L6v-4 - Multi-period Discount Factors Denition A nominal...

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Lecture 6: Interest Rates and Bond Yields Valentyn Panchenko Multi-period Discount Factors Defnition: A nominal discount factor, df ( t ) ,isthep resentva lue of one unit of currency to be paid with certainty at time t . Notation: ° Discount function { 1 × periods } : dF = ° df ( 1 ) df ( 2 ) df ( 3 ) ± ° Vector of cash Fows known to be certain { periods × 1 } : cF = cf ( 1 ) cf ( 2 ) cf ( 3 ) Discounted present value of the cash Fows: pv = dF · cF Example: Three coupon bonds with di f erent maturities ° The Payment Matrix { periods × bonds } : B1 B2 B3 Q = 103 4 3 0 104 3 0 0 103 Year 1 Year 2 Year 3 ° The Price Vector { 1 × bonds } : B1 B2 B3 p = ° 100 101 98 ± ° The Question: What are the real discount factors for Year 1, Year 2, Year 3? Inferring the discount function ° The price of each bond should equal its discounted present value: 100 = df ( 1 ) · 103 101 = df ( 1 ) · 4 + df ( 2 ) · 104 98 = df ( 1 ) · 3 + df ( 2 ) · 3 + df ( 3 ) · 103 ° In matrix notation: p = dF · Q ° Assuming Q 1 exists, the discount function can be inferred as dF = p · Q
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Discount function and replicating portfolio ° Since Q 1 exists, the discount function is df ( 1 × Years) = ° 100 101 98 ± 103 4 3 0 104 3 0 0 103
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L6v-4 - Multi-period Discount Factors Denition A nominal...

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