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Q7. Today is May 15, 2000. (a) Compute (bootstrap) the discount curve Z(0,T) for T -6 month, 1 year, and 1.5 years from the following data:
. A 6-month zero coupon bond priced at $96.80 (issued 5/15/2000) . A 1-year note with 5.75% coupon priced at $99.56 (issued 5/15/1998) . A 1.5-year note with 7.5% coupon priced at $100.86 (issued 11/15/1991) (b) Once you get the discount curve Z(0, T) you take another look at the data and you find the following 1-year bonds: i. A 1-year note with 8.00% coupon (semi-annual) priced at $101.13 (issued 5/15/1991) ii. A 1-year bond with 13.13% coupon (semi-annual) priced at $106.60 (issued 4/2/1981) Compute the prices for these bonds with the discounts you found. Are the prices the same as what the market says? Is there an arbitrage opportunity? Why?
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Answer #1

a) Bootstrap the discount factors Z(t,T)ans substituting the values for 6 month, 1year and 1.5year bonds we have -

For zero coupon bond -

Z(t,T) =$96.80= 100*Z(t-T1)

=0.9680

Likewise we will calculate for 1 year and 1.5 year

T-t Coupon Price Issued Z(t-T)
0.5 0.00 $96.80 5/15/2000 0.9680
1 5.75 $99.56 5/15/1998 0.9407
1.5 7.50 $100.86 11/15/1991 0.9032

b)Now compute the no-arbitrage price of the two bonds given the discount function obtained above.

The prices of the two bonds are: P =101.71 and P =106.60.

For bond P= 101.71, the price is higher than the market prices. There is an arbitrage opportunity. We can make riskless profit by buying the underpriced bond at the traded price and selling the corresponding portfolio of zeros that replicates the cash flows from the bond.

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