R31.P1.T4- Tuckman Chapter 2-Learning SpreadsheetT4.c: Sheet-Matury vs Price

[gargi.adhikari]

Hi,
In reference to:
R31.P1.T4- Tuckman Chapter 2-Learning SpreadsheetT4.c Bundle : Sheet-Maturity vs Price
Please refer to the section circled in orange in the screenshot below. The Coupon is = 4.875% based on which the Cash Flows have been derived. But to calculate the discounted Prices for this bond, we have used the discount factors for another bond which has a different set of Coupons (highlighted in yellow) from which the discount factors have been deduced. How are we using discount factors for one bond for another bond...? I am probably missing a point or two here... Any insights on this would be much appreciated !

 

[David Harper CFA FRM]

Hi @gargi.adhikari I love your engagement, thank you for diving deep to understand This illustrates Tuckmans "Maturity vs Price" (aka, Maturity and Price or Present Value). The exercise (exhibit) is an attempt to answer Tuckman's question:

• Tuckman's Question: "When are bonds of longer maturity worth more than bonds of shorter maturity, and when is the reverse true? (in the third edition he edits this for greater sophistication at the cost of some clarity: "If the term structure of rates remains completely unchanged over a six-month period, will the price of a bond or the present value of the fixed side of a swap increase or decrease over the period?")
• His answer, proved by the exhibit, is: "More generally, price increases with maturity whenever the coupon rate exceeds the forward rate over the period of maturity extension. Price decreases as maturity increases whenever the coupon rate is less than the relevant forward rate." (which becomes in the third edition:" Returning to the original question, then, if the term structure of rates remains unchanged over a six-month period, the present value will rise as the swap matures if its fixed rate is less than the forward rate corresponding to the expiring six-month period. The present value will fall as the swap matures if its fixed rate is greater than that forward rate. Appendix G in this chapter proves this general result.")

So the context is (my variation): given a coupon-bearing bond, what does the passage of time (ie., shorter maturity) do to the price? Our standard answer is, the bonds pulls to par. And it does, but this shows has it may not take a straight line, so to speak!

In yellow are the inputs: these are five bonds, with maturities of 0.5 to 2.5 years, where we observe the market prices (eg, $101.40, $108.98 ...). These five bonds determine the theoretical spot rate curve and its necessary sibling, the discount function (set of discount factors).

Then the exercise is: we assume a bond that pays a 4.875% coupon, and we determine its price at all five maturities, using the derived discount factors. We are not given this bond's price, we are solving for its price under the assumption (the "law of one price" actually) that it's price will not conflict with the existing discount function.

Tuckman's entire point is to create a situation where the price abruptly jumps above par at 1.5 years. Or put another way, the bond starts at 2.5 years at a price of $99.971, then after six months the price does pull slight toward par at $99.977. Importantly, notice the assumption here of unchanged term structure! But then from 2.0 to 1.5 years, the price is unexpected. Hence Tuckman's answer. I am really glad you asked because this exercise actually manages to cover a lot of Tuckman in a single bundle; e.g., when doesn't pull to par apply, if ever? what are the assumptions? Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Thank You Thank You Thank You for that deep insight ! The more nuances I learn from these discussions, the more I realize how much I don't yet know so its both satisfying and terrifying at the same time ... The above was truly an AHA moment for me and cant thank you enuf for that...

A Quick follow up question though on the above and going back to the excel Learning spreadsheet...

Apologies for the dumb question on the Coupon and the discount factor...I get it now...I was missing the same term structure....
But having said that, I learnt that:-
If Coupon/Fixed Swap rate > Forward Rate, Bond Price increases with maturity and Bond price > Par Value.
Coupon/Fixed Swap rate < Forward Rate, Bond Price decreases with maturity and Bond price < Par Value.

So with Par = $100 and Coupon = 4.875%,
Coupon =4.875% > Forward rate @ BOTH 1 year(= Forward Rate being 4.851%) as well as 1.5 years(= Forward Rate being 4.734%)
But the Bond price exceeds the Par Value only at 1.5 years. Why isn't the Bond Price > Par Value @ 1 Year as well...? Is the Bond Price simply "Pulling to the Par" but not yet exceeded the Par because of the shorter Maturity Time than the 1.5 year Bond..?

Also, irrespective of whether the Coupon/Fixed Rate exceeds the Forward Rate or not in all cases, the PV seems to be always decreasing with maturity...please refer below...
So how does this align with the statement in the above explanation :-
"More generally, price increases with maturity whenever the coupon rate exceeds the forward rate over the period of maturity extension. Price decreases as maturity increases whenever the coupon rate is less than the relevant forward rate." (which becomes in the third edition:" Returning to the original question, then, if the term structure of rates remains unchanged over a six-month period, the present value will rise as the swap matures if its fixed rate is less than the forward rate corresponding to the expiring six-month period. The present value will fall as the swap matures if its fixed rate is greater than that forward rate. Appendix G in this chapter proves this general result.")

 

[David Harper CFA FRM]

Hi @gargi.adhikari

Sure, I added the following summary rows (in greeen, see screenshot below) to the XLS, here is my revised at http://trtl.bz/tuckman-maturity-return

Tuckman's concept with respect to maturity and bond price is illustrated by:

• as maturity increases from 0.5 year to 1.0 years, because the coupon rate of 4.875% exceeds the forward rate, f(0.5, 1.0), of 4.851%, the bond price increases from 99.935 to 99.947
  
• as maturity increases from 1.0 year to 1.5 years, because the coupon rate of 4.875% exceeds the forward rate, f(1.0,1.5), of 4.734%, the bond price increases from 99.947 to 100.012
  
• as maturity increases from 1.5 years to 2.0 years, because the coupon rate of 4.875% is less than the forward rate, f(2.0,2.5), of 4.953%, the bond price decreases from 100.012 to 99.977

Pull to par is an effect; it is embedded in some of the movement but it's diluted due the fact the yield is changing. Take the 2.5 year bond: as its yield (slightly) exceeds its coupon rate (4.887 > 4.875), it trades at a discount. Pull to par says that, given an unchanged yield, the price will pull to par as maturity shortens. Indeed, at 2.0 years, the price has pulled to par by increasing to 99.977. But as we go shorter, the yield is changing so the pull to par effect still applies but it's not a pure effect as the yield is not unchanged. For example, at 1.0 years, yield > coupon such that price (99.947) trades at discount. If the yield were to remain unchanged at 4.930%, then this bond's price would increase to $97.97 in six months at 0.5 years (ie, pulling to par), but the price actually decreases (not a pulling to par) but that's because the yield increases to 5.008%. I hope that explains!

 

[gargi.adhikari]

@David Harper CFA FRM Thank you as always for being patient with my ignorance and putting this together ! It is clear as the sky now. I do see the point I was missing earlier on.