Consider the future value of a single sum formula given as:
(2.1)
FV PV APR
t k
k t k
= +
×
1
This chapter coauthored with Timothy Crack and Nelson Lacey.
Bond Price under Continuous Compounding 17
where tis the holding period given in number of years, APRkis the annual percentage rate with kcompounding periods over one year. Federal regula- tions require that all quotes of interest rates be given by using an APR.Ob- viously, the APR quote must be given together with the compounding frequency k,as the compounding frequency affects the future value in equa- tion 2.1. To appreciate the importance of compounding frequency, consider a student who in a desperate moment borrows $1,000 for one year from a pawnbroker at an APR12of 300 percent. Not having read the small print carefully, she goes back to return the sum of $4,000 ($1,000 principal plus
$3,000 interest) to the pawnbroker at the end of the year. To her dismay, she finds that she owes $14,551.92 instead, which is $10,551.92 more than what she thought. The pawnbroker shows her the following calculation:
Since interest on interest, and interest on interest on interest, and so on are higher with more frequent compounding, it leads to a higher future value in equation 2.2. Of course, the reason monthly compounding fre- quency makes such a huge difference is because we assumed an unusually high APR12equal to 300 percent. At more reasonable values of APR12like 10 percent, the compounding frequency would not have made such a big difference.
The most common compounding frequencies in the fixed-income mar- kets are annual, semiannual, monthly, and daily. It is always possible to find equivalent APRsunder different compounding frequencies, such that the fu- ture value remains the same in equation 2.1. To allow mathematical tractability, it is often easier to use an APRwith continuous compounding, where interest on interest is paid out continuously, or with infinite com- pounding intervals in a year. Let yrepresent the APRassuming continuous compounding. Then by using the compounding rule, equation 2.1 can be rewritten as:
Since kgoes to infinity, applying the exponential constant e,equation 2.3 can be rewritten as:
(2.4) FVt =PV×eyt
(2.3)
FV PV y
t k k
t k
= +
→∞
×
lim 1
(2.2) FVt = +
× =
(
+)
=1000 1 300
12 1000 1 0 25
1 12
% 12
. $114 551 92, .
If an APRis quoted with a compounding frequency k,then an equiva- lent APRunder continuous compounding can be given as follows:
Equation 2.5 follows by equating the right sides of equations 2.1 and 2.4, and then taking logarithms of both sides of the equation.
As an example, given APR12(i.e., an APRwith monthly compounding), the continuously compounded APRis given as:
By dividing both sides of equation 2.4 by etywe get the present value of a single sum as follows:
By applying the present value rule (equation 2.7) to every cash flow of a bond, the price of a bond with a periodic coupon Cpaid ktimes a year, and face value F,is given as follows:
where t1, t2, t3, . . . , tNare the Ncash flow payment dates of the bond. As- suming the bond matures at time tN=T, and the time intervals between all cash flow payments are equal, then N=Tk, and t1=1/k, t2=2/k, t3=3/k,. . . , tN=N/k.Substituting these in equation 2.8, the bond price can be expressed by the following formula:
where i=y/kis the continuously compounded APRdivided by k.
(2.9)
P C
e e
F
i Ni eNi
= − −
+ 1 1 1
(2.8)
P C
e C e
C e
C e
F
yt yt yt ytN eytN
= 1 + 2 + 3 + . . . + +
(2.7) PV FV
e
t
= yt
(2.6)
y APR
= +
× ln 1 1212 12
(2.5)
y APR
k k k
= +
× ln 1
Bond Price under Continuous Compounding 19
Unlike bonds, annuities like the mortgage loans do not make a lump- sum payment at the maturity date. Setting F=0 in the above equation, the annuity formula is given as follows:
Perpetuities are annuities with infinite life. Setting N=infinity, in equation 2.10, the perpetuity formula is given as:
Note that equations 2.9, 2.10, and 2.11 assume that all variables are defined in periodic units. For example, if a fixed-income security, such as a bond, annuity, or a perpetuity, paid out coupons semiannually (the most common scenario for U.S. bonds), then equations 2.9, 2.10, and 2.11 can be used with the variables defined as follows:
where C=Semiannual coupon
i=y/2 =Continuously compounded APRdivided by 2 N=Number of semiannual coupon payments
Similarly, if a fixed-income security paid out coupons monthly (e.g., mortgage loans or MBS), then equations 2.9, 2.10, and 2.11 can be used with the variables defined as follows:
where C=Monthly coupon
i=y/12 = Continuously-compounded APRdivided by 12 N=Number of monthly coupon payments
Example 2.1 Consider a 30-year home-equity loan with 360 monthly pay- ments (i.e., 30 × 12=360) of $100. Suppose that the quoted APR with monthly compounding for the loan is 6 percent and we wish to calculate y, the continuously compounded APR.Using equation 2.5, this yield is calcu- lated as:
y=ln(1+APRk/ )k × =k ln(1 0 06 12+ . / )×12=0.05985005
(2.11)
P C
ei
= −1
(2.10)
P C
ei eNi
= − −
1 1 1
The present value of the loan can be computed in two different ways. Using the discrete monthly rate=APR/12=0.06/12=0.005, the loan’s present value is given as:
Using the continuously compounded yield, y=0.0598505, the loan’s pres- ent value is given by equation 2.8 as follows:
Since both approaches give identical answers, we can use the second ap- proach based upon continuous compounding, which turns out to be more tractable mathematically. Throughout this chapter and for much of this book, we will use continuously compounded yields.
We do not have to do a summation of the 360 terms as shown above.
The present value of the mortgage loan can be computed directly by using the formula in equation 2.10, with C=$100, i=y/12=0.0598505/12= 0.00498754, and N=360, as follows: