Bonds II  and bond duration: a continuation of Part I

We want to measure - somehow - the sensitivity of bond prices to changes in current interest rates.

Is a 10-year bond with a 5% coupon more sensitive to interest rate changes than a 10-year bond with an 8% coupon or a 5-year zero-coupon bond or a 7-year 6% bond or a ...

>Okay, I get the idea.

Recall (from Part I) our magic formula for the Value of a bond, namely:

V = B { 1/(1+R)N + (Cr/R) (1 - (1+R/m)-mN)}
where
    N = number of years to maturity
    Cr = annual Coupon rate
    m = number of coupons per year
    B = value of Bond at maturity,
    R = Annual Yield
where it's assumed that the coupons are re-invested at the current yield.

Now, if we take the derivative of V with respect to R ...
>Beg pardon?
Pay attention. It's a bit of calculus. Just take a snooze and go here when I'm finished.

Recall that our magic formula for the Value of a bond is the Present Value of the maturity value
PLUS the sum of the Present Values of all the coupons.

If we let:
B = maturity value of the bond (in dollars)
y = yield per period (period = years or months between coupons)
c = the coupon rate per period (which may be years or months)
n = the number of periods to maturity (n = years x coupons_per_year)
and we rewrite the magic formula above as:

(1)     V = B { 1/(1+y)n + Σ c/(1+y)k } = B Σ M(k)/(1+y)k

where Σ means the sum of terms, from k = 1 to k = n, and BM(k) is the coupon value at time k, namely cB when k = 1, 2, ..., n-1 but we've added the maturity value when k = n, so BM(n) = (c+1)B.

Now consider the rate at which V changes when the yield, y, changes:
dV/dy = B Σ {-k M(k)/(1+y)k+1} = -B/(1+y)Σ { k M(k)/(1+y)k}

As a fraction of the bond price, V, this is:

(2)     (1/V) dV/dy = -1/(1+y)[ Σ { k M(k)/(1+y)k} / Σ { M(k)/(1+y)k } ]

If we write M(k)/(1+y)k = W(k), the second factor, inside the brackets [ ], is just:

Σ { k W(k)} / Σ W(k) = the weighted sum of times, k,
with the weights W(k) being the Present Value of the coupon rates (including the maturity value of our bond).

We can also rewrite formula (2) like so:

(1/V) dV/dy = -1/(1+y) { n(1+y)-n + c Σk (1+y)-k } / { (1+y)-n + c Σ(1+y)-k }

>That's terrible!
Yeah? Well, in 1938, Frederick R. Macauley (see Duration Measures) generated a formula for Bond Duration.
(See Bond Duration for a derivation of the formula as the weighted average of the times when all the coupons AND the value-at-maturity are received).

Macauley's Bond Duration was an attempt to measure the sensitivity of a bond's price to changes in interest rates, namely:

Macauley Bond Duration: BD = (1+y)/y - {1+y + n(c-y)} / {c[(1+y)n - 1] + y}

where c is the coupon rate, y is the yield and n is the number of years to maturity

>That's just as terrible!
Okay, here's a calculator to play with. You just stick in the parameters, per period, and it gives the above formula:
Yield per coupon period (example 6) =%
Number of periods to Maturity (example 12) =
Coupon Rate per period (example 3.25)=%
Macauley Duration = periods

Here's something interesting:
We can rewrite our dV/dy formula (2), above, like so:

(3)     dV/dy = -V/(1+y) BD

where BD = { n(1+y)-n + c Σk (1+y)-k } / { (1+y)-n + c Σ(1+y)-k }

Now, let's write BD* = BD/(1+y) (called the Modified Bond Duration), then we have another magic formula:

dV/V = -BD* dy

where BD* = BD/(1+y) = Modified Bond Duration, and
dV/V is the fractional change in Bond Value, and
dy is the change in yield.

>Mamma mia! Do you expect anybuddy to understand what ...?
Actually, if you can use this formula then you needn't understand its derivation.

>Then let's just use it!
Okay. Let's first make note of how we'd use Bond Duration.

If BD = 15 (measured in years), and the current yield changes by +0.01 (meaning a 1% increase), then the fractional change in V is dV/V = -BD * dy = -15(0.01) = -0.15 or -15%. That means that the current bond price would decrease by 15% if interest rates increased by 1%.

>You used BD. Did you mean BD*, the modified ...?
Well ... uh ... usually one just uses BD. Anyway, let's look at some Bond Durations BD versus Time to Maturity, remembering that the Bond Duration will give the percentage increase in bond price for a 1% decrease in yield.:

For example, consider a bond which matures in 20 years. If the coupon rate is 6.0% and the current yield is 4.0% then the Bond Duration is about 13 years. See the big green triangle, above?

>I could use the calculator, above, right?
Yeah ... it'd look like this
where a period is a year.

Anyway, that 13% duration means ...

>A 1% decrease in yield, from 4% to 3%, would increase the Bond price by 13% ... the value of the Bond Duration. Right?
Right.
>Approximately.
Right again.
>It looks like the Bond Duration is less than the Time to Maturity.
Always.
>Always?
Well, almost always. In fact, for a zero-coupon bond the Bond Duration is the same as the Time to Maturity. However, with coupons, the Duration is always less.
>To be relatively immune to changes in interest rates, I'd pick a short term bond. See the small Duration?
Yes, but if you're sure that interest rates will go DOWN, then you'd want a long term bond. See the big Duration?
>I assume there's a spreadsheet to ...
Yes. It looks like this:


>I assume you've used the Bond Duration ... or maybe the Modified ...
Actually, the spreadsheet uses neither. It just recalculates the Bond Price using the modified Yield or Coupon Rate. With a spreadsheet, that's trivial.
>Which explains why the spreadsheet picture, above, gives a 13.7% increase instead of the 13% as suggested by the Bond Duration formula.
Exactly. Besides, Bond Duration is a generated from a derivative which implies an infinitesimal change in yield ... not a relatively large change, from 4% to 3%. In any case you can Right-click on the picture above and download the spreadsheet in .ZIPd format.
>It'd be nice if the spreadsheet could find the Yield if I tell it ...
Actually, if there is just a single annual coupon, the spreadsheet will also determine the yield if you give it the Time to Maturity and Coupon Rate and ..

>You're assuming I keep the bond for twenty or thirty years, until it matures. I could be dead by then. What if I sell and ...?
Aah, good point. I'm glad you asked that question ...

for Part III