The ΣPrn( Command
SIGMAPRN.GIF

Command Summary

For an amortization schedule, calculates the principal amount paid over a range of payments.

Command Syntax

ΣPrn(paymentt1, payment2, [roundvalue])

Menu Location

On the TI-83, press:

  1. 2nd FINANCE to access the finance menu.
  2. 0 to select ΣPrn(, or use arrows and ENTER.

On the TI-83+ or higher, press:

  1. APPS to access the applications menu.
  2. 1 or ENTER to select Finance…
  3. 0 to select ΣPrn(, or use arrows and ENTER.

Calculator Compatibility

TI-83/84/+/SE

Token Size

2 bytes

The ΣPrn( command calculates, for an amortization schedule, the principal amount over a range of payments: the portion of those payments that went toward paying off the principal. Its two required arguments are payment1 and payment2, which define the range of payments we're interested in. However, it also uses the values of the finance variables PV, PMT, and I% in its calculations.

The optional argument, roundvalue, is the number of digits to which the calculator will round all internal calculations. Since this rounding affects further steps, this isn't the same as using round( to round the result of ΣPrn( to the same number of digits.

Usually, you will know the values of N, PV, and I%, but not PMT. This means you'll have to use the finance solver to solve for PMT before calculating ΣPrn(; virtually always, FV will equal 0.

Sample Problem

Imagine that you have taken out a 30-year fixed-rate mortgage. The loan amount is $100000, and the annual interest rate (APR) is 8%. Payments will be made monthly. How much of the principal amount was paid in the first five years?

We know the values of N, I%, and PV, though we still need to convert them to monthly values (since payments are made monthly). N is 30*12, and I% is 8/12. PV is just 100000.

Now, we use the finance solver to solve for PMT. Since you intend to pay out the entire loan, FV is 0. Using either the interactive TVM solver, or the tvm_Pmt command, we get a value of about -$733.76 for PMT.

We are ready to use ΣPrn(. We are interested in the payments made during the first five years; that is, between the 1st payment and the 5*12=60th payment. ΣPrn(1,60) gives us the answer: -$4930.14 (the negative sign simply indicates the direction of cash flow)

Formulas

The formula that the calculator uses for ΣPrn( is in terms of bal(:

(1)
\begin{align} \definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \operatorname{\Sigma Prn}(n_1,n_2)=\operatorname{bal}(n_2)-\operatorname{bal}(n_1) \end{align}

When the roundvalue argument isn't given, we can substitute the explicit formula for bal( and simplify to get the following formula:

(2)
\begin{align} \definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \operatorname{\Sigma Prn}(n_1,n_2)=\left(\operatorname{PV}-\frac{\operatorname{PMT}}{I\%/100}\right)\left[\left(1-\frac{I\%}{100}\right)^{n_1}-\left(1-\frac{I\%}{100}\right)^{n_2}\right] \end{align}

Error Conditions

  • ERR:DOMAIN is thrown if either payment number is negative or a decimal.

Related Commands

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