4

They are different because there are different conventions in different places. Whilst it would make the maths more consistent to use discount (or accrual) factors to describe interest, the foremost concern historically has been absolute clarity on amounts of money, so having a simple way to calculate the coupon payment on a given bond. So a UK Gilt paying 5%...


3

I grappled with different compounding conventions for a while when I first joined the world of finance. After some time, I came to the conclusion that the best analogy is that of different units of speed in different countries: i.e. in the UK, the unit of speed is miles, whilst in continental Europe, it is kilometers. A car travelling at 100mph is travelling ...


3

Hint: Let $$z = \mathrm{e}^{-y} $$ That way you get a quadratic equation in $z$ (note that $z$ is positive) and then you can get back to $y$ using: $$ y = -\ln (z) $$


3

Continuous-time formulation is much easier for some basic asset pricing theories. In continuous-time you will have to deal with integrals rather than sums which makes your life much easier. And for those, you will need continuous discounting. Here's an excerpt from John Cochrane's Asset Pricing: The choice of discrete vs. continuous time is one of ...


3

You add 1 to every monthly return of a given quarter, take the product of those returns, and then subtract 1. In R (without any package): Suppose r are the monthly returns, and dt are the timestamps. r <- rep(0.01, 12) dt <- seq(from = as.Date("2020-1-1"), to = as.Date("2020-12-1"), by = "1 month") tapply(r, paste(as....


2

In continuous compounding, a nominal (or an index value) in time $t$ is given by formula $$ N_t = N_0\mathrm{e}^{rt}, $$ where $r$ is return (or interest) rate per annum. Based on the equation above, the $r$ can be calculated as $$ r = \frac{1}{t}\ln\frac{N_t}{N_0}. $$ So, for $t = 1$ we have the annualized return: $$ r_{t=1} = \frac{1}{1}\ln\frac{4086}{...


2

Sometimes it is easier to work with continuos compounding in some models, especialy when you compound interest daily. Moreover, it can come from history when it was more difficult to calculate higher powers than tabulated exponential function. Take for example annual interest rate $i = 5 \%$. Then effective annual interes rate based on daily compounding is $...


2

Depends on which OIS you are referring to. For EUR OIS Swaps, the EONIA Swap rate is calculated via the usual compounding formula (notice that in the example below, the rate $r_i$ is updated every night): Example is shown here: For USD OIS Swaps, the link to Investopedia that you shared is correct: it is pretty much the same formula as for the EUR swap ...


2

Let's add a time variable to extend to non-annual periods $$1 + r_d t = e^{r_c t}$$ The taylor expansion of exponential is \begin{align} e^{r_c t} &= \sum_{n=0}^\infty {\frac {(r_c t)^n} {n!}}\\ &= 1 + r_c t + {\frac 1 2}(r_c t)^2 + \cdots \end{align} so by equating the two equations, we see that $$r_d = r_c + {\frac 1 2}r_c^2 t + O(t^2)$$ Two things ...


1

Your formula in the first example is on the right track. Standing at time step $i$, your value at next time step $i+1$ is $V_{i+1} = (V_{i} + c_i)(1+r_{i+1})$, i.e. your previous portfolio value plus an influx of $c_i$ in cash (SIP) are yielding a one step return of $r_{i+1}$. Explicitly you have \begin{align} V_0 &= V_0 \\ V_1 &= (V_0 + c_0)(1+r_1) \...


1

They're not equivalent, but you can use log identities to derive something similar after applying a log. Eg, $ln\left(\left(\frac{1+E_2}{1+E_1}\right)^{\frac{d}{360}}\right)$ $\frac{d}{360}*\left(1+E_2-\left(1+E_1\right)\right) $ $\left(E_2-E_1\right)*\frac{d}{360}$


1

Let's assume that the collateral rate on cash equals the overnight rate, that we have a schematic (lined/tiled up accrual periods and payments dates) strip of dates/times $T_0<T_1<\ldots <T_n$, accrual factor $\tau_t := \tau(t-1,t)$, and $c_t$ collateral rate at $t$ (overnight $t-1$ to $t$). The floating coupon is then: $$ \prod_{s=T_{i-1}}^{T_i}\...


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