Introduction to Personal Finance Calculations Using the Finance Package
The Finance package assists you in performing financial calculations and financial modeling. This example worksheet introduces the personal finance commands in the Finance package. With these commands, you can calculate the present value and the accumulated value of annuities, growing annuities, perpetuities, growing perpetuities and level coupon bonds. Moreover, it can also help you compute the yield to maturity of a bond. You can construct an amortization table, determine the effective rate of interest for a given compound interest rate, and find the present value and the future value of a fixed quantity for a given compound interest rate.
Note: All examples use dollars ($) and all interest rates are in terms of percent (%). The default setting for floatingpoint precision is 10.
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$\mathrm{with}\left(\mathrm{Finance}\right)\:$


Amortization Method


The most common method of repaying interestbearing loans is the amortization method. This procedure is used to liquidate an interestbearing debt by a series of periodic payments, usually equal, at a given interest rate. Maple can determine how many payments are required to pay off the loan. You can also create amortization tables.
Consider a debt of $100, with interest at 10% per annum, which is to be amortized by payments of $50 at the end of each year for as long as necessary.
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$A\u2254\mathrm{amortization}\left(100\,50\,0.10\right)$

${A}{:=}\left[\left[{0}{\,}{0}{\,}{0}{\,}{}{100}{\,}{100}\right]{\,}\left[{1}{\,}{50}{\,}{10.00}{\,}{40.00}{\,}{60.00}\right]{\,}\left[{2}{\,}{50}{\,}{6.0000}{\,}{44.0000}{\,}{16.0000}\right]{\,}\left[{3}{\,}{17.600000}{\,}{1.600000}{\,}{16.000000}{\,}{0.}\right]\right]{\,}{17.600000}$
 (1.1) 
The list object returned from the above command is displayed in a Matrix below, along with descriptive headings. We see that you must make three payments: $50, $50, and $17.60. The second object returned above, $17.60, is the cost of the loan.
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$\mathrm{amortization\_table}=\mathrm{Matrix}\left(1+\mathrm{nops}\left(A\left[1\right]\right)\,5\,\left(i\,j\right)\→\mathrm{if}\left(i\=1\,\left[\'n\'\,\'\mathrm{Payment}\'\,\'\mathrm{Interest}\'\,\'\mathrm{Principal}\'\,\'\mathrm{Balance}\'\right]\left[j\right]\,A\left[1\right]\left[i1\right]\left[j\right]\right)\right)$



Annuities


Maple can find the present value of ordinary simple annuities. Suppose that you want to find the present value of an annuity paying $100 per annum for 5 years, starting 1 year from now, at an annual interest rate of 10%.
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$\mathrm{annuity}\left(100\,0.10\,5\right)$

To find the accumulated value of the same annuity at the end of 5 years, take the result and multiply it by ${1.10}^{5}$.
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$\cdot {\left(1.10\right)}^{5}$

Consider a growing (increasing) annuity that pays $100 at the end of the first year, then grows at 5% per annum. It is a 5year annuity and the annual interest rate is 10%.
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$\mathrm{growingannuity}\left(100\,0.1\,0.05\,5\right)$

If the interest rate changes to ${j}_{12}\=10$% and the growth rate is unknown (call it $g$), then the future value is given by the formula below.
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$\mathrm{growingannuity}\left(100\,\frac{0.1}{12}\,g\,5\cdot 12\right)$

$\frac{{100}{}\left({1}{}{\left({0.9917355375}{\+}{0.9917355375}{}{g}\right)}^{{60}}\right)}{{0.008333333333}{}{g}}$
 (2.4) 
As a final example, analyze the case in which the payments per time period are not fixed. Suppose that you want to find the present value of variable revenues expected from a project. The project expects $200 in revenue in year 1, $150 in year 2, and $100 in year 3. The opportunity cost of capital is 7.8%.
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$\mathrm{cashflows}\left(\left[200\,150\,100\right]\,0.078\right)$

You may generalize the above result. If the discount rate is $r$%, then the present value of the benefits earned from the project is given by the command cashflows.
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$\mathrm{cashflows}\left(\left[200\,150\,100\right]\,r\right)$

$\frac{{200}}{{1}{\+}{r}}{\+}\frac{{150}}{{\left({1}{\+}{r}\right)}^{{2}}}{\+}\frac{{100}}{{\left({1}{\+}{r}\right)}^{{3}}}$
 (2.6) 


Bonds


When a corporation or government needs to borrow a large sum of money for a reasonably long period of time, they issue bonds that they sell to investors. The bond's yield rate is the income divided by the amount invested.
A $1000 bond that pays interest at ${j}_{2}\=10$% (the bond rate) is redeemable at par at the end of 5 years. Suppose you want to find the purchase price of the bond to yield an investor 14% compounded semiannually. (Note: The yield rate always comes before the coupon rate.)
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$\mathrm{levelcoupon}\left(1000\,\frac{0.14}{2}comma;\frac{0.10}{2}comma;5\cdot 2\right)$

The result above shows that the bond is purchased at a discount, because the opportunity cost of capital is higher than the bond rate.
Try a more complicated example. A $5000 bond, maturing on September 1, 2017, has semiannual coupons at 13%. Find the purchase price on March 1, 1996, to guarantee a yield of ${j}_{2}\=12.5$%. (Note: There are 43 payment periods.)
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$\mathrm{levelcoupon}\left(5000\,\frac{0.125}{2}\,\frac{0.13}{2}\,43\right)$

We see that the bond was purchased at a premium.
Suppose that you want to find the yield rate to maturity of a bond. Suppose that a large corporation issues a 15year bond that has a face value of $10,000,000, and pays interest at a rate of ${j}_{2}\=10$%. If the purchase price of the bond is $11,729,203.32, the yield to maturity for the bond is found by the yieldtomaturity command.
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$\mathrm{yieldtomaturity}\left(11729203.32\,10000000.00comma;\frac{0.10}{2}comma;30\right)$

That is, approximately 4% per halfyear, or ${j}_{2}\=8$%.


Effective Interest Rates


For a given nominal rate of interest ${j}_{m}$ compounded $m$ times per year, the annual effective rate of interest is the rate j which, if compounded annually, will produce the same amount of interest per year.
Suppose that you want to calculate the annual equivalent rate $j$ corresponding to ${j}_{2}\=10$%.
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$\mathrm{effectiverate}\left(0.10\,2\right)$

which is 10.25%.
Compute the annual effective rate of interest to ${j}_{365}\=13.25$%.
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$\mathrm{effectiverate}\left(0.1325\,365\right)$

which works out to be about 14.17%.
The effective annual rate of interest corresponding to ${j}_{m}\=r$% is
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$\mathrm{effectiverate}\left(r\,m\right)$

${\left({1}{\+}\frac{{r}}{{m}}\right)}^{{m}}{}{1}$
 (4.3) 
As another example, to find the rate ${j}_{4}$ equivalent to ${j}_{2}\=10$%.
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$4\cdot \mathrm{effectiverate}\left(\frac{0.10}{4}comma;\frac{2}{4}\right)$

which is approximately 9.88%.
Recall that ${j}_{m}$ is the annual interest rate that is compounded m times per annum. The continuous compound rate is the nominal interest rate that is compounded without limit, or continuously. Typical notation for this is ${j}_{\infty}$. For instance, the annual effective rate of interest equivalent to ${j}_{\infty}$=15% is
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$\mathrm{effectiverate}\left(0.15\,\mathrm{\infty}\right)$

You may determine the rate ${j}_{12}$ equivalent to this rate in the following manner.
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$12\cdot \mathrm{effectiverate}\left(\frac{0.15}{12}comma;\frac{\infty}{12}\right)$

The future value $S$, of an amount $P$, given that it is compounded continuously at a rate ${j}_{\infty}\=r$ over $t$ years, is given by $S\=P{\ⅇ}^{rt}$. The accumulated value of $5000 over 15 months at a nominal rate of 18% compounded continuously is given by
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$5000{\ⅇ}^{0.18\cdot \left(\frac{15}{12}\right)}$



Interest Formulas


If $P$ is the principal at the beginning of the first interest period, $S$ is the accumulated value at the end of $t$ periods, and $r$ is the interest rate per time period, then $S\={P\left(1\+r\right)}^{t}$. Use the futurevalue command to find $S$ and the presentvalue command to determine $P$.
Suppose that you deposit $100 in the bank, and earn interest at 10% per annum. The following command finds the accumulated value of the deposit at the end of four years.
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$\mathrm{futurevalue}\left(100\,0.10\,4\right)$

If you want $146.41 four years from now, then how much money must you invest now at an interest rate of 10%?
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$\mathrm{presentvalue}\left(146.41\,0.10\,4\right)$

You may extend the first example to the fundamental compound interest formula. If $P$ is the principal at the beginning of the first interest period, $S$ is the accumulated value at the end of $n$ periods, and $i$ is the interest rate per conversion period, then $S\={P\left(1\+i\right)}^{n}$. Again you can use the futurevalue and presentvalue command, but you must modify the arguments, because you are dealing with compound interest here.
Going back to the first example, suppose that you invest $100 at an annual interest rate of 10% compounded monthly for 4 years. This means that, for each compound period, the interest is $\frac{0.10}{12}$ (conventionally written as ${j}_{12}\=10$%). Since the number of compound periods per year is 12, the total number of periods is (4)(12). The following command finds the accumulated value.
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$\mathrm{futurevalue}\left(100\,\frac{0.10}{12}comma;4\cdot 12\right)$

Change the original investment to $($100\+a$). The interest rate may only be a proportion ($b$) of what the current compounded interest rate is from the above problem, or $\frac{0.10b}{12}$.
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$\mathrm{futurevalue}\left(100+a\,\left(\frac{.10}{12}\right)\cdot b\,4\cdot 12\right)$

$\left({100}{\+}{a}\right){}{\left({1}{\+}{0.008333333333}{}{b}\right)}^{{48}}$
 (5.4) 


Perpetuities


A perpetuity is an annuity whose payments begin on a fixed date and continue forever.
Suppose that you want to establish a scholarship fund paying scholarships of $1500 each year. How much money must you invest at an annual interest rate of 9% if the endowment is to pay its first scholarship one year from now?
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$\mathrm{perpetuity}\left(1500\,0.09\right)$

If the first scholarship is to be given out 3 years from now, you must modify the above command slightly. Notice that you should use ${1.09}^{2}$, as opposed to ${1.09}^{3}$, since you discount only 2 periods. As a result, the present value of the perpetuity is in 2 years from now:
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$\frac{\mathrm{perpetuity}\left(1500\,0.09\right)}{{1.09}^{2}}$

Just like simple annuities, perpetuities can grow. Suppose that you buy some shares for a company. You expect the first dividend payment to be $235 one year from now, and these payments are expected to grow at $g$% per annum, continuing indefinitely. Money is worth 7.5%. The following command determines the present value of these payments.
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$\mathrm{growingperpetuity}\left(235\,0.075\,g\right)$

$\frac{{235}}{{0.075}{}{g}}$
 (6.3) 

For more information, consult the Overview of the Finance package help page. You can also refer to the following help pages: amortization, annuity, growingannuity, cashflows, levelcoupon, yieldtomaturity, effectiverate, futurevalue, presentvalue, perpetuity, and growingperpetuity.
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