Question about Yield and Modified Duration concepts and using a real world example

Having passed Level II however, still somewhat not clear about bond’s yield concept and its use in duration calculation. Here I have a real transaction of a zero-coupon bond:

Issue date: Mar 08, 2018

Maturity date: Sep 09, 2018

Basis: ACT/ACT

Settlement date: Mar 13, 2018

Purchase price: 97.60771522

Yield: 4.9699%

Notional: 150,000,000

Here are my question:

For example, I wish to calculate the Modified Duration as of Aug 30, 2018, so I have to know the YTM. The YTM is just the discount rate when price is set equal to the market price, so I retrieved the historical price on Bloomberg Terminal:

Date, Mid Line

Fr 08/31/18

Th 08/30/18 97.612

We 08/29/18 97.619

Tu 08/28/18 97.648

Mo 08/27/18 97.656

So I used 97.612 as current market price to calculate a new YTM for the bond using the Excel function Yield:

=YIELD(“8/30/2018”,“9/09/2018”,0,97.612,100,1,1), but I get 89.29%!

So here are my questions:

1. I’m not sure whether this process for calculating Modified Duration is correct or not? Usually the question will just ask you to calculate a Modified duration given YTM, or ask if interest rate changes, what’s the change in price, etc. But here the problem is what’s the impact on duration as passage of time. A bond was bought, and after sometime, what would the bond modified duration be?

2. I don’t know if it’s correct to calculate a new YTM as if I were to buy such a bond today in order to calculate the Modified Duration, because YTM is the yield you get if you buy the bond and hold it to maturity, so it should not change, and so a “new YTM” is meaningless? Or should I use some sort of interest rate from the market to calculate the Modified Duration?

3. I usually get confused when it says “the Yield of the bond” on the book or financial news, or someone asks what’s the yield on the bond, I think that’s just the YTM but I’m not sure… Sometimes the book also mentioned “the market interest rate”, seems that’s also referring to YTM when doing practice questions, but YTM always depends on the market price, looks like that’s not a rate that’s from the market, it’s an implied rate, and it needs not to be estimated, it’s a always a certain value because you can always find the market price for the traded bond. On the other hand, when using spot rate to discount the bond to get a price, that’s valuation purpose, to assess if the market price is overvalued or undervalue, and that has nothing to do with YTM. Is my understanding about YTM correct?

4. The per hundred price is 97.6077, and I think the face value may be just 100, but there is a notional of \$150,000,000, before when I saw notional I would think of it as the synonym of face value, but here looks like it’s not, but the total amount of money invested, so does that mean 150,000 bonds were bought?

There’s clearly something wrong with the prices you’re quoting: a 2.4% discount with only 10 days left to maturity?

Yes . . . that’s a 90% YTM.

Let’s figure that one out, then move on to computing duration.

(By the way, the duration (Macaulay, modified, effective, whatever) of a bond that has only 10 days to maturity is going to be very, very short; like 0.027 years.)

Thank you for answering! I didn’t think about the price quote too much because that’s what BBG showed after entering the cusip or isin…

However, even I just use the insane YTM of 90% to compute the ModDur, still gives something like 0.02… probably that’s because Duration is already so small, like you said, and divided by (1 + YTM) would not make a huge difference whether YTM is 0.9 or let’s say 0.05…

But other than the 90% YTM, if let’s say the price I got has no problem and I get a reasonable YTM in the same way, is my reasoning for the calculation of YTM and ModDur above correct? Is my understanding of YTM (yield, market rate, spot rate, etc.) concepts correct? Very much appreciate!

As I understand duration, it is a measure of interest risk.

Remember that any risky bond faces 2 kind of risks: Credit Risk (default) and Interest Rate Risk

Also remember that we may talk about Macaulay Duration, Modified Duration and Effective Duration.

If you want to calculate Modified duration, why are you calculating the YTM? You just need the remaining cash flows, the market yield as of 30th august and calculate the Macaulay duration, then adjust it to arrive to the Modified Duration using the formula we know.

Effective duration is another story, the calculation process is different but should arrive near the value of above. In this case we assess the changes in the bond price at small changes in the YTM. For this, you will also need the remaining cash flows.

Effective duration should be used when the bond has an embedded option. Not your case this time tho.

Hope this helps.

Thanks for replying! However when discounting the future CF to calculate the MacDur for example, we need a discount rate, and what would it be? I think it’s the YTM, not the YTM when you bought the bond but the newly calculated YTM based on current market price. Also, from MacDur to ModDur, you also need YTM… or if the appropriate discount rate is not YTM and my understanding was wrong, please correct me, thanks again!

Yes, you need to YTM on the day of your calculation. I assumed you were using Bloomberg or similar, so you can have the YTM of the current day.

I told here.

It is a non sense to use the YTM at issuance. Remember that a bond burns its life every day (approaches maturity) and its credit risk changes, so YTM is a dynamic variable (so price). If you want to calculate duration as of each point in time, you need to pair YTM and its unique Price. A YTM can’t throw 2 different prices or viceversa.