# Tag Info

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You seem to have two distinct problems: How to generate random portfolios How optimal portfolios are structured Ad 1) A straightforward way to simulate the weights of random portfolios is to use the Dirichlet distribution $Dir(\alpha_1,\ldots,\alpha_n)$. This is a distribution on the Simplex (i.e. on $S=\{x\in\mathbb{R}^n | \sum x_i =1, x_i\geq 0\}$, ...

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No need to invent your own algorithm for random portfolio weights. There is a very simple algorithm to generate a random point in a simplex (i.e. to generate $e_i,i=1,k$ such that $e_i\ge 0$ and $\sum_{i=1}^k e_i=1$). It is due to Rubinstein and Melamed (1998): Generate k independent exponential random variables $Y_1,\cdots,Y_k$ (for example they can be ...

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The equation to be solved should be $w_1 D_1 + w_2 D_2=0$ where $D_1$ and $D_2$ are the respective durations of the two bonds. However you need an investment constraint to fix the values of $w_1$ and $w_2$. Hence you also need $w_1 P_1 + w_2 P_2 = \Pi$ where $\Pi$ is the amount invested. You can then subsitute $w_1 = - w_2 D_2 /D_1$ into the second ...

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First assume you have been given/you know the shocks scenarios. Ideally you would have these scenarios in term of shifts/movements- e.g., curve shifts by $a+bT$. So what I would do is to price the products using the current market interest rate data. Then apply the shifts to the curve and then re-price the products. The change in price is the main object of ...

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Do you have correctly formulated the problem for the solver ? If you want to maximise a function (the sharpe ratio) $f$, it is equivalent to minimise $-f$. This kind of confusion (minimising instead of maximising) would basically lead to a similar outcome as yours.

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Carry is most often defined as the effect on the bond if the yield curve does not change. Roll down is seen as a component of carry that results from changes on the position of the yield curve. See this reference on the topic.

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You do it the same way as with long only as weighted sum of the durations of each position. You have two possibilities for calculating the weights: long/short with respect to a benchmark: then take as basis the dollar value of your portfolio P USD, set the benchmark to the same dollar value and calculate each weight as fraction of the position $w_i = P_i/... 1 One way might be to calculate a proxy yield based on peer group metrics such as credit rating and currency. This won't however make any allowance for the liquidity premium, but nonetheless, it might still be a useful approximation. If the credit rating history is not available, then you might have to use something like the KMV model (part of Moody's ... 1 Assume that there are two zero coupon bond with maturities$N_1$and$N_2$with prices$P_1 = \frac{CF_1}{(1+y)^{N_1}}$and$P_2 = \frac{CF_2}{(1+y)^{N_2}}$respectively. If we construct a bond portfolio by purchaing one each of the two ZCB, the price of the portfolio is$P=P_1+P_2$. Now, the convexity of the portfolio is$\begin{align} {Convexity}_p &= ...

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Ok, so I think you are just asking what is the dv01 of the bond. So if the yield goes up one bp what's the new price? And if it goes down, what's the new price? That's the simple way that people look at it. For a bond that can't be called or converted in any way it's pretty easy. Let's assume that's what you have. Here's the process: So first you need ...

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Your question depends on the discount factor you wish to use for pricing. If u use the risk-free rate (from the bond), it wouldn't be in line with the no-abitrage condition to assume an risk neutral agent can't/wouldn't invest in bonds to carry money into next period. To understand this: just assume a 1 period model with two outcomes for S, where both ...

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This question complicates a simple issue. Model e.g. Italian sovereign as a credit, and then treat the spread either against Germany or IRS curve, and stress as you would any financial/non-financial credit risk. The underlying benchmark curve would naturally fall onto the interest rate risk.

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