Hello, I am having some troubles understanding the formulas for corporate finance on this document. I would be great to have someone who could explain them to me with some practical examples. Thanks

Victor

Appendix: Formulae Statictical formulas for expected value, variance and covariance: (1) E ( X ) = pi X i i Var ( X ) pi [ X i E ( X )] 2 2 X i XY Cov( X , Y ) pi [ X i E ( X )][Yi E (Y )] i where pi is the probability of state i and pi 1 . i Formulas for average return, standard deviation and covariance between two rates of return for historical data: (2) 1 T r = rt T t 1 r2 Var ( r ) 1 T ( rt r ) 2 T 1 t 1 12 C ov ( r1 , r2 ) 1 T (r1t r1 )( r2t r2 ) T 1 t 1 Expected return and variance of a portfolio consisting of n assets: n (3) E ( rp ) = w1 E ( r1 ) w2 E ( r2 ) ... wn E ( rn ) wi E ( ri ) i 1 2 p 2 1 2 2 1 2 2 2 = w w2 ... wn 2 n (4) 2 w1 w2 12 2 w1 w3 13 ... 2 w1 wn 1n 2 w2 w3 23 2 w2 w4 24 ... 2 w2 wn 2 n ... p2 = n w w i j ij i 1, j 1 where is the variance of asset i, is the ij i2 covariance between the returns on asset i and asset j, and i,j = 1,2, w1 w2 ... wn 1 ...,n. wi is the proportions invested in asset i, Optimal weights wi for market portfolio for economy with 3 risky assets (n = 3): E ( r1 ) rf 1 12 2 12 3 13 E ( r2 ) rf 1 12 2 22 3 23 E ( r3 ) rf 1 13 2 23 3 32 i wi 1 3 3 Page 1 of 2 (5) Annuities: (6) n (1 r ) 1 C C C ... = C 1 2 n n 1 r 1 r 1 r r (1 r ) where is fixed payment and is the discount rate. r C Perpetuities: PV C C C PV ... = 1 2 r 1 r 1 r where is fixed payment and is the discount rate. r C (7) Macaulay duration of the bond: (8) 1 C 2 C n (C FV ) 1 duration = ... 1 2 1 y n P 1 y 1 y where is fixed coupon payment, is yield to maturity, is time to maturity and is the n y P C price of the bond. Modified duration of the bond: duration modified duration = 1 y where y is yield to maturity. Page 2 of 2 (9)