4

There is no possibility to convert any two of your mentioned variables into the remaining one. For the compound and arithmetic return you can derive an inequality, but that's the best you can do. The definitions for your statements are: $$r_{\mathrm{compound}}= \prod_{t=0}^{n}{\left( 1+r_t \right)}$$ $$r_{\mathrm{arithmetic}}=\frac{1}{n} \sum_{t=0}^n{r_t}$...


2

Look this is just a geometric sum: Assume interest is paid monthly at rate $r = 0.08/12$ (you can use the exact monthly equivalent if you want) and let $x_n = $total after $n$ months (including that month's interest and deposit). So $x_0= 100$ and $x_{n+1} = x_n(1+r) + d$, where $d = 5$ is your deposit amount (added at the end of the month). Applying the ...


2

Often the choice of discount curves are dependent upon the context of the assets/liabilities being valued. Banks, for example who have active and to-the-second accurate valuations of these due to active trading desks reporting daily mark-to-market should have more accurate curves, so that all of their operations are consistently valued and do not lead to ...


2

By no arbitrage, market participants need to agree on the values of the discount factor, even if they are using different conventions (day count, compounding period) to convert the discount factor into a rate. For example, consider two discount factors computed using continuous compounding, where one is computed using the 30/360 day count (year fraction $t_{...


2

Convention for most currencies is flat compounding. This includes the most liquid G4 basis swaps: EUR 3m vs 6m, 3m vs 12m, 1m vs 3m USD 1m vs 3m, 3m vs 6m GBP 3m vs 6m, 1m vs 3m JPY 3m vs 6m, 1m vs 6m, 1m vs 3m Typically the spread is quoted in terms of the leg with the shorter tenor. Two notable exceptions are EONIA v 3m and the (relatively new) SOFR ...


2

Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional expectations. Since $t < t_{n-2}$, we can write using the tower property of conditional expectations: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{...


2

So, you: 1- take your daily return series. I've used the SPY ETF including divis 2- take a log return series, ln(1) 3- add ln(1.015)/261 to 2, given 261 trading days on average each year 4- do a running sum series of 2 and 3 5- exp(4) to give you a price Gives you: The ratio between the two is 1.35 = 1.015^20, ie your 150bps compounded over the ...


1

Well let's just do the math. 10% p.a. with semi annual means 10%/2 for 6 months, so you get $1.05*1.05=1.1025$ That is, for 1 dollar you'll have 1.1025 in 1 year, i.e. 10.25% p.a. if it was annualy compounded. What should be the rate for continuous compounding (annual)? Well: $e^{r\times1}=1.1025$ gives $r=0.0975803$ or 9.758% as stated in your book.


1

This would be specified in the ISDA or term sheet. There are four alternative methods: No compounding: Meaning neither the rate nor the spread get compounded. Compounding: Meaning both the spread and the libor rate get compounded. Spread exclusive compounding:Meaning the libor get compounded but the spread does not. This was not covered by ISDA 2006 but ...


1

Let $\Delta P = P_t - P_{t-1}$ and expand the continuously compounded return in a Taylor series $$ r = \log\left(\frac{P_t}{P_{t-1}}\right) = \log\left(\frac{P_{t-1}+\Delta P}{P_{t-1}}\right) = \log\left(1+\frac{\Delta P}{P_{t-1}}\right) \approx \frac{\Delta P}{P_{t-1}} - \mathcal{O}\left(\left(\frac{\Delta P}{P_{t-1}}\right)^2\right) = \frac{P_t - P_{t-1}}{...


1

Since the simple interest $r_{s}$ and the continuous compounded interest $r_{c}$ are connected by $$(1 + r_{s} \cdot (T_{2}-T_{1})) = e^{r_{c} \cdot (T_{2}-T_{1})}$$ it follows for the continuous compounded interest: $$r_{c} = \frac{1}{T_{2}-T_{1}} \cdot \ln{(1+r_{s} \cdot (T_{1}-T{2}))}$$ your convexity formula becomes than: $$ \frac{1}{T_{2}-T_{1}} \cdot \...


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