Derivatives09Week10 - Interest Rates

Derivatives09Week10 - Interest Rates - Derivatives Interest...

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1.1 Derivatives Interest Rates
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1.2 Where we are Last Session: Fundamentals of Forward and Futures Contracts (Chapters 2-3) This Session: Introduction to Interest Rates and the Value of Future Cash (Chapter 4) Next Session: Continuation of Interest Rates, FRAs & Forward and Futures Prices (Chapters 4-5)
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1.3 Plan for This Session Review some items left over from last time Interest Rates Present Value & Future Value Compounding frequency; Continuous Compounding Spot Rate & Discount Function Yield measures The Yield Curve & Zero Curve Duration & Convexity
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1.4 Interest Rates - Types of Rates Treasury rates LIBOR rates Repo rates Risk Free Rate: LIBOR, Eurodollar futures, and the swap market
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1.5 Measuring Interest Rates The compounding frequency used for an interest rate defines the units in which an interest rate is measured The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers
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1.6 Measuring Interest Rates Future Value of a Present Dollar FV(t 0 ,T) = 1 x (1 + R) n where T is a term of n years and R is the interest rate per annum, on today, t 0 , with compounding once per annum More generally, FV(t 0 ,T) = 1 x (1 + R/m) nm where T is a term of n years and R is the interest rate per annum, on t 0 , with compounding m times per annum We might want to denote R as R(m) to emphasize compounding frequency For m=1, R = R(1), the equivalent annual interest rate
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1.7 Measuring Interest Rates Equivalently, we think of the Present Value of a Future Dollar PV(t 0 ,T) = 1 / (1 + R/m) nm Where a Dollar is received on t 0 +T And the future unit of currency is discounted by 1 / (1 + R/m) nm In general, the PV of A dollars received on t 0 +T PV(t 0 ,T) = A / (1 + R/m) nm
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1.8 Measuring Interest Rates Note the effect of compounding frequency Look at the FV of $100 deposited today Compounding Frequency FV(1 year) Annual (m=1) $110.00 Semiannual (m=2) $110.25 Quarterly (m=4) $110.38 Monthly (m=12) $110.47 Weekly (m=52) $110.51 Daily (m=365) $110.52
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1.9 Continuous Compounding In the limit as we compound more and more frequently we obtain continuously compounded interest rates FV of P dollars, invested today, t 0 , at rate R for n years, and compounding k periods per year, is given by FV(t 0 ,n) = P x (1 + R/k) kn = P x [(1 + R/k) k ] n = P [ f(k) ] n where, f(k) = [ 1 + R/k ] k To find the limit of f(k) when the number of compounding periods, k, increases without bound, take the natural logarithm both sides of f(k) above, ln [f(k)] = k x ln[ 1 + R/k ]
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1.10 Continuous Compounding Let h = R/k ln [f(k)] = k x ln[ 1 + R/k ] = R/h x ln(1+h) = R x ln(1+h)/h So as, h->0, k->oo and as a consequence of L’Hopital’s rule (when lim f(x) & lim g(x) are both 0 lim f(x)/g(x) = lim f’(x)/g’(x), if RHS is finite) so, As k->oo, the compounding frequency becomes arbitrarily often, and lim ln[f(k)] = R x lim [ 1 / 1+h ] = R (where the lhs lim is oo, and the rhs lim is 0) So from the previous slide, where there is continuous compounding FV(t 0 ,n) = P x (1 + R/k) kn = P x e Rn where R is the rate with continuous compounding
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1.11 Continuous Compounding $100 grows to $ 100e RT when invested on t 0 at a
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