# Bond is maturing in 10.25 years, YTM calculation

Bond is maturing in 10.25 years and has an annual coupon rate 4.15% paid semiannually and price 92-12+

I need to calculate yield to maturity

Ok so I know that 92-12+ is basically 92 + 12/32 + 1/64 = 92.390625.

Now here comes the yield to maturity confusion. Normally I'd expect to have the maturity time to be divisible by the compounding rate, but 10.25 is not divisible by 0.5 which confuses me.

The formula for YTM (let $$y$$ be the YTM) if my memory serves me well is this:

$$\sum_{n=1}^{20} \frac{2.075}{(1+0.5y)^n} + \frac{100}{(1+0.5y)^{20.5}} = 92.390625$$

But the last summand confuses me abit. Is this actually correct?

The first coupon occurs 0.25 years from now, the next 0.75 years from now, the twentieth and last 10.25 years from now. So no, the equation is not correct. You can only use your formula on a coupon date and today is not a coupon date. You could calculate PV at time 0.25 and then discount to the present (i.e. half a period):

$$\frac{1}{1+0.25y}\left(\sum_{n=1}^{20} \frac{2.075}{(1+0.5y)^{n-1}} + \frac{100}{(1+0.5y)^{19}}\right) = 92.390625.$$

• Thank you very much for your reply. Let me just make sure of something: is it correct to discount like that provided that our rate isn't continuous? As in, I was sure that it only worked for continuous compounding, I guess I should go back to the textbooks ... I mean, I remember that there is a relation for rates, like $1 + x = (1 + 0.5x)^2$ if we want to find the semi-annual rate from the annual rate. Shouldn't we apply the same here if we want to discount by a quarter using the semiannual rate $y$? – Makina Dec 16 '18 at 18:17
• The way I did it is the US convention, I believe. For other countries it may be different.... These yield calculations are complex, best to use standard, tested, software if you can. – Alex C Dec 16 '18 at 18:37
• Ok, also, I believe if first coupon occurs 0.25 years from now, the 20th will occur 9.75 years from now, which turns into 9,5 (hence, 19th power for the last term) and the 0.25 discount outside. – Makina Dec 16 '18 at 19:24