Can someone explain me after the underlined portion.. If the NZD curve remains the same….
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Aaj ke NZD interest rate term structure ko dekhe toh 4.5 years ka rate approximately 4th and 5th year ka average hoga = 2.76+2.92/2 = . Yaani, agar hum ye assume karte hai ki 6 mahine baad term structure change nahi hoga, toh 6 mahine baad 4.5 years ka rate 2.84 hi hoga. Yaani humare aaj ke kharide 2.92% rate ke 5 year bond ka 6 mahine baad rate 2.84% ho jaayega (kyuki woh tab 4.5 year bond ban jaayega). Iska matlab humein rate girne se faayeda hoga. Rate gira hai 2.92-2.84=8bp. This translates to 33bp of price increase based on the then duration figure.
Abb second case mei hum assume kar rahe hai ki 6 mahine baad ka rate hoga woh jo aaj ka term structure bata raha hai 6 mahine baad hona chahiye. This relates to the basic concept of forward rates. Agar 1 year spot is 2% and 2 year spot is 4%, this means ek saal baad ek saal ka rate is: aaj ke spot rates se calculate kiya gaya forward rate i.e., (1.04^2/1.02)-1.
Just like this, aaj ka 6 month rate is 2.03%. Aaj ka 5 year rate is 2.92%. This means 6 mahine baad 4.5 saal ka rate is: the current forward rate. The calculation is shown in the image below. This comes out to be 3.03%, which is higher than the current 4.5 year rate of 2.84%.
If this second case holds, then iska ye matlab hai ki hum agar 6m ka bond ko pura 6 month tak hold kare yaa agar 5 year bond kharid kar 6 mahine baad use bech de, effectively dono humein same hi return denge. Recall this explanation in Level 2 fixed income curriculum.
I get the second case where the expected spot rates evolve as per the implied forward rate that is PET holds good. Whatever is our horizon we should buy bonds for that only, buying long term bonds would entail the same return.
But can you please help me understand how did we arrive at 33 BP. (Then duration * 8 Bp).
How to get the then duration is it (4.5/1+r) but I am not getting the same answer.
Thanks
The 4.5 year receiving leg implies that for the remaining 4.5 years the receiver will get the 2.92% pa fixed rate of the swap every six months. Let’s take the notional amount to be 100. This means for the next 9 (4.5*2) periods, the receiver will get 1.46(2.92/2) each period and in the final period he will get 100+1.46 back (obviously, notionally). This means it is like a 2.92% coupon paying bond with 4.5 years to maturity. We know that a 4.5 yr ZCB has a Macaulay duration of 4.5 and a 4.5 yr Coupon paying bond has a Macaulay duration of less than 4.5. The writers must have calculated the Macaulay Duration based on the specific details of this receiving leg of the swap and arrived at the value 4.2. Note that for you to calculate the duration in the exam there will obviously be clear information given.