Examples of Periods (t or n) in the following topics:
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- Since the number of periods (n or t) is one, FV=PV(1+i), where i is the interest rate.
- PV=100 and i=5% (or .05) so FV=100(1+.05).
- The number of periods can be represented as either t or n.
- We will address these later, but note that when $t=1$ both formulas become $FV = PV \cdot (1+i)$.
- Since this is a single-period investment, t (or n) is 1.
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- Interest Rate (i or r) [Note: for all formulas, express interest in its decimal form, not as a whole number. 7% is .07, 12% is .12, and so on. ]
- 't' in this equation would simply be 1, simplifying this equation to FV = PV(1+r).
- With multi-periods in mind, interest begins to compound.
- This means that the interest rate of a given period may not be the same percentage as the interest rate over multiple periods (in most situations).
- Normalizing expected returns in present value terms (or projecting future returns over multiple time periods of compounding interest) paints a clearer and more accurate picture of the actual worth of a given investment opportunity.
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- Also, up to now, we have avoided any special notation for the Fourier transform of a function, simply observing whether it was a function of space-time or wavenumber-frequency.
- So for this section we will follow the convention that if $h=h(t)$ then $H=H(f)$ is its Fourier transform.
- $h_k \equiv h(t_k), \hspace{10mm} t_k \equiv k\Delta, \hspace{10mm} k=0,1,2,\dots , N-1,$
- $\displaystyle{H(f_n) = \int _ {- \infty} ^ {\infty} h(t) e ^{-2\pi i f_n t} ~dt \approx \sum _ {k=0} ^ {N-1} h_k e^{-2\pi i f_n t_k} \Delta . }$
- However it is clear that the DFT is periodic with period $N$ :
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- $\displaystyle{u(x,t) = A \sin\left(\frac{\pi n}{l} x\right) \cos\left(\frac{\pi nc}{l}t\right) + B \sin\left(\frac{\pi n}{l} x\right) \sin\left(\frac{\pi nc}{l} t\right). }$
- $\displaystyle{\sum _ n A_n \sin\left(\frac{\pi n}{l} x\right) \cos\left(\frac{\pi nc}{l}t\right) + B_n \sin\left(\frac{\pi n}{l} x\right) \sin\left(\frac{\pi nc}{l} t\right)}$
- $\displaystyle{u(x,t) = \sum _ n A_n \sin\left(\frac{\pi n}{l} x\right) \cos\left(\frac{\pi nc}{l}t\right) + B_n \sin\left(\frac{\pi n}{l} x\right) \sin\left(\frac{\pi nc}{l} t\right)}$
- $\displaystyle{\frac{\partial u(x,t)}{\partial t} = \sum _ n B_n \frac{\pi nc}{l} \sin\left(\frac{\pi n}{l} x\right) . }$
- Just compute the energy (kinetic + potential) and integrate over one complete period of the motion. )
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- Since the units have to be consistent to find the PV or FV, you could change one period to one month.
- That calculates an EAR of .0824 or 8.24%.
- In this equation, A(t) corresponds to FV, A0 corresponds to Present Value, r is the nominal interest rate, n is the number of compounding periods per year, and t is the number of years.
- The equation follows the same logic as the standard formula. r/n is simply the nominal interest per compounding period, and nt represents the total number of compounding periods.
- Finding the FV (A(t)) given the PV (Ao), nominal interest rate (r), number of compounding periods per year (n), and number of years (t).
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- Perpetuities don't have a FV because they don't have an end date.
- where m is the amount amount, r is the interest, n is the number of periods per year, and t is the number of years.
- $\displaystyle{P_0 = \frac{P_n}{(1+i)^n} = P \cdot \sum_{k=1}^n \frac{1}{(1+i)^{n+1-k}} = P \cdot \frac{1-(1+i)^{-1}}{i}}$
- Solving for the interest rate or number of periods is a bit more complicated, so it is better to use Excel or a financial calculator to solve for them.
- It is just a matter of when the first and last payments occur (or the size of the payments for perpetuities).
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- For an annuity-due, the payments occur at the beginning of each period, so the first payment is at the inception of the annuity, and the last one occurs one period before the termination.
- For an ordinary annuity, however, the payments occur at the end of the period.
- For both of the formulas we will discuss, you need to know the payment amount (m, though often written as pmt or p), the interest rate of the account the payments are deposited in (r, though sometimes i), the number of periods per year (n), and the time frame in years (t).
- where m is the payment amount, r is the interest rate, n is the number of periods per year, and t is the length of time in years.
- Provided you know m, r, n, and t, therefore, you can find the future value (FV) of an annuity.
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- We have: $N = N_o e^{-t/\tau} \text{ where } N/N_o=4/14≈0.286 $, $\tau = t_{1/2}/ln2 \approx 8267 \text{ years, } t=−\tau lnN/N_o≈10360 \text{ years.}$
- The half-life is related to the decay constant by substituting the condition $N=N_o /2$ and solving for $t = t_{1/2}$:
- Since radioactive decay is exponential with a constant probability, each process could just as easily be described with a different constant time period that (for example) gave its 1/3-life (how long until only 1/3 is left), or its 1/10-life (how long until only 1/10 is left), and so on.
- These marker-times reflect a fundamental principle only in that they show that the same proportion of a given radioactive substance will decay over any time period you choose.
- Mathematically, the nth life for the above situation would be found by the same process shown above -- by setting $N = N_{0}/n$ and substituting into the decay solution, to obtain:
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- Hence the sampling period is $T_s \equiv 1/f_s$ .
- $\displaystyle{f(t) = \frac{1}{4\pi f_s} \sum _{n= -\infty}^{\infty} f(n/2f_s) \int _{-2\pi f_s} ^{2 \pi f_s} e^{i(\omega n/2f_s - \omega t)} ~d\omega }$
- Suppose we sample this signal at a sampling period of $T_s$ .
- $x[n] \equiv x(nT_s) = A \cos(2\pi f n T_s + \phi).$
- Show that $x[n] = y[n]$ .
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- Suppose an object is at positions $\vec{x}(t)$ at time $t$ and $\vec{x}(t+\Delta t)$ time $t + \Delta t$.
- Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
- On the other hand, the equation for an object's position can be obtained mathematically by evaluating the definite integral of the equation for its velocity beginning from some initial period time $t_0$ to some point in time later $t_n$.
- That is $x(t) = x_0 + \int_{t_0}^{t} v(t')~dt'$, where $x_0$ is the position of the object at $t=t_0$.
- For the simple case of constant velocity, the equation gives $x(t)-x_0 = v_0 (t-t_0)$.