420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
438
439
440
441
442
443
449
450
451
452
453
454
455
456
460
461
462
463
464
469
476
477
478
A B C D E F G H I J K
N3
I0.1
PMT 100
Time period 0 1 2 3
CFt100 100 100 0 Annuity PV
PV3100.00 90.91 82.64 0.00 = $273.55
I = 10%
Time period
0 1 2 3 4
0100 300 300 -50
Using the function wizard, we follow the same procedure as above, except remember to enter a “1” to
tell Excel that in this problem the payments occur at the beginning of the periods.
To find the present value of the annuity due, this problem is solved just like the previous problem,
except that the payments occur in periods 0 through 2.
482
483
484
485
486
487
488
489
492
493
494
496
497
498
499
500
507
508
509
510
511
516
517
518
519
520
521
522
523
524
525
360 or 365 for annual compounding.
528
529
530
531
532
533
538
540
541
542
543
544
A B C D E F G H I J K
2300 247.93
3300 225.39
4-50 -34.15
$530.09
Or
Inputs
m=periods/yr 2This is the number of periods per year, m.
IPER = inom/m
IPER = 10% / 2
IPER = 5%
EFF% = (1+ INOM/M)M
EFF% = (1 + (10%/2))^2 – 1
h. (1.) Identify (a) the stated, or quoted, or nominal rate (iNom) and (b) the periodic rate (iPER).
With, the financial calculator, we could enter each of these cash flows and the discount rate, and
simply press NPV for the present value of the cash flow stream. In Excel, we can perform a similar
time period zero. However, the “NPV” function interprets the first data entry as being the cash flow in
time period one. Therefore, the initial cash flow must be added seperately. In this particular example,
the initial cash flow is zero.
Larger, because interest is earned on interest.
PV of CF stream =
compounding periods.
546
547
548
549
550
551
552
553
554
555
556
557
559
What is the FV with semiannual compounding?
560
561
562
563
564
565
566
567
568
569
570
573
What is the FV with monthly compounding?
574
575
576
577
578
579
What is the FV with daily compounding?
581
582
583
584
585
586
587
588
589
590
591
592
593
594
596
597
598
599
600
601
602
603
604
605
607
608
A B C D E F G H I J K
EFF% = 10.25%
SEMIANNUAL AND OTHER COMPOUNDING PERIODS
h. (3.) What is the future value of $100 after 5 years under 12% annual compounding?
N 3
I0.12 FV = $140.49
PV 100
N (years x 2) 6
I (I per year/2) 0.06 FV = $141.85
PV 100
What is the FV with quarterly compounding?
N (years x 4) 12
I (I per year/4) 0.03 FV = $142.58
PV 100
N (years x 12) 36
I (I per year/12) 0.01 FV = $143.08
PV 100
N (years x 365) 1095
I (I per year/12) 0.00032877 FV = $143.32
PV 100
NBeg. Amt. Payment Interest Principal End. Amt.
1 $1,000.00 $402.11 $100.00 $302.11 $697.89
2 $697.89 $402.11 $69.79 $332.33 $365.56
3 $365.56 $402.11 $36.56 $365.56 $0.00
Note: See Columns M
Now, construct an amortization table for the loan described above.
j. (1.) What would the required payment be on a $1,000 loan that is to be repaid in three equal installments at the
end of each of the next three years if the interest rate is 10%?
I. Will the effective annual rate ever be equal to the nominal (quoted) rate? Only if the compounding period is equal
to 1 year.
j. (2.) What is the annual interest expense for the borrower, and the annual interest income for the lender, during
Year 2?
$450.00
Payment
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619
620
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625
626
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628
634
635
636
637
638
639
640
646
647
648
649
650
655
Periods 0 1.0 2 3.0 4 5.0 6
FV of CF $121.55 $110.25 $100.00
657
658
659
660
663
664
Annual effective rate = 10.25%
FV = $331.80
A B C D E F G H I J K
through R for a 30 year
mortgage example.
I0.00031054
N273
FV $108.85
Years 0 0.5 1 1.5 2 2.5 3
Periods 0 1.0 2 3.0 4 5.0 6
Cash Flow 0 100 0100 0100
Total FV = Σ = $331.80
k. On January 1, you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with
interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later? (273
days)
l. (1.) What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10%,
Alternatively, you could calculate the annual effective rate and use this to find the future value of a 3-year annuity.
$0.00
$50.00
$100.00
$150.00
$200.00
$250.00
$300.00
$350.00
$400.00
$450.00
1 2 3
Year
Principal
Interest
666
667
668
669
670
671
672
673
674
676
677
678
679
680
681
682
684
685
686
687
688
689
690
691
See which provides the greater future wealth
698
699
700
701
702
703
704
705
708
709
710
711
712
713
715
716
717
See which has the higher effective rate of return, EFF%
720
721
722
723
724
A B C D E F G H I J K
Periods 0 1 2 3.0 4 5.0 6
PV of CF $90.70 $82.27 $74.62
In the second approach, we use the annual effective rate to find the present value of a 3-year annuity.
PV = $247.59
0 1 2 3 4 5456
850
I0.00018538
0 1 2 3 4 5456
1000
0 1 2 3 4 5456
850 1000
what condition holds? (Hint: Think of annual compounding, when iNOM = EAR = iPER.) What would be wrong with
your answer to questions l(1) and l(2) if you used the nominal rate (10%) rather than the periodic rate (iNOM/2 = 10%/2
l. (2.) What is the PV of the same stream?
See which has the greater present value
m. Suppose someone offered to sell you a note calling for the payment of $1,000 in 15 months (or 456 days). They
offer to sell it to you for $850. You have $850 in a bank time deposit that pays a 6.76649% nominal rate with daily
compounding, which is a 7% effective annual interest rate, and you plan to leave the money in the bank unless you
buy the note. The note is not risky–you are sure it will be paid on schedule. Should you buy the note? Check the
l. (3.) Is the stream an annuity? No, because we don’t have a payment for each compounding period.
Using the first approach, we find the present value of each individual cash flow using the periodic rate
and the number of periods.
727
728
729
A B C D E F G H I J K
I 0.035646% per day
EAR 13.89% > 7% so buy the note.
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