Understanding the Mathematics of Personal Finance


Analogous to the present value is the future value. Present value is the value today of some amounts of money that are known on different dates. Future value is the value at some future date, which must be specified, of some amounts of money that
are known on different dates. The point here is that in various situations, you will be interested in the value of some amounts of money on some date and you must calculate, using known or at least estimated interest rates, what this value is. The only real difference is that present value means the value today (a unique date), whereas future value means the value at some date in the future that must be speci­fied; “in the future” is not a unique date. Future value calculations are identically compound interest calculations, so I don’t have to generate a new spreadsheet cal­culator. If I put $1 in the bank today at some APR and come back 10 years from today, my bank balance will be the future value of this $1 deposit.

In Chapter 5 i I have presented several different possible prepayment penalties for loans, including the infamous rule of 78. I’d like to reexamine the last example of the chapter, that is, various prepayment penalties for a $300,000, 15iyear loan, amortized monthly at 8% annual interest. Figure 5.3 shows the prepayment penalties versus the payment number when you decide to prepay the balance of the loan for four cases:

1. the rule of 78;

2. PP1: 6 months’ interest on 80% of the balance;

3. PP2: 2% of the balance;

4. PP3: 3% of the balance for the first year, 2% for the second year, 1% for the third year, then 0 for the rest of the term of the loan.

In Figure 7.1, I redid Figure 5.3. I replaced each value with its present value at payment day 0 (the start of the loan) using the same 8% interest number as in the


Payment number

Figure 7.1 Present value of the prepayment penalties example in Chapter 5.

loan itself. I used the same horizontal and vertical axes for Figure 7.1 as I did for Figure 5.3 to make it easy to compare the two figures.

For low payment number values, the two figures look almost identical. This makes sense; the present value of an amount quoted only a few months away (at 8% annual interest) isn’t much less than the quoted amount.

After about 60 payments, you start seeing the differences between the two figures. The worst prepayment penalty scheme, the rule of 78, peaks at about $8,000 rather than the almost $12,000 of Figure 5.3. Also, it peaks a bit sooner. Then, further out in time, all of the penalties (except PP3, which has already gone to 0), start falling very rapidly. At 10 years out (payment number 120), the amounts have dropped from about $3,000, $2,000, and $1,200 for rule of 78, PP1, and PP2, respec­tively, to about $6,000, $4,400, and $3,000. Compared with the balance of the loan at that time (about $139,500), these are getting to be not-too-significant nuisances.


Note: Since the principal purpose of Chapter 7 was to introduce the topic of present values so that later chapters could use this topic, there are very few problems below. Many problems will be presented in these later chapters that require calculating present values and using this information for various purposes.

1. I take a $22,000 auto loan, payable monthly at 6.00% APR for 5 years (60 payments). If I can save money at 4.00%, what is the present value of this loan the day that I take it?

2. A TV store dealer will “let me have” a $2,300 TV for 24 monthly payments of $100.00, the first payment due on the day I take the TV out of his or her store. Under what circum­stances is this a good deal for me; under what circumstances is it a bad deal for me?

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