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13

The PCA analysis does not really tell you what the bonds do but it tells you how the rates move together. The variations of $n$ rates (i.e. 1 y, 2y, ...) are split up in (at first) abstract factors like $$ \Delta R_i = \sum_{j=1}^n e_{i,j} f_j $$ where $\Delta R_i$ is the change in the rate $i$ and $f_j$ is factor $j$ and $e_{i,j}$ is the (factor loading=) ...


12

Let $P_t$ be the price of the overall market index at the end of quarter $t$ Let $D_t$ be the dividend for the overall market in quarter $t$ Let $X_t = \frac{D_t}{P_t}$ be the dividend to price ratio. Two key concepts in time-series statistics are stationarity and ergodicity. If the dividends to price ratio is a stationary, ergodic process, then dividend ...


9

Maybe I am a little bit late to the party, but I want to give a shot. As in Campbell and Shiller, start from the identity $R_{t+1}\equiv\frac{P_{t+1}+D_{t+1}}{P_t}$ where $R_{t+1}$ is the gross return between time $t$ and $t+1$, and $P_t$ is the price at time $t$. Rearrange the relationship as $R_{t+1} =\frac{D_{t+1}}{D_t}\frac{\left(1+\frac{P_{t+1}}{D_{t+1}}...


7

Just to elaborate on the comments above to include some visuals. As you pointed out, the high coupon, seasoned 10.625s traded at a steep discount. The first chart below shows the yield spread against 4.25s; the spread blew up to 80 bps at one point in 2008: This phenomenon was not unique to these two bonds. Toward the end of 2008, many Treasuries traded out ...


5

For a US investor to hedge the bonds the investor would (1) Buy EURUSD in the Spot market, (2) Buy the German bonds with the EUR proceeds, (3) Short EURUSD in the forward market to provide a guaranteed repatriation rate when the bonds mature (thus avoiding FX risk). Currently the two year forward exchange premium/discount for the EURUSD is 532 forward ...


4

In practice, bonds of the same maturity will have yields that vary slightly from each other. Several possible reasons (a) a bond with a higher coupon is effectively shorter maturity than a bond with lower coupon, because a higher percentage of the cash flows are returned earlier. So if the yield curve is upward sloping, high coupon bonds will yield a bit ...


4

Unless all of your yields are par yields (yield of bonds trading at par), you'll get very unreliable results if you fit your curve using yields alone. This is because yields can be distorted by the coupon effect – given two bonds maturing on the same day and assuming the yield curve is upward sloping, a higher coupon bond will always have lower yield. What ...


4

It's simpler to just think of the yield to maturity as the internal rate of return of the bond given the current price. It's like the discount rate you would apply to the final payout and coupons, such that the result is the market price. A short paper by Forbes, Hatem, and Paul explains that yield to maturity ignores reinvestment. Strictly speaking, yield ...


4

Carry and roll-down are two different measures. The carry is the PNL resulting from holding a position. However, even if you don't finance the bond in repo, you can still measure your carry as the yield-to-maturity of maturity of the bond vs the yield of the alternative default investment you would have made with your cash (for example 0% if sitting on ...


4

The formula you quote (forward minus spot) is the yield carry for a financed position. The problem is that different people use the word carry to mean different things. The most commonly used convention, at least when we prepare analytical reports and quote sheets, is to use the word "Carry" to refer to the breakeven measure – it tells us how much yield ...


4

Let $P$ denote the dirty price, $F$ the face value and $i$ the YTM. Using the geometric sum we get \begin{align} P &= \sum_{j=1}^n \frac{C}{{(1+i)}^j} + \frac{F}{(1+i)^n}\\ &= C\frac{1-\frac{1}{{(1+i)}^n} }{i} + \frac{F}{(1+i)^n} \end{align} and thus \begin{align} P=F \Leftrightarrow & F= C\frac{1-\frac{1}{{(1+i)}^n} }{i} + \frac{F}{(1+i)^...


4

When the market enters a risk-off period the investors proceed to a rotation between more risk assets (commodities, equities etc...) to the less risky ones. At this point there is just a lot of supply/demand imbalance on the bonds which drives the yield of the 10y down When investors proceed to "flight to quality" they want to protect themselves against ...


4

Let's assume we have yearly cash flows, and let's focus on just two years - year 1 and year 2. Let $R_1$ and $R_2$ represent the zero rates of year 1 and year 2. So if you want to borrow for one year, you pay $R_1$ percent, and if you want to borrow for 2 years, you pay $R_2$ percent per year. So in an upward sloping scenario, these will look like this: Now,...


4

I would not say that this is universally acknowledged but here is my view: Instead of considering par rates, i.e. 10Y and 20Y, consider forward rates, such as 10y and 10y10y. The useful difference here is that forwards do not 'overlap' and therefore incorporate aspects of each other into the price. A 20Y is >50% directly dependent upon the 10Y price for ...


4

Suppose 40yr bond and 30yr bond have the same yield. It is a mathematical fact as @attack68 has pointed out, that the convexity of the 40yr is greater than the convexity of the 30yr bond. So consider the following strategy ; long the 40 yr bond and short the 30yr bond with the same dv01. Then every time the market moves, you make money (get longer when ...


4

It's an interesting question. The fundamentally devout macro wannabe-strategist within cries out for a long-term growth/inflation expectation narrative. However, the cynical realist within reminds that although the market does make long-term predictions thus because it has to create prices then, there is no latent consensus that the world will really look so ...


4

To get the bond yield from the price: import QuantLib as ql maturity = ql.Date(30, 1, 2030) coupon = 0.03 issueDate = ql.Date(30, 1, 2019) frequency = ql.Semiannual dayCount = ql.Thirty360() price = 104.5 bond = ql.FixedRateBond(2, ql.TARGET(), 100.0, issueDate, maturity, ql.Period(frequency), [coupon], dayCount) yld = bond.bondYield(price, dayCount, ql....


3

He's talking about central bank intervention in the longer maturities, not the short end. The Fed bought a lot of long dated Treasuries, which helped flatten the curve. Hence an inverted curve may reflect all the securities bought by the Fed.


3

For clarity, I'll use two expressions, "liquidity premium" and "illiquidity premium": "Liquidity premium" arises when investors value the liquidity profile of an instrument so much that they are willing to pay for the enhanced liquidity, thus pushing the price of the instrument above fair value (and its yield below fair value). "Illiquidity premium" arises ...


3

I did some tinkering with the numbers... It seems to me that Wilmott is expressing his answer as a continuous time interest rate. Notice that $e^{0.0658}=1.03347\times2$. That is how your answer 3.347 per period and his answer 6.58 for 1 year can be reconciled. He is working with $e^{rt}$ and you are working with $(1+r)^n$. Your answer is the industry ...


3

Due to the leap year 366 days need to be used here to match UST conventions (which is ACT/ACT). In this case it doesn't matter whether your interest period extends to only 1 day after the 29th of February or, e.g., 200. In fact if you look at the daycount description of the bill it says: "the day count basis for price and yield calculations is 365 ...


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

For simplicity, let us assume continuously compounded zero rates and periodically compounded par yields. If you have to work with continuous rates, you may adapt the formulas accordingly. Using the zero rate discount factors $D(T) \equiv e^{-r(T)T}$, the present value of a coupon bearing bond is \begin{equation} PV=\sum_i^N c D(t_i) + D(t_N) \end{equation} ...


2

Some models do use ln(r_t), like Black–Derman–Toy and the Black–Karasinski models. Mainly to avoid negative interest rates in low rates / high volatility environments through the use of the log-normal distribution. Negative rates can wreak havoc in option premiums for example. They are interest rates indeed, that we call short rates, not yield on treasuries....


2

If you are not able to find a data set, containing the dividend yield information for all the companies listed in ASX20/50/100/200/300, the only way is for you to make it by researching the companies. However I found this dividend yield scan to get you started. Once you have the dividend yield rate for all the stocks in the given index, it is just a matter ...


2

Have you looked at Quantlib.net? We use it both in the back office and some soft realtime trading system for pricing bonds. There are a few questions on this site that deal with using it for pricing bonds. See here: https://quant.stackexchange.com/questions/tagged/quantlib+bond


2

A couple quick thoughts. Do the PCA on changes or log-changes in your series. That is often how PCA is conducted in fixed-income settings. You're large move in wights corresponds to outlier moves in the blue series. Given the assumptions of a PCA, I would consider whether your dataset has suffered from any breakpoints, regime changes or other rare events ...


2

The IRR cannot be used to move the Present Value to a Future Value, because it doesn't represent the rate for that interval of time. It is a complicated average of rates for 1,2,3 years, not the rate for 3 years which you require.


2

while it is true that $$\lim_{T\to\infty} Z(t, T) = \lim_{T\to\infty} e^{-r(T-t)} = 0$$ this is when $r$ is independent of time to maturity, a flat and constant yield curve. In practice, we use yield curves which vary depending on what day they are estimated and what maturity the ZCB is. If in fact $r(t, T)$ depends on today and the maturity then the ...


2

This is something that banks don't do very well (in my opinion), but we can look to the insurance industry for help. Insurance liabilities often span decades, and the regulation has come up with something called the Ultimate Forward Rate (or UFR). It's currently a hotly debated topic with the advent of Solvency II (insurance regulation) coming into effect ...


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