Can someone explain what a portfolio is in financial math?

Question

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of questions re confusing me because I don't quite understand financial terminology and I guess my professor thinks that we had taken finance classes in the past. Can someone explain what a portfolio is and what $V(O)$, $V(T)$, and $K_v$ is referring to in this question?

Let $A(0)=90$, $A(T)=100$, $S(0)=25$ dollars and let
$$S(T) = \begin{cases} 30, & \text{with probability } p \\ 20, & \text{with probability } 1-p \end{cases}$$

where $0 < p < 1$. For a portfolio with $x=10$ shares and $y=15$ bonds, calculate $V(0)$, $V(T)$, and $K_V$.

I know what a random variable is and how to solve for expectation because I learned that in probability, but I just don't know what these finance terms are refering to?

Answer 1

I agree that @BCLC is right on saying that I have used risk neutral information

The Edited Answer is

$ V(0) = 15\times90+ 10\times25 = 1600$

Now compute V(T)

$$V(T) = 1800, \text{ if stock goes up}$$

$$1800 = 30\times 10 + 100\times 15$$

$$V(T) = 1700, \text{ if stock goes down}$$

$$1700 = 20\times 10 + 100\times 15$$

$V(T) = 15\times A(T) + 10\times S(T)$ where SS(T) = 30 or 20

hence the return on the portfolio is defined as

$$K_V = \frac{V(t)-V(0)}{V(0)}$$

So $$K_V = .125, \text{ if stock goes up}$$ $$K_V = .0625, \text{ if stock goes down}$$

Thus $K_V$ is 12.5% or 6.25%.

Answer 2

There seem to be two times here. $t=0$ and $t=T$.

A portfolio is a collection of financial instruments. For instance, I could have a portfolio consisting of 3 stocks and 1 bond. Its value today is the sum of the individual values of the instruments today.

$V(0)$ is the value of the portfolio at time 0 (today?)

$V(T)$ is the value of the portfolio at time T (at maturity?)

$A(0)$ is the value of some instrument/s (bond/s?) in the portfolio at time 0 (today?)

$A(T)$ is the value of some instrument/s (bond/s?) in the portfolio at time T (at maturity?)

I'm guessing bonds because that is what is stated later on. So, we might have:

$V(0) = S(0)x + A(0)y = 25*10 + 90*15$

$V(T) = S(T)x + A(T)y = S(T)*25 + 100*15$

$S(T)$ is random so that's the most we can do.

However,

$E[V(T)] = E[S(T)]*25 + 100*15$

where $E[S(T)] = 30p + 20(1-p) = 10p + 20$

This book suggests $K_V$ is the return on the portfolio (simple return? log return?). We might have:

$$K_v = \frac{V(T) - V(0)}{V(0)}$$

$$ = \frac{(S(T)*25 + 100*15) - (25*10 + 90*15)}{25*10 + 90*15}$$

Also random. However, we can calculate the expected (simple?) return:

$$E[K_v] = \frac{E[V(T)] - V(0)}{V(0)}$$

$$ = \frac{(E[S(T)]*25 + 100*15) - (25*10 + 90*15)}{25*10 + 90*15}$$

In case you're computing log returns, be careful:

$$E[\ln X] \ne \ln E[X]$$

See more:

1. Jensen's Inequality

2. NNT

3. More NNT

P.S. NNT's account