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4

So lets look at the what we mean by compound interest. Let use $r$ as a rate for a year. If we want to define a semiannual compound rate we do the following $$ \left(1+\frac{r}{2}\right)\left(1+\frac{r}{2}\right) = \left(1+\frac{r}{2}\right)^{2} $$ for quarterly we have $$ \left(1+\frac{r}{4}\right)^4 $$ since at the end of the year we have $4$ growths where ...


3

There are a few missing (but standard) assumptions underlying this. As a technical point, you have to assume that the market is liquid and arbitrage free. in particular, you have to assume that there is a rational market available into which you could sell the unexercised option at any time. Or, similarly, a market wherein you can short the asset liquidly ...


2

The formula rewritten as a transformation of $p_2$ looks like $$p_2'={p_2\over 1+p_1-p_1\cdot p_2}$$ and is not immediately recognizable as a "typical" result in any particular field that I am aware of. The effect is, as you probably know, that $p_2$ is modified according to the current value of $p_1$. Moreover, this effect is that $p_2'$ is smaller than ...


2

The costs are fix costs. They are payed at t=0. Interest rates and the inflation rate are not taken into account. The profit after n years is $P=\left( \text{electricity per year [kWh]} \right) \times \left( \text{feed rate}\right) \times \left( \text{price per kWh}\right) \times \left( \text{future value annuity factor}\right)-\text{fix costs}$ ...


1

It is well known that $$d_2 = d_1 - \sigma\sqrt{T}\text{.}$$ To see why this is true, notice that (with $\delta = 0$, the dividend rate): $$\begin{align} d_1 &= \dfrac{\ln\left(\dfrac{S_0}{x}\right) +\left(r +\dfrac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}} \\ &= \dfrac{\ln\left(\dfrac{S_0}{x}\right) +\left(r ...


1

Suppose the currently expected price in six months time is $F$. So an agreement now to buy in six months at $F$ (a forward purchase) has zero current expected value. But this should be the same as buying a call option with strike price $F$ and selling a put option with strike price $F$. This means the call and put options at strike price $F$ have equal ...


1

$\tau$ here strikes me as a Stopping Time. In that case, it is not a process but a random variable a) defined on the entire space of paths, b) measurable with respect to the filtration $\mathcal{F}_t$ in the sense that: $$\{\tau \leq t\}$$ is $\mathcal{F}_t$ measurable for all $t$. It is strange for me to think of it as a standard adapted process. Say for ...


1

*you are forgetting the interest on the payments * Assuming payments at the end of each year, interest of 4% for 3 years will accrue for 1st 200 when payment is due, for 2 years for 2nd payment , etc. The formula you used was devised taking this into account. You can check that exactly 48 interest would have accrued at time of payment, you have paid the ...


1

Two conditions can obtain at $t_1$: either $S_1 \leq K_1$, and thus it is not optimal to exercise, or $S_1 \gt K_1$. In this case, one could exercise, but one can build a zero cost portfolio that yields a positive payoff with probability greater than zero (using the usual assumptions of no arbitrage: short selling, borrowing and lending at rate $r$, etc.). ...


1

As the frequency of compounding increases, the amount you have after any specified time increases, but not without bound. For any specified time period and interest rate, there are amounts that cannot be exceeded by the final balance no matter how frequently you compound. The smallest of those amounts that cannot be exceeded by discrete-time compounding is ...


1

When compounding for the amount not annually, not bi-annually, monthly or daily or by the hour or by the second... the amount does not go to $\infty$ but mathematically tends to definition of the exponential function. $$ A = P e^{r\,t} \leftarrow A = P + P \,r\, t $$ for annual compounding, where we have taken first two terms in an infinite series as an ...


1

The other replies offer great explanations for how the formula is generated and e's purpose, but this is the simple formula you will use for interest that is compounded continuously: $ A=Pe^{rt} $ Where A is Amount, P is Principal (money down), e is a constant (~2.718281828), r is rate, and t is time. As far as what compound interest is, think of it like ...


1

Let $P$ the periodic payment (deposit), $n$ the number of years over which payments are made and $r$ the annual interest rate. The payment in the first year at the end of $n$ years will produce the value $S_1=P(1+r)^n$. The payment in the second year at the end of $n-1$ years will produce the value $S_{2}=P(1+r)^{n-1}$. And so on. The payment in the ...


1

Here we want the value of the investment on 31st December 2029, which will be the same as the value on the 1st Jan 2030 So you start with... On 31st Dec 1990, the value of his investment is $100\times 1.1$ On 31st Dec 1991 the value is $(100\times1.1+100)\times 1.1$ On 31st Dec 1992 the value is $((100\times1.1+100)\times1.1+100)\times 1.1$ This is ...


1

In practice I always like to balance the equation at a certain time (usually the beginning or the end) when I did a problem about interests while I am still in high school. Let us consider the values at the end of $20$ years. How much does the $k$-th payment worth at the end of $20$ years? Imagine you put it back into deposit right after you get the payment, ...


1

By the way, $20$ years is $240$ months. I will continue with your formulation. Let $r=1+\frac{0.05}{12}$. After $1$ month, the bank has $$xr-20000$$ After $2$ month, the bank has $$(xr-20000)r-20000=xr^2-20000r-20000$$ After $3$ months, the bank has $$xr^3-20000r^2-20000r-20000$$ Following the pattern, after $240$ months, the bank has ...



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