3

You are correct on both questions. 1 you answered yourself. It is the correct rate to close out the trade. 2 you use a dollar discount rate because you are discounting dollars. The (1-K/X) term represents one dollar from the first trade and K/X dollars from the close out trade.


3

I performed spectral analysis on the stock market for disaggregated returns. If $\mu$ is the center of location and anything away from $\mu$ is an "error", then the stock market is in equilibrium once every 20-21 years as an aggregate whole. But, like a musical instrument, the periods of the individual firms could be relatively small. Still, that would ...


3

That formula is algebraically equivalent to saying different, stochastic assets can have different expected returns. $$ \mathbb{E} \left[ R_i \right] = r_f + \gamma_i $$ Some simple algebra Let $X_i$ be a random variable denoting a risky cash flow, $p_i$ be today's price of that risky cash flow, $r_f$ be the risk free rate, and $\gamma_i$ be some risk ...


3

Carry is typically only associated with known cashflows - its closely related cousin, roll, is typically associated with unknown cashflows, assuming the state of the world is unchanged. Given this, carry is typically only analyzed for the current period of the swap or bond. If we assume your swaps are fixed Semi vs a 6m Ibor index, then the natural period ...


3

You have to discount the future cash flows with your discount rate (= yield) The coupon is paid quarterly, yet you discount annually. You will miss all other coupons in your calculation.


3

As long as your market is complete and $\tau$ is measurable w.r.t. the filtration generated by the market the continuous cash flow paid until $\tau$ is a hedgeable contingent claim and you have to work under the risk neutral measure.


3

I think you have a little misunderstanding about treasury futures. I would get this book: http://www.amazon.com/Treasury-Bond-Basis-Depth-Arbitrageurs/dp/0071456104?ie=UTF8&psc=1&redirect=true&ref_=oh_aui_search_detailpage It is the absolute best guide to this product. A few important things to understand: Every treasury future has ...


3

If the cash flows depend on the (random) interest rates, then the $C_i$ are random variables and so would be the sum $\sum\limits_{i=1}^n C_id(i)$. However, initial market prices need to be constants, they cannot be random (because you need to know how much this claim is worth right now). The $d(i)$ are real numbers though since they are discount factors (...


2

Why don't you calculate the IRR of each investment? (aside from all the issues with IRR).


2

I have laid out below one way of solving this kind of problem. You have your timeline right and I have reproduced it with the correct amounts. The way to discount your 30Ks is the same as discounting 1,500K if you do it this way. Basically, you need to compute a discount factor. To calculate this discount factor, you need to de-annualize your interest rate ...


2

We are comparing two situations: (1) an all equity firm, versus (2) the same firm which has decided to do a predefined amount of borrowing. Because the tax shields arise as a result of the borrowing decision (i.e. they are a potential advantage of borrowing) they are discounted at the borrowing rate. It is treated almost as a separate project: to borrow ...


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

Pricing always takes place under the risk neutral probability measure. In fact, this would make the price more conservative (i.e. lower) with respect to risk; if you priced it under the true measure you would be putting a smaller hazard rate for this random time. Completeness make the risk neutral probability measure unique. In your case you might have ...


2

The conversion factor associated with each bond the futures' delivery basket is constructed such that the invoice prices of the bonds are identical under the assumption that the yield curve is flat at the level of the futures' notional coupon. Therefore, the bond with the highest duration will be the CTD when yields are above the notional coupon and the bond ...


2

First, I am not sure which exact statement was made. Also, you cannot just say "without CF" because you are essentially creating an artificial market with messed-up utility. In summary the cheapest-to-deliver bond is: The bond that results in the smallest loss or greatest profit for the futures seller. Futures sellers have to buy the bonds they are going ...


1

Let's consider a single cash flow CF $PV = (\frac{1}{1+i})^n CF$ As you wrote $v = -\frac{1}{PV} \frac{d PV}{di}$ Taking the derivative of PV with respect to i and plugging it in: $v= - \frac{(1+i)^n}{CF} n \frac{1}{(1+i)^{n-1}}\frac{-1}{(1+i)^2}CF$ after simplifying we get $v = \frac{1}{1+i}n$ (which is easy to remember, no need to derive it every ...


1

Another way to write: IRR(A) = x and IRR(0,0,A) = x is: PV(A;x)=0 and PV(0,0,A;x)=0 where PV=present value, and x is the discount rate. Since we are using the same discount rate x, we can just add these up: PV(A,0,A;x)=0 which means that IRR(A,0,A) = x Is it clear?


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

“You can't compensate for risk by using a high discount rate." - Warren Buffett at the 1998 Berkshire Hathaway Shareholder Meeting The simple answer to your question is, “yes, many implementations of discounted cash flow analyses which adjust the discount rate for risk are double counting”. This practice is pervasive in academia, but has no basis in the ...


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

Yes, the IRR is the same regardless of the prepayment date. The only thing that varies is the life of the loan.


1

Your thought process on valuing the tax savings to the whole enterprise at the cost of capital makes perfect sense to me. While I get the thought process on tax savings being funded by debt, the cash flows due to tax savings do not flow back to debt, but rather to the enterprise whether or not the firm is highly leveraged. I think a simple scenario ...


1

I know this question has been answered but I am giving a complete derivation to assist others as well as serve as my notes. We will take a firm with a free cash flow $\mathcal F$ every year perpetually. This is a simplification that can easily be generalized to variable payments. Unlevered Firm The earnings by the stockholders are reduced by taxes. Thus ...


1

So, you have this: $$ \sum_{k=1}^{k=15} 0.8 \cdot 1.06^{2k} = 5.3456\ldots $$ And you want to know if there's a formula, or closed form. Yes there is. $$ \sum_{k=1}^{k=n} x^{k} = x \frac{x^{n+1}-1}{x-1} $$ Where, we're going to set $x=1.06^{-2}$, and sum from 1 to 15 (since you're not including the first period in the valuations). $$ 0.8 \sum_{k=1}^{k=...


1

In basic instruments, one can ignore the past fixings and price purely on the future cashflows. Hence the term Net Present Value. With more exotic stuff such as range accruals, the past fixings are used to calculate the future payoff. In this case, to find the NPV at an intermediate date between the start date and expiry, you typically need to enter the ...


1

The idea of considering past cash flows into an NPV calculation is rather adventurous, in my opinion. If an investor wants to invest money into a financial instrument that has already generated positive cash flows before making his investment (e.g. investing into a bond when some interest payment dates have already passed), the investor would not consider ...


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

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 ...


1

The annuity method is the correct method. I am not familiar with wolframalpha but I assume it is correct. Look at it this way: in the second case (take out 481,000, repay 500,000 after 100 days) you have full use of the borrowed 481,000 for 100 days. In the first case (takeout 481,000, repay with an annuity of 10 payments over 100 day) you effectively ...


1

In a case like this, where the settlement date is in the middle of the coupon period, it is not right to use PV = -110 (minus the purchase price) in Step 3. Instead you should increase the purchase price by the accrued interest, which is a fraction of the coupon based on how far the settlement date is within the current coupon period. (So for ex if you are ...


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