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I have the following problem: I am supposed to decide, if there exists an arbitrage,and if so, I am supposed to construct it.

1.Bond

  • Maturity: 1 year

  • Nominal value: 1$

  • Annual coupon: 0%

  • Market price: 0.95$

2.Bond:

  • Maturity: 2 years

  • Nominal value: 1000$

  • Annual coupon: 10%

  • Market price: 1050$

3.Bond:

  • Maturity: 2 years

  • Nominal value: 5000$

  • Annual coupon: 5%

  • Market price: 4790$

I find out, that there exists arbitrage in all of the cases. In the first the price of bond is 0.95$, in the second the price of the bond is 1260$, and in the third, the price of the bond is 5269,05$.

But I do not know how to construct arbitrage. Can anyone help me?

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    You need to figure out if some combination of buying and selling above bonds will give you arbitrage, that is risk free money. You cannot have arbitrage on one bond alone, it always needs to be a combination of buying and selling
    – ssn
    Dec 6 '19 at 15:26
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For an arbitrage you would cancel out all cash flows with a combination of buying and selling bonds and see how much is left over.

So start by buying bond 2 and selling bond 3 since those are the only ones that have a year 2 cash flow. Buy 1 bond 2 and sell enough of bond 3 to cancel out the nominal value and coupon of band 2 (hint: it would be the nominal + coupon of bond 2 divided by the nominal + coupon of bond 3). Then buy/sell the amount of bond 1 that would cancel out the net year 1 cash flow of bonds 2 and 3 (which would just be the coupons).

Sum up the amounts times the prices of each buy/sell and see how much is left over. If it's a positive amount (meaning you profit) then buy/sell those amounts. If it's negative, then sell/buy those amounts.

I show that you would sell 1 bond 2, buy 0.21 bond 3, and buy 47.62 of bond 1. That combination would give you an arbitrage profit of $1.143.

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