You get a loan for $100,000 today and will pay it back with yearly payments of $10,000 each year in years 1 to 10. In addition, you will make a single dollar payment in year 3. How big must the single payment be, if the loan charges 6.00% APR (compounded annually)

$31,442

Explanation:

The computation is shown below:

Years             Cash flows              Discount factor         Present value

0                     $100,000.00              1                      $100,000.00

1                      $10,000.00        0.9433962264

2                     $10,000.00        0.88999644

3                                                 0.839619283

4                      $10,000.00         0.7920936632

5                      $10,000.00          0.7472581729

6                      $10,000.00          0.7049605404

7                       $10,000.00          0.6650571136

8                       $10,000.00           0.6274123713

9                       $10,000.00           0.5918984635

10                     $10,000.00            0.5583947769

Total                                                  7.3600870514

Now the present value is

= $10,000 × 7.3600870514

= $73,600.87

The single payment is

= $100,000 - $73,600.8705

= $26,399.1295

After considering the 6% APR, it is

= $26,399.1295 ÷ 0.839619283

= $31,441.72

The discount factor should be computed below

= 1 ÷ (1 + rate) ^ years

where,  

rate is 6%  

Year = 0,1,2,3,4 and so on

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