1 2
2 2 2.1 IS-LM 1 2.2 1 1 (interest) 100 (net rate of interest) (rate of interest) ( ) = 100 (2.1) (gross rate of interest) ( ) = 100 (2.2) 1 1. 2. 1 1 ( )
2.3. 3 2.3 1 (yield to maturity) (rate of return) 2.3.1 1 1 n 1+x = x 100 (2.3) n r (1 + r) n =1+x r = np 1+x 1 y c n c x r r y = c (1 + r) + c (1 + r) + + c + x 2 (1 + r) n (1 + r) n 1+r y(1 + r) n c(1 + r) n 1 c(1 + r) n 2 (c + x) =0 (2.4) n 5 0 y>0 n c 1 0;c 2 0; ;c n 1 0;c n > 0
4 2 1 y = c 1 (1 + r) + c 2 (1 + r) 2 + + c n (1 + r) n (2.5) r ( ) y (2.5) i c i (1 + r) i 0 r i c i x = 1 (2.5) 1+r y = c 1 x + c 2 x 2 + + c n x n (2.6) n f(x) f(0) = 0; f 0 (0) > 0; f 00 (x) > 0(n 2 ) f 0 (x) > 0; (x >0) f 00 (x) > 0; (x >0) f(x) x 0 (2.6) x>0 x>0 x<1 3. i c i >y 0 <x<1 1. g(x) =c i x i y = g(x) y = f(x) 1. r r>0 () nx i=1 c i >y [ ] (2.6) f(x) P n i=1 c i >y f(1) >y f(x) x 0 f(0)=00 <x<1 x y = f(x) P r>0 0 <x<1 n y = f(x) <f(1) = i=1 c i 1. 1 cn > 0 cn =0 n
2.3. 5 2.3.2 y c 1 ;c 2 ; ;c n (2.6) 4 2 2 g(x) =f(x) y g(x) =0 x>0 g(0) = y g(x) x =0 x g(x) =0 x 1 x<1 x =1 x r =0:5 x =2=3 P n x i=1 c i y 1 x =1 x r r x Excel 2. y =100, c 1 =10;c 2 =11;c 3 =9;c 4 =15;c 5 =25;c 6 =33 visual BASIC 2.3.3 2 g(x) =0 2 g(x) =0 [a; b] a b P n i=1 c i >y 0 <x<1 1. a 0 =0;b 0 =1 2. 2 z =(a i + b i )=2 g(z) 3. g(z) 0 3.No g(z) < 0 a i+1 = z; b i+1 = b i 2 3.No g(z) > 0 a i+1 = a i ;b i+1 = z 2 2 3 4
6 2 3.Yes g(z) ß 0 z ( ) 3. 2 2 2 g(x) =0 g(x) 2 1 x 0 x =1 x n+1 = x n g(x n) g 0 (x n ) (2.7) g(x n+1 ) ß 0 4. (2.4) r c + x y n y + n 1 (x y) 2n c n c x y r (2.4) y @y @r @y r @r x 4. y r @y @r ; @ 2 y @r 2 y c @y @c ; @ 2 y @c 2
2.4. 7 2.4 (amortization) y i s n y 1 y + iy s 2 (y + iy s)+i(y + iy s) s = y +2iy + i 2 y 2s is n k P k k P k+1 P k P k+1 = P k + ip k s (2.8) P 0 = y P k = y(1 + i) k sf(1 + i)k 1g i P n =0 iy s = (2.9) 1 (1 + i) n 1 (1 + i) n i amortization factor 5. y n s s 2.5 (credit risk) 3 1 1 ( ) (discount bond) 3
8 2 T 1 t v(t; T ) t» T 1 (present value) (discount function) ( 0 ) t T (t; T ) v(t; T ) 5. v(t; t) =1 1. v c (t; T ) t =(T t)=n v c (t; T )= nx i=1 cv(t; t + t) +xv(t; T ) 2.6 r 1 (t; T )= 1 v(t;t ) 1 T t (2.10) 1 v(t; T ) =1+(T t)r 1(t; T ) T t 1 r 2 (t; T ) ρ ff 1 v(t; T ) = 1+ (T t)r 2 2(t; T ) 2 T t n 1 r n (t; T ) ρ ff 1 v(t; T ) = 1+ (T t)r n n(t; T ) (2.11) n
2.6. 9 ρ ff v(t; T )= 1+ (T t)r n n(t; T ) (2.12) n n!1 e x = lim 1+ x n n!1 n (2.12) y(t; T )= lim n!1 r n(t; T ) (2.13) v(t; T )=e (T t)y(t;t ) (2.14) y(t; T ) 1 (yield to maturity) ln v(t; T ) y(t; T )= ; (T >t) (2.15) T t 6. y(t; T ) t T t T (term structure) 6. (2.15) 1. 2. 3. 4.
10 2 2.7 T! t (2.15) fi ln v(t; T ) fififi r(t) = lim = @ T!t T ln t @T v(t; T ) T =t (2.16) (spot rate) 1 t T t B(t) 0 B(0) = 1 db(t) = r(t)dt dt R T r(fi )dfi B(t) =e t (2.17) 7. 2.8 t <T <fi T fi ( ) (2.15) f(t; T; fi) = ln v(t;fi ) v(t;t ) fi T (2.18) 8. (forward) (futures) fi! T ln v(t; T + h) ln v(t; T ) f(t; T ) = lim h!0 h = @ ln v(t; T ) @T
2.9. 11 r(t) =f(t; t) 9. v(t; T )=e R T t f (t;s)ds (2.19) r(t )=f(t; T ); (T >t) T (2.19) v(0;t)=e R T 0 r(s)ds (2.17) 1 B(T )= v(0;t) 7. 8. R T f(t; s)ds t y(t; T )= T t 2.9 (