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# ecg590i_lecture02 - ECG590I Asset Pricing Lecture 2 Present...

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ECG590I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving \$100 now or \$100 one year from now, then you would rather have your money now. If you have to decide between paying \$100 now or \$100 one year from now, then you would rather pay one year from now. This section is about computing the value today of something in the future. John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 2: Present Value 2 2.1 Present value: the future. 2.2 Simple case One payment of face amount (principal) X , one period in the future. ± First, suppose you had V now and could save it at interest rate R . In one period, you would have X = V (1 + R ) . ± Rearrange terms to get V = X 1+ R . ± V is the present value of X . John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 2: Present Value 3 Two periods in the future: X = V (1 + R 1 )(1 + R 2 ) = V 2 Y i =1 (1 + R i ) from which we get: V = X 2 4 2 Y i =1 (1 + R i ) 3 5 ± 1 = X 2 4 2 Y i =1 (1 + R i ) ± 1 3 5 Many periods: V = X 2 4 I Y i =1 (1 + R i ) ± 1 3 5 Special case when R 1 = R 2 = ²²² = R I = R : V = X (1 + R ) I John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 2: Present Value 4 2.3 V = C 0 + C 1 1 + R 1 + C 2 (1 + R 1 )(1 + R 2 ) + + C I (1 + R 1 )(1 + R 2 ) (1 + R I ) + X (1 + R 1 )(1 + R 2 ) (1 + R I ) = I X i =0 2 4 C i i Y j =0 (1 + R j ) ± 1 3 5 + X I Y i =0 (1 + R j ) ± 1 R 0 ² 0 . John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 2: Present Value 5 2.4 Continuous time Compounding: 1 + R ! (1 + R 2 )(1 + R 2 ) = (1 + R 2 ) 2 1 + R ! (1 + R n ) n lim n !1 (1 + R n ) n ± e R (1 + R ) t ! [(1 + R n ) n ] t = (1 + R n ) nt lim n !1 (1 + R n ) nt = e Rt Present value of X to be paid at time T : V = Xe ² RT John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 2: Present Value 6 Continuous-time vs. discrete-time interest rates: Once per annum: 1 + R = e ~ R n times per annum: (1 + R n ) n = e ~ R John Seater, North Carolina State University, Fall 2007
7 R constant: V = Z T 0 Ce ± Rt dt = C Z T 0 e ± Rt dt = C ± 1 R e ± Rt ± T 0 = C ± 1 R e ± RT + 1 R ± = C R (1 ± e ± RT ) Note that V ! C

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ecg590i_lecture02 - ECG590I Asset Pricing Lecture 2 Present...

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