Use Matlab to Understand Bonds Valuation

Use Matlab to get an intuitive understanding of bonds valuation.

1. Basic knowledge:
1.1 The price equation and its six contributing factors
As we know, there are six factors that determine the expected price of bonds: the par value(F), the maturity(n) the yield to maturity(y), the coupon interest(CF), the interest payment frequency(m), and the interest rates for each period(ri).

We assume that the coupon interest is fixed, then the price of bonds(P)is the discounted cash flows of each period:
P=i=1nm(CF(1+ym)i)+F(1+ym)mn=i=1nm(CF(1+rim)i)+F(1+rnm)mn

Indeed, to intuitively understand this equation is not easy. Firstly, we can see that the yield to maturity(y) is a kind of “average” of the interest rates(ri) for each period.
Thus, when one of the interest rates goes up, so does the YTM(yield to maturity).

We can simplify this equation: P=fy,n,m,F,CF=fri,n,m,F,CF.

So, there are at last five variables in each equation. In each bonds, the par value(F), the interest(CF), and the way to pay interest(m and n)are predetermined. Thus the change in price is mainly due to the change in YTM, in other words, due to the change in interest rate, thus we can simplify this equation to: P=fy|n,m,F,CF= fri|n,m,F,CF or
P=fy= fri

1.2 For example (This example will be used throughout the paper):
The Pembroke Co. wants to issue the bonds that have:
Par value(F):$100
Coupon interest(CF): $10
Coupon interests are paid annually (m=1),
Maturity(n): 5years

Thus P=fy=i=15(10(1+y)i)+100(1+y)10

2 The trend of YTM:
2.1 relation between YTM and interest rates in future periods:
The relation between the YTM and periodic interest rates is not so easy to understand, from the equation:
P=i=1nm(CF(1+ym)i)+F(1+ym)mn=i=1nm(CF(1+rim)i)+F(1+rnm)mn

We can see that the YTM have a positive relation with each period’s interest rate, the YTM is a kind of “average” of each period’s interest rate. In order to see how YTM changes in future, we consider