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The advantage of cash-settled swaptions is that the payoff only depends on one variable: the corresponding swap rate which is directly observable in the market: $$\mathrm{Payoff}(T) = f(S_T) = A^{\mathrm{Cash}}(S_T)\max(S_T - K,0)$$ The payoff of a physical swaption on the other hand depends on the physical annuity which is not directly observable. You ...

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The market standard formula approximation for cash settled swaptions applies Black/shifted Black/Bachelier around the forward swap rate so that with this formula parity between payer and receiver swaptions occurs around the forward swap rate, and in particular the zero wide collar struck at the forward swap rate is worth zero (a zero wide collar is the ...

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To build intuition, let us consider the underlying swap itself rather than a swaption. Conceptually, you can think of the swap annuity factor as the present value of gaining 1 unit every period of the underlying swap. Scaled appropriately, the swap annuity factor is the PV01, i.e. the Present Value of a Basis Point. Adjusting for convexity gives you the ...

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To lighten notation, we assume a constant accrual factor $\tau$, a swap rate $S_n(T)$ which fixes at $T$ and pays at $T_p$ (e.g. $T_p-T=\text{3 months}$) and a simple CMS payoff of the form: $$\Phi(S_n(T))=(S_n(T)-K)$$ fixed at time $T=T_m$. We are interested in pricing under a measure for which the underlying risk factor of interest (i.e. the swap rate) is ...

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ok so if you sell a CDS for 100bp and then the market moves to 90bp, you have a profit of 10bp. But how much is that actually worth in dollar terms? Suppose you then buy the CDS for 90bp, what have you got? You have 10bp per annum until the reference entity defaults, which is worth 10bp * the Risky pv01 of the contract. Hope that explains it. The ...

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you can also solve this by using the PMT function and goalseek in excel. The answer is as follows: What I did is, I first set the interest rate to 0% and calculated the monthly payment using the PMT function. Then I goalseeked the monthly interest rate such that the monthly payment would be 300.

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The annuity expression $a_{4}^{(12)}$is written as: $$a_{4}^{(12)}= \frac{1-(1+i)^{-4}}{i^{(12)}} = \frac{i}{i^{(12)}} a_4$$ where, $i$ is the effective annual rate of interest and $i^{(12)}$ is nominal rate of interest convertible monthly, which is equal to $$i^{(12)}=12((1+i)^{1/12}-1)$$ There is no closed formula to get the interest rate, you have to ...

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There are 48 monthly payments. You can use the formula for the Present Value of an annuity: $12000 = 300 \frac{1}{i/12}[ 1-\frac{1}{(1+i/12)^{48}}]$ to find the interest rate However there is no explicit solution for i, it is solved by trial and error. The value I get is 9.2418%

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Interest rate conversions can be confusing, so an exact answer depends on the convention rate being used. However, I can get you close. Given a general solution to a series summation: $$\sum_{n=1}^{N} \frac{xn}{(1+r)^n} = \frac{(1 + r - (1 + r)^{-N} (1 + r + N r)) x}{r^2}$$ We can rewrite the value present of annuity which pays 2n units per period as: $$... 2 Another approach is to utilize a full yield curve dynamics such as BGM in order to model the spread between the two types of swaption. This will help determine the zero collar valuation and also will show you that the valuation difference depends on the correlation between different parts of the yield curve : specifically , between the rates used to ... 1 All else being equal, the original debt is simply being repaid over a longer time period. Thus a larger interest income will accrue to the bank as the debtor now pays a higher fraction of his regular cashflows towards interest payments (instead of amortization). At an extreme, an annuity level of A=N/r would mean that the debtor will never amortize the ... 1 The bond engine uses the CashFlows::npv() method internally, so you can do the same after stripping the principal payment from the bond cashflows: cashflows = fixedRateBond.cashflows() coupons = cashflows[:-1] # all except the last includeSettlementDateFlows = False annuity = CashFlows.npv( coupons, spotCurveHandle, False, spotCurveHandle.... 1 This is already answered but you should be more clear as I can see this is a problem for the FM exam. You probably meant i^{(4)}=2\% and the annuity is due (first payment is due immediately), you also need to clarify the amount of each payment which is probably 1 annually payable quarterly (0.25 dollars every 3 months).You can use the formula \ddot a_{n,j}... 1 The sum of 2 annuities of the same length is still an annuity. You have a monthly annuity with PV = 600000+400000 = 1000000 Number of payments N = 24 Amount of monthly payment PMT = 27136.37+19205.15 = 46341.53 Then using the RATE(N,-PMT,PV)*12 function in Excel or similar Annuity function in a financial calculator you find the IRR to be 10.42543% 1 Assuming the annuity pays K every year from year 1 to n, you can write it’s PV as follows: PV=K \left( \frac{1}{1+r} +\frac{1}{(1+r)^2 }+ \dots + \frac{1}{(1+r)^n} \right) And FV, by noting that the first K is invested for n-1 periods, and the last one is received at n: FV=K \left( (1+r)^{n-1} + (1+r)^{n-2}+ \dots+ 1 \right)  Now one just needs to ... 1 I do not follow your analysis. In the case of either type of annuity the FV is equal to the PV times (1+r)^n. This factor is simply the factor which translates any amount in period 0 into an equivalent amount in period n. For an ordinary annuity:$$PVA=PMT \frac{1}{r}[1-\frac{1}{(1+r)^n}]$$When this value is "transferred" to period n by multiplying ... 1 In Wolfram Alpha language ... 2.7∗10^6*(1+x)^12 -75000*(1+x)^8 +50,000 -3.1∗10^6 = 0 ... gives x≈0.0124671 per month, which is 16.03% per annum. I.e. All incomings and outgoings must add up to zero, after adjusting for the monthly interest rate "x" over the number of months invested. 2.7m remains in the fund for the full 13-1 = 12 months, and would ... 1 A simple query on google could have given you the answer... Let's define lumpsum q periodic contribution a y periods a periodic rate i$$q*(1+i)^y + a( ((1+i)^y-1) / i ) - a = f$$Suppose we do not want an initial investment q=0. 2040 - 2016 = 24 years. As you want to know the monthly contribution, everything needs to be converted to months. Thus 24 ... 1 Assume that "I" was born on 1st of January. I pay for 40 years and receive payments for 36 years (65 to 100). So I pay when I am$$25, 26 \dots 64$$and receive payments when I am$$65, 66,\dots 100. Hence the PVs of the annuities are: \$\begin{equation} PV_{paid} = \frac{2000}{0.05}\left(1-\left(\frac{1}{1.05}\right)^{40}\right) = 34318.1727 \...

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