Q 18 - GARP 2010 Practice Exam Here are the prices for 2 out of 3 US Treasury Notes for settlement on August 30, 2008. All 3 bonds will mature exactly 1 year later on August 30, 2009. Assume annual coupon payments and that all three bonds have the same coupon payment date. Coupon Price 2 7/8 98.4 4 1/2 ? 6 1/4 101.3 What is the price of the 4.5 US treasury Note? Solution: 2.875x + 6.25(1-x) = 4.5; x = 52% implying that the 4.5 is 52% 2 7/8 and 42% 6 1/4 ie P = 52% * 98.4 + 48% * 101.3 = 99.8 Could anyone provide intuition for this?

I just think of this question very simply... First, assume the no arbitrage situation... Value of coupon 2 7/8 with price 98.40 equals that of coupon 4 1/2 with price ? or that of coupon 6 1/4 with price 101.30 second, in this situation we dont care of buying between #0.6(weighed quantity) 98.40 plus #0.4(weighed quantity) 101.30 and #1.0(whole buying) price ?(which is 4 1/2 coupon). third, in the context of present value, price depends on yield and future cash-flow...(function of these) In the question, all is same except for coupon(which is varible), we can set no arbitragy strategy, 2.875% * x% + 6.25%*(1-x%) = 4.5% * X (which is the whole percentage we would like to buy) so we can solve for X% (which is 52%) and another is 48% (1-52%). Adding weighting Value of two US treasury notes equals unknown one US treasury notes price... So what we want to know is 99.80 (=52%*98.40+42%*101.30)

…. And just to piggyback on kthwow's no arbitrage explanation, this means to apply (Tuckman's) law of one price: the idea that equivalent cash flows must cost you the same (have the same price to you). If you have $99.80, then one "portfolio" buys the 4 1/2 coupon bond and earns $4.50. The second portfolio, consisting of the other two bonds, if it costs the same 99.80 must produce the same 4.50 cash flow (coupon). The mix solved for is the necessary combination (52% and 48%) that produces an identical cash flow (4.50) so it should cost the same (99.80). The solution breaks this into two steps: 1. If i have 100% to invest, what mix of bonds #1 and #3 give me the same cash flow (coupon) as 100% of bond #2 2. Knowing the mix (52%/48%), there is only one bond #2 price that satisfies the "law of one price." David