# Arbitrage opportunity sum

Jorgen Welsher, CFA obtains the following quotes for zero coupon government bonds all with a par value of \$100.

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Welsher can earn arbitrage profits by:

A) selling the 2-year bond in the spot market, going short the forward contract and buying the 3-year bond in the spot market.

B) buying the 2-year bond in the spot market, going long the forward contract and selling the 3-year bond in the spot market.

C) buying the 2-year bond in the spot market, going short the forward contract and selling the 3-year bond in the spot market.

IS IT LIKE,
Buying 2 yrs bond @\$92.45 { Payoff- 100-92.45= 7.55}
long F(2,1) @ \$94.55 {payoff- 100-94.55= 5.45}
TOTAL= 7.55+5.45 = 13
and sell 3yrs bond @\$91.51 {100-91.51= 8.49}
ARBITRAGE PROFIT= 13-8.49= 4.51
CAN SOMEONE GUIDE IF I APPLIED CORRECT CONCEPT?

You have 2 choices:

1. Invest at the 3 year spot rate
2. Invest at the 2 year spot rate and reinvest at the forward rate for another year.

To avoid arbitrage, both should produce the same ending value, or equivalently, they should both have the same PV.

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but the question says by using which option can he earn arbitrage profit

Yes, if they both have the same PV, then there is no arbitrage opportunity. So do options 1 and 2 have the same PV???

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IS IT LIKE,
Buying 2 yrs bond @\$92.45 { Payoff- 100-92.45= 7.55}
long F(2,1) @ \$94.55 {payoff- 100-94.55= 5.45}
TOTAL= 7.55+5.45 = 13
and sell 3yrs bond @\$91.51 {100-91.51= 8.49}
ARBITRAGE PROFIT= 13-8.49= 4.51

For option 1, we need \$91.51 today in order to receive \$100 at time 3. Simple and easy.

For option 2. we have to work backwards. We need \$94.55 at time 2 in order to receive \$100 at time 3. Working backwards yet again, we need \$94.55 * 0.9245 = \$87.11 at time 0 to receive \$94.55 at time 2 in order to receive \$100 at time 3.

Options 1 and 2 should have produced the same PV if the forward price was priced properly. For arbitrage, we “buy low, sell high”, so we buy option 2 and sell option 1 to valued counterparties

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