Securities

There are several types of securities, each with its own features and scheduled cash flows. Cash flows can be scheduled at the end of every coupon period or just at the end of the security’s life. If we see a security as an investment, its yield can be viewed as the internal rate of return. The cash flows of a security can consists of periodic payments (equal to a certain percentage of the par value), the coupons, and the future value of the security. In general, the general cash flow equation

\[v_p(1+r)^N + p\sum_{i=1}^N (1+r)^{i-1} + v_f = 0\]

where \(v_p\) is the present value, \(v_f\) is the future value, \(N\) the number of periods, \(p\) is a constant periodic payment and \(r\) is the constant interest rate, holds. AIMMS provides functions the most common types of securities like treasury bills and bonds. However, the present value, future value, periodic payments, number of periods and interest rate are different for each specify security type.

Security Types

We distinguish three main types of securities:

  • securities with zero coupon periods (discounted securities),

  • securities with one coupon period (at maturity), and

  • securities with multiple coupon periods

Discounted Securities

In the case of discounted (or zero coupon) securities such as treasury bills, there are no periodical payments. The only positive cash flow is a fixed redemption at the end of the security’s life. Therefore, only the value of this redemption and the investment made for the security determine its yield. In this case, the present value is equal to the price \(-P\), the price at which the security is bought at the settlement date, there 0 periods (so no periodic payments), and the future value at the maturity date is equal to the redemption \(R\). Thus the general cash flow equation reduces to

\[-P(1+r_yf_{SM}) + R = 0\]

where \(r_y\) is the annual yield of the security, and \(f_{SM}\) is the difference (in fractions of years) between the settlement and maturity date, computed with respect to the specified day count basis method.

Discount Rate

Commonly with discounted securities, the yield is not expressed in terms of the price, but in terms of the fixed redemption. The discount rate is the increase in value per year as a percentage of the redemption. The relationship between the yield \(r_y\) and the discount rate \(r_d\) is given by

\[1 + r_yf_{SM} = \frac{1}{1-r_df_{SM}}\]

which leads to the following equivalent relation between price and redemption

\[-P + R(1-r_df_{SM}) = 0\]

Treasury Bills

A treasury bill is a discounted security with less than one year from settlement until maturity, the number of days in one year is fixed at 360 and redemption is fixed at 100.

Functions for Discounted Securities

AIMMS supports the following functions for securities with zero coupon periods:

One-Coupon Securities

Securities that only pay interest at maturity can be seen as securities with only one coupon period, where the accrued interest increases linearly in time until it is paid (when the security expires), and the redemption equals the par value of the security. In the general cash flow equation,

  • the present value

    \[v_p=-P - v_{\textit{par}}r_cf_{IS},\]

    where \(P\) is the price of the account at settlement and \(f_{IS}\) is the difference between the issue and settlement date (in fraction of years) with respect to the specified day count basis method, to account for the accrued interest from the issue date until settlement,

  • the periodic payment

    \[p =v_{\textit{par}}r_yf_{IM},\]

    where \(r_y\) is the annual yield and \(f_{IM}\) is the difference between the issue and maturity date (in fraction of years) with respect to the specified day count basis method, and

  • the interest rate

    \[r=r_yf_{SM},\]

    where \(f_{SM}\) is the difference between the settlement and maturity date (in fraction of years) with respect to the specified day count basis method.

This results in the following equation for securities with one coupon period:

\[(-P - v_{\textit{par}}r_cf_{IS})(1+r_yf_{SM}) + v_{\textit{par}}r_yf_{IM} + v_{\textit{par}} = 0\]

Functions for One-Coupon Securities

AIMMS supports the following functions for securities with one coupon period:

Multi-Coupon Securities

For securities with multiple coupon periods, interest will be accrued linearly during and paid at the end of each coupon period (i.e. at the coupon date). In the general cash flow equation

  • the number of periods

    \[N=\lceil ff_{SM}\rceil,\]

    where \(f\) is the coupon frequency (number of coupon periods per year), and \(f_{SM}\) the difference between settlement and maturity date (in fraction of years) with respect to the specified day count basis method,

  • the present value

    \[v_p = -P -v_{\textit{par}}\frac{r_c}{f}\frac{f_{PS}}{f_{PN}},\]

    where \(P\) is the price of the security at settlement, \(v_{\textit{par}}\) the par value of the security, \(r_c\) the annual coupon rate, \(f_{PS}\) the difference (in fraction of years) between the previous coupon and settlement date, and \(f_{PN}\) the difference between the previous and next coupon date, both with respect to the specified day count basis method,

  • the periodic payment

    \[p=v_{\textit{par}}\frac{r_c}{f}\]

  • the interest rate

    \[r=\frac{r_y}{f},\]

    where \(r_y\) is the annual yield.

This results in the following equation for securities with multiple coupon periods:

\[\left(-P -v_{\textit{par}}\frac{r_c}{f}\frac{f_{PS}}{f_{PN}}\right)^{N-1+\frac{f_{SN}}{f_{PN}}} + \sum_{i=1}^{N}v_{\textit{par}}\frac{r_c}{f}\left(1+\frac{r_y}{f}\right)^{N-i} + R = 0\]

Functions for Multi-Coupon Securities

AIMMS supports the following functions for securities with multiple coupon periods: