1. The yearly log returns on a stock are normally distributed with mean 0.1 and standard deviation 0.2. The stock is selling at $100 today. What is the probability that one year from now it is selling at $110 or more? \[P(x>=110)=1-P(x<110) \\ Retornos ~ IID Normal\\ Retorno del registro = log(PF/PI)=log(110/100)\\ P(x>=110) => 1-pnorm(log(110/100),mean =0.1,sd =0.2)\]
1-pnorm(log(110/100),mean =0.1,sd =0.2)
## [1] 0.509354
  1. Suppose the price of a stock at times 1, 2, and 3 are P1 = 95, P2 = 103,and P3 = 98. Find r3(2).

\[r_3(2)=r_3+r_2 ; r_3(2)=0.031\]

  1. The prices and dividends of a stock are given in the table below.
  1. What is R2?
R2 <- ((54+0.2)/52)-1

El retorno simple ajustado por dividendos en el periodo 2 es 0.04230769 (b) What is R4(3)?

R4 <- ((59+0.25)/53)
R3 <- ((53+0.2)/54)
R2 <- ((54+0.2)/52)

(R4)*(R3)*(R2)-1
## [1] 0.1479588

El retorno simple ajustado por dividendos en el periodo \(R_4(3)\) es 0.147959

  1. What is r3?
r3 <- log((53+0.2)/54)
r3
## [1] -0.01492565
#

El retorno continuo ajustado por dividendos en el periodo 3 es -0.01492565

  1. Let rt be a log return. Suppose that r1, r2, . . . are i.i.d. N(0.06, 0.47).
  1. What is the distribution of rt(4) = rt + rt−1 + rt−2 + rt−3?

Distribucioin Normal

\[N(0.24,1.88)=N(0.06+0.06+0.06+0.06+,0.47+0.47+0.47+0.47)\] (b) What is P{r1(4) < 2}?

pnorm(2,mean = 4*0.06,sd=sqrt(4*0.47))
## [1] 0.9003611

  1. What is the covariance between r1(2) and r2(2)? \[Cov(r1(2), r2(2)) = Cov(r1 + r0, r2 + r1)\\ = Cov(r1, r2) + Cov(r1, r1) + Cov(r0, r2) + Cov(r0, r1)\\ = 0 + V ar(r1) + 0 + 0\\ = 0.47.\]

  2. What is the conditional distribution of rt(3) given rt−2 = 0.6?

\[E[rt(3)] = E[rt +r_{t−1} + 0.6]= 0.06 + 0.06 + 0.6\\ V ar[r_t(3)] =\\ =Var[r_t + r_{t−1} + 0.6] =\\ =Var[r_t] + V ar[r_{t−1}] = 0.47 + 0.47\\ N(0.72,1.94)=N(0.06+0.06+0.6,0.47+0.47) \] 7. The daily log returns on a stock are normally distributed with mean 0.0002and standard deviation 0.03. The stock price is now $ 97. What is the probability that it will exceed $100 after 20 trading days?

1-pnorm(log(100/97),mean =(0.0002*20),sd =(0.03*sqrt(20)))
## [1] 0.4218295