gain Factor)^{3}?
Exactly.
>So?
So, our goal is to see how the value of our bond portfolio grows. And how our annualized
return depends upon the sequence of yields. And whether we should buy 4 or maybe 6 or 10
bonds with increasing maturities. And what kind of average yield ...
>So just do it.

Okay, suppose ...
>Wait! What of we buy bonds with coupons? Then what?
Then we change the Gain Factors from G
to something that includes the Coupon Rate and ...
>What's that?
Have you forgotten already?
We've used V = B/G with 1/G = 1/(1+R).
If we include coupons we'd use:
1/G = {1/(1+R)^{N} + (C_{r}/R) (1  (1+R/m)^{mN})}

V = B { 1/(1+R)^{N} + (C_{r}/R)
(1  (1+R/m)^{mN})}
where
N = number of years to maturity
C_{r} = annual Coupon rate
m = number of coupons per year
B = value of Bond at maturity,
R = Annual Yield to Maturity


Note that, at maturity, you'd get $B as well as mN coupons,
each worth $(C_{r}/m)B, and that makes a total of $B (1 + N C_{r})
which, of course, you could use to buy more bonds.
>You don't have any pictures.
Patience.
Notice that, except for initial transients, the gains are associated with the
longest term bonds ... because of the G^{3} factors which keep piling up, in Table 1.
Hence, although we always have bonds maturing in 1, 2 and 3 years, our gains are more and more
associated with the longest term bonds.
>Is that good?
Yes, so we may want to have a 5 or 10 rung ladder


>If, eventually, the gains are associated with long term bonds, then ...
Then why not just buy long term bonds? Yes, but our objective is to have staggered
maturities ... so we have to start with short and long term bonds. As each matures,
we buy the longer bonds.
>Yeah, but if I like 5year bonds then I'd just buy a 5year bond.
Then, when it matures, you buy another 5year?
>Why not?
What if yields are really high or really low when it comes time to buy another bond? What if ...?
>If I'm afraid of volatility in interest rates, then I'd buy 5year bonds
that mature in 1year, 2years ... uh ...
Exactly. You start by buying 1year and 2year etc., and, as each matures ...
>I buy 5years.
Exactly. Eventually you'll just be buying 5years, but with staggered maturities
... to smooth out the volatility in interest rates.
>Does that really lower volatility?
Averaging the Bond Yields gives a lower volatility. For example


Oneofthesedays I'll have a spreadsheet which looks like
this.
for Part V on Bond Convexity.
