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Replicating Portfolio, Tuckman Table 1.5

arpitasaraswat

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Hi David - I have a question about Tuckman, Chapter 1 - Prices, Discount Factors and Arbitrage, Table 1.5 Replicating Portfolio. This may sound stupid but I wanted to understand the logic behind the calculation of the face amount of the 3 bonds. I have access to your sample spreadsheet so I understand the calculation but am not able to understand the logic. For eg. to calculate the face amount of 4.5% bond, why did Tuckman use 100.375 and discount rate [1+(4.5%/2)]. When you have a min, could you please provide explanation behind the calculation of each of the face amounts? Thanks!
 

David Harper CFA FRM

David Harper CFA FRM
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Hi @arpitasaraswat Sure, my replication of his T 1.5 is below. We are replicating the 0.750% of 11/30/2011 bond shown on orange column (6) which has been identified as "trading cheap" (its traded price of $100.190 is lower than the theoretical price we get by applying discount factors). The replicating portfolio is the three bonds and the solution is to find the face values that produce matching cash flows (shown in blue column 5), we need the right mix of the three bonds so that we produce $0.375, $0.375 and $100.375 because that's the cheap bond's cash flows.

We start with the 4.5% bond in column 4 because it's the only bond that can deliver a 1.5 year cash flow. If we buy $100 face amount of this bond, it's future cash flow in 1.5 years (its final cash flow) will be $100*(1 + 4.5%/2) per the familiar FV CF calculation. But that's a bit too much, we want F(1.5)*(1+4.5%/2) = $100.375 such that F(1.5) = $100.375/(1+4.5%/2) = $98.16626 because that gives us exactly the same cash flow in 1.5 years as our trading-cheap bond.

Now we can solve for the amount of the 1.0 year bond, maturing on 5/31/11 in column 3. If we buy $100 face amount of this bond, it's future cash flow in 1.0 year (its final CF) will be $100*(1+4.875%/2) but that's way too much, if the goal is to replicate the $0.375 paid by the trading-cheap bond's coupon. Further, now that we know we are selling 98.16626 of the 1.5 year bond, the 1.5 bond will be delivering a coupon in 1.0 year equal to $98.16626 * 4.5%/2 = $2.209; so actually, we've already got too much cash replicated in 1.0 year. We need to solve for F(1.0)*(1 + 4.875%/2) + $2.209 = $0.375, or F(1.0) = (0.375 - 2.209)/(1+4.875%/2) = 1.79011. And you can try the final ... I hope that's helpful!

 
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