Chapter011

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

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The Theory of Interest - Solutions Manual 126 Chapter 11 1. A generalized version of formula (11.2) would be 12 n n n n t tt ttn t t t tv R d vR ++ + = + " " where 0 n t <<< < . Now multiply numerator and denominator by () 1 1 t i + to obtain 1 21 1 . n n n n n t t tR tv R R d RvR + = + " " We now have 1 1 1 1 0 lim lim . t iv t dd t R →∞ == = 2. We can apply the dividend discount model and formula (6.28) to obtain ( ) 1 . Di k Pi =− We next apply formula (11.4) to obtain ( ) 2 1 11 .08 .04 . v ik = = − Finally, we apply formula (11.5) 1.08 7 . .08 .04 dv i =+ = = 3. We can use a continuous version for formula (11.2) to obtain 0 0 n t n n t n tv dt I a d a tv dt and then apply formula (11.5) ( ) . 1 n n dv I a v ia +
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The Theory of Interest - Solutions Manual Chapter 11 127 4. The present value of the perpetuity is 1 . a i = The modified duration of the perpetuity is () 1 1 1 /1 . 1/ t t t t vt v dv I a v ia v vi d v v id i v i = = == = + = = 5. Applying the fundamental definition we have ( ) ( ) ( ) ( ) ( ) 8 8 8 8 10 800 at 8% 10 100 10 23.55274 800 .54027 5.99. 10 5.74664 100 .54027 Ia v di av + + + + 6. ( a ) We have ( ) 1 P v P ii =− = + s o t h a t 650 100 1.07 d = and 6.955. d = ( b ) We have ( ) ( ) [ ] 1 Pi h Pi h v +≈ s o t h a t ( ) ( ) [ ] [] .08 .07 1 .01 100 1 .01 6.5 93.50. PP 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 ( ) 12 3 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 vR −− ++
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The Theory of Interest - Solutions Manual Chapter 11 128 8. We have () ( ) ( ) ( ) 1 2 3 1 1 21 Pi R i P iR i P i =+ =− + ′′ 3 2 12 2 and 2 1 1.715. 11 . 0 8 i c i 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 ( ) ()() ( ) ( ) ( ) ( ) 23 34 1000 1 1 1000 1 2 1 1000 2 1 6 1 . i i i i i i −− ⎡⎤ + + ⎣⎦ + + + + Now, 2 1.1 2 1.1 1.1 2 1.1 1.1 1.1 1.1 v ++ = = and 1.1 2 3.1 . 4 8 . 1.1 1 2.1 dv i + = + ( b ) We have 61 P ii i c +++ + and multiplying numerator and denominator by 4 1 i + ( ) ( ) 32 3 2 2 1 6 2 1.1 6 8.2 3.23. 2.541 1 . 1 1 . 1 i + = + + 10. When there is only one payment d is the time at which that payment is made for any force of interest. Therefore, 2 0. dd dv dd σ δδ = 11. ( a ) ( ) 123 1000 1 2 1 3 1 1000 1.25 2 1.25 3 1.25 $3616. i i i −−− + + + + + = ( b ) 1000 1.25 4 1.25 9 1.25 7968 2.2035. 3616 1000 1.25 2 1.25 3 1.25 d =
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The Theory of Interest - Solutions Manual Chapter 11 129 ( c ) 2.2035 1.7628. 11 . 2 5 d v i == = + ( d ) () 34 5 123 21 121 361 12 13 1 2 1.25 12 1.25 36 1.25 4.9048. 3.616 Pi i i i c iii −− − −−− ′′ ++ + +++++ 12. Per dollar of installment payment, we have ( ) ()() () () ( ) ( ) ( ) ( ) 34 2 1 231 . n n i i i i i nn i −− =+ ++ +++ =+ + + + + " " If 0, i = the convexity is ( ) ( ) ( ) ( )() ( ) ( ) 22 2 01 2 2 3 1 1 1 1 2 12 1 1 12 1 3 1 62 6 1 241 . 63 Pn n c P n n n n n ⋅+⋅+ + + ++ + +++ + + = + + + + " " "" 13. We have ( ) ( ) 1 3 2 Di k P iD i k =− so that 3 2 2 1250. .08 .04 D ik c i k = = = 14. From formula (11.10) 2 or 800 6.5 or 800 6.5 842.25. dv vc c c di = = + =
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The Theory of Interest - Solutions Manual Chapter 11 130 Now applying formula (11.9b), we have ()( ) () 2 2 1 2 .01 .08 100 1 .01 6.5 842.25 2 97.71.
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Chapter011 - The Theory of Interest - Solutions Manual...

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