Chapter011 - The Theory of Interest - Solutions Manual...

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The Theory of Interest - Solutions Manual Chapter 11 1. A generalized version of formula (11.2) would be 1 2 1 2 1 2 1 2 1 2 n n n n t t t t t n t t t t t t t t v R t v R t v R d v R v R v R + + + = + + + L L where 1 2 0 n t t t < < < < K . Now multiply numerator and denominator by ( 29 1 1 t i + to obtain 1 2 1 1 2 1 2 1 1 2 1 2 . n n n n t t t t t t n t t t t t t t t t R t v R t v R d R v R v R - - - - + + + = + + + L L We now have 1 1 1 1 0 lim lim . t i v t t R d d t R →∞ = = = 2. We can apply the dividend discount model and formula (6.28) to obtain ( 29 ( 29 1 . D i k P i - = - We next apply formula (11.4) to obtain ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 1 .08 .04 . P i D i k v P i D i k i k - - - - - = - = - = - = - Finally, we apply formula (11.5) ( 29 1.08 1 27. .08 .04 d v i = + = = - 3. We can use a continuous version for formula (11.2) to obtain ( 29 0 0 n t n n t n tv dt I a d a tv dt = = and then apply formula (11.5) ( 29 . 1 n n d v I a v i a = = + 126
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The Theory of Interest - Solutions Manual Chapter 11 4. The present value of the perpetuity is 1 . a i = The modified duration of the perpetuity is ( 29 1 1 1 / 1 . 1/ t t t t v tv d v Ia v i a v v id v v i d iv i = = = = = + = = = = 5. Applying the fundamental definition we have ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 8 8 8 8 10 800 at 8% 10 100 10 23.55274 800 .54027 5.99. 10 5.74664 100 .54027 Ia v d i a v + = = + + = = + 6. ( a ) We have ( 29 ( 29 1 P i d v P i i = - = + so that 650 100 1.07 d = and 6.955. d = ( b ) We have ( 29 ( 29 [ ] 1 P i h P i hv + - so that ( 29 ( 29 [ ] ( 29 ( 29 [ ] .08 .07 1 .01 100 1 .01 6.5 93.50. P P v - = - = 7. Per dollar of annual installment payment the prospective mortgage balance at time 3 t = will be 12 .06 8.38384 a = . Thus, we have ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 3 1 2 3 1.06 2 1.06 3 9.38384 1.06 1.06 2 1.06 9.38384 1.06 26.359948 2.71. 9.712246 t t t t tv R d v R - - - - - - + + = = + + = = 127
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The Theory of Interest - Solutions Manual Chapter 11 8. We have ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 3 1 1 2 1 P i R i P i R i P i R i - - - = + = - + ′′ = + ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 2 1 2 2 1 2 and 2 1 1.715. 1 1.08 P i R i c i P i R i - - - ′′ + = = = + = = + 9. ( a ) Rather than using the definition directly, we will find the modified duration first and adjust it, since this information will be needed for part ( b ). We have ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 2 3 3 4 1000 1 1 1000 1 2 1 1000 2 1 6 1 . P i i i P i i i P i i i - - - - - - = + + + = - + - + ′′ = + + + Now, ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 3 1 2 2 1.1 2 1.1 1.1 2 1.1 1.1 1.1 1.1 P i v P i - - - - + + = - = = + + and ( 29 1.1 2 3.1 1 1.48. 1.1 1
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Chapter011 - The Theory of Interest - Solutions Manual...

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