How Does One Determine Yield To Maturity On TIPS?

[Bob_Smith]

I want to verify Vanguard's math regarding Yield to Maturity on a recent purchase of long term TIPS. The number I get is fairly close to theirs, but far enough off that I can't attribute it to rounding. I must be doing something wrong. So, what is the formula for determining YTM on long term TIPS purchased on the secondary market? Here are the numbers pertaining to a purchase I made on 4/29/2004 (yesterday):

--39 bonds ($39,000)
--Price - $124.63
--Maturity Date - 4/15/2029
--Index Ratio on 4/29/04 - 1.13224
--Coupon - 3.875%
--Accrued Interest - $70.14
--Commission - $40.00 (YTM is inclusive of commission)

Here's what Vanguard comes up with:

Principal amount: $55,041.32 (but I get $55,033.32)
They add the Accrued interest and commission for a total of $55,151.46.
They have a yield to maturity of 2.532%.

What is the formula to determine yield to maturity? Are they correct on this one? JWR1945? Anyone?

[wabmester]

YTM is is a series expression, so your best bet is to use a spreadsheet or a calculator like this one:

http://www.moneychimp.com/calculator...calculator.htm

Just plug in your price of $124.63, a par value of $100, your coupon rate of 3.875, and years to maturity of 25 and you get a YTM of 2.533%.

[Bob_Smith]

Thanks Wabmester! That's exactly what I was looking for.

[Bob_Smith]

Wabmester, any idea how to handle TIPS in Quicken? I went to the Quicken newsgroup and the question was posted several times by others over the years with no responses. Since the principal is adjusted regularly, there doesn't seem to be a good way to handle it in Quicken. Suggestions?

[wabmester]

Sorry, Bob_Smith, I gave up on Quicken years ago when I decided it was too hard to keep it in sync. I just use a simple spreadsheet that lets me keep track of arbitrary data, so I can include the inflation index for each TIPS bond separately.

Never used QUICKEN myself. I rely on the backsides of
well used envelopes. I brush off the coffee grounds
and cigarette ashes. Then, they provide all the space
I need to plan my financial future.

John Galt

[JWR1945]

Since Bob_Smith has mentioned me and since I am not able to provide a better answer than wabmester, I will at least explain where the Yield-To-Maturity formula comes from. (BTW: wabmester's link also has a tutorial.)

Notice first that if you invest an amount P at an interest rate r for N years, it grows to P*(1+r) after the first year. It grows to [P*(1+r)]*(1+r) = P*(1+r)^2 after the second year. It grows to [P*(1+r)^2]*(1+r) = P*(1+r)^3 after the third year. And so on.

The best way to derive the formula is to look at the balances at maturity. Whenever there is any source of money with time remaining, it is assumed to grow at the Yield-To-Maturity rate r for the remaining number of years.
a) The original price P will have grown to P*(1+r)^N where r is the Yield-To-Maturity.
b) The first bond coupon arrives at the end of the first year. It has (n-1) years to draw interest at the rate r. It grows to c*(1+r)^(n-1) where c is the coupon dollar amount.
c) The second bond coupon arrives at the end of the second year. It has (n-2) years to draw interest at the rate r. It grows to c*(1+r)^(n-2).
d) Similarly for the third coupon: it grows to c*(1+r)^(n-3).
e) And so on. The last coupon arrives at the end of n years with no time remaining. It contributes the dollar amount c.
f) Finally, there is the par value B. It is delivered at maturity. It does not grow.
g) Add all of the coupons along with their growth to maturity and the par value. This equals the amount that the price P has grown to (with interest compounded annually at the rate r).
h) The final step is to divide all of the terms by (1+r)^N. This final form is most convenient when you need to calculate the price P when the coupon c, the Yield-To-Maturity r and the par value B are specified.

I do not know of any advantage to be gained by simplifying the geometric series related to the coupons. [If SUM=1+x+x^2+..+x^(N-1), then SUM is a geometric series and (1-x)*SUM=1-(x^N). This comes in handy when investing fixed dollar amounts at fixed interest rates. It is helpful for formulas with periodic payments as well.]

There is almost always a final adjustment in financial formulas. For example, when making monthly payments, you divide the interest rate twelve (when using a specified annual percentage rate) and multiply N by twelve (for the number of months). In this case there is no additional adjustment.

Have fun.

John R.

[charlie]

Hey John R.

I am still confused. It seems to me that you just
described how to compute YTM for a conventional
bond with a fixed par value at maturity. In the case
of TIPS you don't know the end value since the
bond value increases (or decreases) each year
depending on inflation. Is it possible that the YTM
for a TIPS assumes that the value at purchase
remains constant until maturity? I know (I think)
that the end value of a TIPS could be just the
issue price if we ran into a long period of deflation.
Right? If that happens you are going to have a lot
of dissapointed not to mention pissed off retirees
who bought at a higher than issue price in the secondary market. Buyers of new issues would not
have that potential problem.

I am still holding off purchase of TIPS until I understand
them better.

Thanks for your wisdom,

Charlie

[wabmester]

I don't mean to keep stealing thunder from JWR1945, but the YTM for the *real* portion of TIPS is computed the same as conventional bonds. The inflation adjustment is factored out separately. In Bob_Smith's original example above, the past inflation adjustment is reported as the "index ratio."

As you'd expect, you don't know the total yield for a TIPS until it matures or you sell it. At that point, it's an interesting exercise to compare the yield to what you would have got had you bought a conventional long-term bond. This will tell you how accurately the market predicted long-term inflation.

[JWR1945]

Chuck-Lyn

Quote:

I am still confused. It seems to me that you just described how to compute YTM for a conventional bond with a fixed par value at maturity.

All of the calculations are in terms of real dollars. That keeps the par value constant in the calculation.

When you apply the formula, we generally prefer to speak in terms of today's dollars. That means that the par value today is not $1000, but it includes inflation. Of course, you can still use today's price and today's coupon amount. [The coupon's yield applies to the original bond amount of $1000 plus inflation. It excludes any premium.]

There is a theoretical possibility, which is highly unlikely, that extended deflation would bring the principal down below $1000 (in nominal dollars) at maturity. In that case, you would get a bonus. The final payout is the greater of the final principal amount (including adjustments for inflation and deflation) and $1000 in nominal dollars.

wabmester is making things easy for me by providing the right answers. All that I have to do is figure out what he is saying (which is not difficult).

Have fun.

John R.

[charlie]

John &Wab,

I feel like gummy's sidekick ........?? ZZZzzzzz

Thanks anyway. Maybe the light will ....ZZZZZzzzz

When in doubt, sack out .... my new motto.

Cheers,

Charlie

[wabmester]

Don't nap yet, Charlie. *If a smart guy like you doesn't understand something, then most everybody else is probably confused as well.

I assume you've got a good handle on conventional bonds, so you're comfortable with the idea that paying a premium over par value is equivalent to buying a new issue at a lower yield (the only significant difference is some tax accounting at maturity).

So, the only "weird" thing about TIPS is the inflation adjustment. * Bob_Smith bought bonds that were issued on 4/15/1999. * Bob_Smith paid three premia over par value when he bought on the secondary market:

1) He paid a yield premium of $124.63 for each $100 par value. * That knocked his yield down from 3.875% to the market yield at the time, 2.533% (remember, that's the real yield above inflation).

2) He paid accrued interest, just like you would for any bond on the secondary market. * He gets that back at his next coupon payment.

3) He paid an inflation adjustment to the original owner. *That's the 1.13224 factor. *This is multiplied by the price he paid in (1) to get his actual cost per $100 par value. * Basically, that means inflation was 13.224% over the last 5 years. * He gets that back at maturity (or when he sells) in *addition* to his inflation adjustment going forward.

So, there's no more risk buying a secondary TIPS issue than there is for any other treasury bond.

[charlie]

Thanks Wab,

I think I understand that YTM is calculated based on
the market price (exclusive of inflation), the original
issue price, the remaining years to maturity and
the coupon yield. That clears up question No. 1.

As for the accrued inflation, you say that Bob
will get that back at maturity or when he sells,
regardless of inflation or deflation in the future.
Right? This is the point that is still a little fuzzy
to me. I suppose it would have to be that way or
nobody would buy on the secondary market.

If I am correct in my understanding, then Bob's risk
is that if real yields rise the $124.63 price will fall
just like in a conventional bond (and visa versa).

If there is 0 inflation or net deflation until maturity,
then Bob gets the original issue price plus the inflation
that had accrued at the time he bought the bonds.

If Bob sells after x years and there has been more
inflation, he will get the market price plus the amount
he paid for accrued inflation plus the new inflation.

If Bob sells after x years and there has been net
deflation, he gets the market price plus the original
accrued inflation minus the net deflation after he
purchased the bonds. Right? I assume the market
would not let the bonds drop below the value of a
newly issued TIPS of the same duration in the event that deflation was greater than inflation.

Right? ZZZZZZZzzzzzzz

Charlie

[wabmester]

OK, I left out a subtle risk, but I see now that was your main point. Yes, you're right that buying on the secondary market subjects you to some additional downside risk associated with the accrued inflation adjustment. Net deflation can cause the principal value to erode down to par value, so there is a small risk that all or part of the 13% inflation premium paid by Bob_Smith could be lost.

However, in Bob_Smith's case, he would need to see net deflation over a 25-year period before taking a hit. I seriously doubt we've ever had net deflation over 25 years, but anything's possible I suppose. If that happens, we'll all have much more to worry about than losing some money on TIPS.

[eytonxav]

Relative to these longer term tips, what happens to such tips if inflation stays low, and interest rates rise?

Doug

[JWR1945]

DFW_M5

Quote:

Relative to these longer term tips, what happens to such tips if inflation stays low, and interest rates rise?

You are asking what happens if the real interest rate increases.

It is similar to conventional bonds and nominal interest rates. A rise in real interest rates would cause the price of existing TIPS to fall.

Have fun.

John R.

[Peter]

Quote:

It is similar to conventional bonds and nominal interest rates. A rise in real interest rates would cause the price of existing TIPS to fall.

But, if you assume that interest rates are generally correlated to inflation, then some of this drop in TIPS price would surely be offset by the inflation adjustment?

Peter

[JWR1945]

Peter

Quote:

But, if you assume that interest rates are generally correlated to inflation, then some of this drop in TIPS price would surely be offset by the inflation adjustment?

I would be very cautious in this regard. There is insufficient history to be very confident about such details. There is a speculative element that could swamp out other effects, at least on a temporary basis.

There is a natural competition between regular bonds and inflation protected bonds. You could see people moving into and away from TIPS depending on whether they want to lock in a real interest rate or whether they want to lock in a nominal interest rate.

Have fun.

John R.

[eytonxav]

This is my biggest concern with tips; if inflation stays low, but other interest rates CD/Bonds rise; you could really suffer with long term tips.

Doug

[wabmester]

For one thing, it would be unusual to see a big increase in nominal interest rates without seeing a big increase in inflation as well.

For another thing, the average real yield of long-term treasuries over the past 50 years has been about 2.5%, so if you ignore whatever rates happen to do, you can take solace in knowing that you're still getting an average real return from TIPS.

And for a third thing, if nominal rates shoot to the moon, other bond holders will be hurt just as much as TIPS owners, and probably more. You get inflation protection with TIPS, which nominal bonds don't give you. So other bonds carry both inflation risk and interest rate risk. TIPS are simply less risky.