When giving a particular class in financial mathematics for a student I saw a problem in a list of exercises that says:
How to calculate the price at 15 December 2010 of a bond paying a coupon of 11.04 % 2 times a year knowing that the coupon payment days are
Coupon date
1 15 March 2011
2 15 September 2011
3 15 March 2012
4 15 September 2012
5 15 March 2013
6 15 September 2013
7 15 March 2014
8 15 September 2014
9 15 March 2015
10 15 September 2015
11 15 March 2016
12 15 September 2016
13 15 March 2017
14 15 September 2017
15 15 March 2018
16 15 September 2018
17 15 March 2019
18 15 September 2019
19 15 March 2020
and having the following Yield curve:
Yield curve
Term (D) Rate
1 0.0051505
9 0.0051179
16 0.005344728
23 0.005602964
33 0.00560621
64 0.006237435
92 0.006553657
124 0.006818637
153 0.00702155
184 0.00721352
215 0.007444632
245 0.007695142
278 0.007965138
306 0.0082047
337 0.008453748
369 0.008692272
460 0.009484277
551 0.010313827
642 0.011160763
733 0.01202446
1098 0.015414247
1463 0.018648449
1828 0.02158657
2196 0.02408674
2560 0.026161802
2924 0.027907447
3289 0.029391256
3655 0.030649823
My background is more theoretical (probability and etc) and don't have to much knowledge of financial products as you can see by my question. Could you help me to understand how to compute it in a practical way simple to explain for a student. But first for me, because I lack of practical knowledge.
I know that if we have a interest rate process $(r_t)_{t\geq 0}$ then the Zero-Coupon bond price at date $t$ of maturity $T$ is given by $P_t = \mathbb E_t^{\mathbb Q}\left [exp(-\int^T_t r_s ds ) \right ]$ where $\mathbb Q $ is risk neutral measure. Then the Coupon-bearing bond price is given by $$ P_0 = \sum_{i=1} ^n cP_0(T_i) + F P_0(T) =\sum_{i=1} ^n \rho F P_0(T_i) + F P_0(T) $$
where $F$ is the face value, $\rho $ is the pre-assigned interest and $T_1\leq \cdots \leq T_n \leq T$ are the dates where coupon are paied.
Also I know that the Yelds is given by
$$ R_t(T) = \frac{-ln P_t(T)} {T-t}$$
since $P_t(T) = \exp(-(T-t) R_t(T))$.
So should I just comput $P_o$ by this relations or Am I mistaken about the Yield curve definition? (I think there are other kinds of Yields right ?
Another question comes out. While searching for an answer to this question I reviewed some theoretical material about the models but also some piratical ones. In one of the practical materials I've seen a path simulation for a stock spot price using the forward rate instead of the short rate. In the context they had the same initial date them I have in my problem: an yield curve. Why should be the forward rate a better approximation for the short rate than the yield curve (supposing that you interpolate it if necessary in order to have a more refined and smooth approximation) ?
I know that the short rate $r_t$ is given by $r_t =\lim_{\tau \to t} f_t(\tau) =\lim_{\tau \to t} \nabla_\tau [R_t(\tau)(\tau-t)]$ (supposing that those limits exist!)
Thanks in advance for yours advices.