Formulas for Loans, Mortgages and Savings accounts Bernard Liengme January 2012 The interest rates banks charge on credit card Interest Rate balances is criminal! Somewhere around 26% • When you borrow money you pay interest; when you save money you earn interest. • The interest is computed as a percentage rate and in normally quoted as a yearly (per annum) value. It is then called the APR – annual percentage rate. • But you are generally charged (or earn) on a monthly basis • If rate is the per annum percentage then the monthly percentage is rate/12 Number of Periods (nper) • A period is the interval between which you make a payment on the load (or the bank deposits earned interest into your account) • All of our examples will use monthly periods • If nper is the number of years over which you will pay of the load then in our formulas we replace nper by nper * 12. Principal • The starting amount of a load to distinguish it from the interest • Get the spelling right it is NOT principle • This word does not appear in Excel formulas Present Value PV • A $100 bill is worth (surprise!) $100. That is its present value. • You have won a prize that pays you $100 a month for 10 years. What is it worth? Or what is its present value? • Its PV is the amount of money someone would need to invest and get $100 a month for 10 years leaving nothing in the bank at the end of that time • We have to assume the interest rate remains the same as today’s value. Future Value FV • A $100 bill is worth $100 today and will be worth $100 in 10 years time (It may not buy as many pints of beer but it will still have a value of $100). That is it PV and its FV. • I deposit $100 in the bank, wait 10 years and then withdraw it. How much will I get? Or what will be its future value? • The catch phrase is the time value of money. Payment pmt • You borrow $1,000 from the bank for 1 year. • Every month you give the bank $86.99 (normally this will be deducted from you regular bank account automatically) • After 12 months the loan is paid off • When you began the loan: PV = 1000, FV = 0, rate = 8%/12 and nper = 1 * 12. We will see soon how to compute the payment pmt. How are these thing related? • By the money equation shown here nper 1 rate ) 1 nper fv 0 pv *(1 rate) pmt *(1 rate * type) * rate • But we will not have to worry about it Money has a value and a direction • There is a difference between giving and receiving money. • Money received is a credit and we give it a positive value • Money paid into the bank is a debit and we give it a negative sign Excel formulas • All the formulas for personal finance use the terms PV, FV, rate, nper and pmt • For everything but rate, we can rearrange the money equation to get just one term on the left. nper 1 rate ) 1 nper fv 0 pv * (1 rate) pmt * (1 rate * type) * rate nper pmt * (1 rate * type) 1 rate) 1 fv pv * nper (1 rate)nper (1 rate) rate PV • Present value =PV(rate,nper, pmt, [fv],[type]) • Arguments in brackets [] can be omitted when zero • My prize gives me $100 each month for 10 years, what is its PV? If the rate is 8% pa then we compute the present value with: =PV(8%/12,10*12,100) What if I will also get $500 at the end of the 10 years? =PV(8%/12,10*12,100, 500) And the answer is • Open the workbook Finance and on the PV worksheet note the way we do the calculations. Why are the two PV values shown in red? • Because it someone went to the bank to set up a price like this they would have to give money to the bank. Money would flow away from them. FV • Future value =FV(rate, nper, pmt, [pv],[type]) • I deposit $100 each month into a saving account. The interest rate is 8% pa. • PV = 0 since the saving account had nothing in it before the first payment. What formula should I use to find how much I can expect after 3 years? =FV(8%/12,3*12,-100) Why is pv given the value -100 with a negative sign? See worksheet FV Payment • Payment: =PMT(rate, nper, pv, [fv], [type]) • I borrow $1,000 at 8% pa and pay off the loan in 12 monthly installments. How much must I pay? • See worksheet PMT Number of Periods • I plan to save $250 each month until I have $100,000. Assuming the APR is 8%, how many monthly payments must I make? • Number of periods =NPER(rate, pmt, pv, fv, type) • Rate=APR/12; PMT = -250 (note the negative) • PV = 0, FV = 100,000, type = 0 (end of month) • Worksheet NPER Rate • Rate is found with RATE(nper,pmt,pv,fv,type,guess) • When the syntax is shown like this the bold arguments are required and the others are optional – in the other form we used [] for optional arguments • The money equation cannot be solved for rate, so Excel uses a ‘trial-and-error’ method like GoalSeek. Hence the guess argument: it sometimes helps to give Excel a start! Rate (cont) • I plan to save $400 each month for 25 years to give myself a pension. If my target is $500,000 what must the APR be during this time? • The answer is 9.65% which is unlikely at the present time. See worksheet Rate IPMT and PPMT • Where as PMT computes the payment on a loan, IPMT computes the interest part of this payment and PPMT computes how much went to paying down the principal. • See worksheet IPMT