Wikipedia:Reference desk/Archives/Mathematics/2011 October 17

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October 17

complex analysis

is exp(re z) periodic?if so what is its period,z belongs to complex number 14.139.45.245 (talk) 09:05, 17 October 2011 (UTC)

You mean the value of the function is e^x for point x+iy? No that's not periodic in any normal sense, though of course if you follow a circle round the origin the values you get would be periodic. Dmcq (talk) 14:54, 17 October 2011 (UTC)

It is periodic in the sense that $f(x+i)=f(x)$, so it has period i (and any other nonzero pure imaginary number is a period). But it really boils down to the particular definition of periodicity you're using. -- Meni Rosenfeld (talk) 15:51, 17 October 2011 (UTC)

Must remind myself to check the definitions carefully. :) Dmcq (talk) 16:57, 17 October 2011 (UTC)

If z = x + iy then e^re(z) = e^x. For real a and b, e^a = e^b if and only if a = b. The level sets are the lines x₀ + iy, where x₀ is a fixed real number and y varies over all the real numbers. According to the definition of a periodic function, e^re(z) is periodic with period iy for any non-zero real number y. So e^re(z) is a very strange example, rather like the constant real valued functions (which have arbitrary periods). — Fly by Night (talk) 19:57, 17 October 2011 (UTC)

Bond prices and interest rates

Hi all, I'm trying to get better at estimating changes in bond prices due to changing interest rates, and I'm hoping you can help. I wasn't sure whether to file this under maths or economics. First of all, I am aware that bond prices generally move inversely to interest rates!

Basically, I often come across interview questions about changing bond values with interest rates, and want to get better at estimating a situation such as the following:

A 10 year bond with a par value of £1,000 is issued with coupons of 4% paid every 6 months (i.e. 2% every 6 months), with interest rates also at 4% (so if I understand correctly you should be receiving your £1000 back at the end of the bond's life - maybe slightly more to compensate for the risk you take on with the bond): if interest rates rise from 4% to 4.5%, how much does the price of the bond then change by?

Now most simple examples showing interest rates and prices move inversely would say that you need to have a price of (1000*4/4.5 = £889) to get an effective interest rate of 4.5% out of your bond, equal to the market rate. But is it really that simple? Don't we need to somehow account for the fact that our future cashflows will be discounted by the (new?) interest rate, and the maturity of 10Y of the bond should have some effect on the price? I'd be very grateful if anyone could explain a little to help my understanding. What's the best way to work out the price change approximately in your head, assuming your arithmetic is strong enough? Thanks very much everyone, help much appreciated! Delaypoems101 (talk) 21:26, 17 October 2011 (UTC)

Since no financial expert has replied yet, our article on Discount rate gives an example that might be helpful. Dbfirs 18:53, 19 October 2011 (UTC)

The price of the bond is the present value of the future cashflows, using the bond's yield (what you are calling the interest rate) as the discount rate (actually, that's the definition of the yield - the price is determined by the market and then you work out the yield from that). It's easiest with zero-coupon bonds. In that case, there is only one cashflow - the par value and the maturity date. If we call the par value P, the duration (time until maturity) n and the yield i, then the price V is given by V=P*(1+i)^-n.

If there are coupons, then you need to include those too. For example, if there was one couple of amount C paid half way through the period, the price would be V=P*(1+i)^-n+C*(1+i)^^-n/2. When there are lots of coupons, it becomes easier to consider them as an annuity and there are formulae to calculate their present values (the present value of £1 a year paid at the end of each year for n years is ${\frac {1-v^{n}}{i}}$ where $v={\frac {1}{1+i}}$).

If you know the par value, coupons and yield, then you can just calculate the price. What's more common is that you know the price and want the calculate the yield, which can be a little tricky - for anything more than one payment, it's usually easiest to do it numerically (just try lots of different yields until you get one yield that gives you a price that is slightly too high and one yield that gives you a price that is slightly too low and linearly interpolate between them to get a good approximation of the yield). --Tango (talk) 20:17, 21 October 2011 (UTC)