Simple and compound interest, done right

Two people each put ₹1,00,000 in a deposit at 7.5 % a year for 30 years. One leaves every year’s interest in the deposit and ends with ₹8,75,496; the other has the interest paid out and spends it, and ends with her ₹1,00,000 back and ₹2,25,000 received along the way, ₹3,25,000 in all.

The difference is compounding. This lesson works out simple and compound interest for any compounding frequency, turns a quoted rate into the rate a deposit really earns in a year, checks the rule of 72 against the doubling period India Post states for Kisan Vikas PatraA Government of India savings certificate, bought at post offices and authorised banks, that pays back double the amount invested when it matures., and shows why interest taken out never compounds. It is education only: the rates are examples for the arithmetic, and no product is recommended.

Simple interest pays only on the money you put in

A deposit of ₹1,00,000 at 7.5 % a year pays 7.5 % of ₹1,00,000 at the end of every year, ₹7,500. With simple interestsimple interest: Interest worked out on the principal alone, as principal × rate × time. Interest earned earlier earns nothing, so every year pays the same amount. that never changes: the interest is always worked out on the principalprincipal: The amount put into a deposit or borrowed in a loan, on which interest is worked out. In a deposit of ₹1,00,000 at 7.5 % a year, the principal is ₹1,00,000., the ₹1,00,000 put in, and never on interest already earned.

In words: interest equals the principal times the yearly rate times the number of years. For three years, ₹1,00,000 × 0.075 × 3 = ₹22,500. The time need not be whole years: 18 months is 1.5 years, so ₹40,000 at 6 % earns ₹40,000 × 0.06 × 1.5 = ₹3,600. Year by year the balance climbs by the same step:

The steps are equal because nothing that was earned is put to work. That makes simple interest easy to check, and it is how interest behaves whenever it leaves the deposit, which the last section of this lesson returns to.

Compound interest also pays interest on the interest

Now let the interest stay. With compound interestcompound interest: Interest worked out on the whole balance, interest added earlier included, so each period pays more than the last while the money stays invested. each year’s interest is added to the balance, and the next year’s 7.5 % is worked out on that larger balance. Step through the first three years, simple on the left and compound on the right, and watch where the two columns part:

In year 2 the compound side earned ₹562.50 more, which is 7.5 % of the first year’s ₹7,500: interest on interest. Multiplying by 1.075 once a year gives the balance after any number of years at once:

In words: the balance after t years is the principal times one plus the rate, multiplied by itself once for each year. After three years, ₹1,00,000 × 1.075 × 1.075 × 1.075 = ₹1,24,229.69. The program below works out both kinds of interest over the first three years and then over 30:

"""Simple and compound interest on the same deposit, year by year.

Rs 1,00,000 at 7.5 % a year. Simple interest pays 7.5 % of the first Rs 1,00,000 every year. Compound interest,
added once a year, pays 7.5 % of the whole balance, so the interest of earlier years earns interest too. 7.5 %
compounded yearly is the rate India Post lists for Kisan Vikas Patra; the deposit here is an example, not advice.
Money is worked with the decimal module and rounded half up.
"""

from decimal import ROUND_HALF_UP, Decimal


def inr(amount, places="1"):
    """Rupees with Indian digit grouping: 124229.6875 -> "₹1,24,230"; places="0.01" keeps paise."""
    value = Decimal(amount).quantize(Decimal(places), rounding=ROUND_HALF_UP)
    whole, _, paise = str(abs(value)).partition(".")
    head, tail = whole[:-3], whole[-3:]
    groups = []
    while len(head) > 2:
        groups.insert(0, head[-2:])
        head = head[:-2]
    if head:
        groups.insert(0, head)
    text = ",".join(groups + [tail]) if groups else tail
    return "₹" + text + ("." + paise if paise else "")


PRINCIPAL = Decimal(100000)
RATE = Decimal("0.075")

print(f"{inr(PRINCIPAL)} at 7.5 % a year, interest added once a year")
print()
print("The first three years")
print(f"{'Year':<6}{'Simple interest':>17}{'Simple balance':>16}{'Compound interest':>19}{'Compound balance':>18}")
simple = compound = PRINCIPAL
for year in (1, 2, 3):
    simple_interest = PRINCIPAL * RATE  # always on the first Rs 1,00,000
    compound_interest = compound * RATE  # on everything in the deposit
    simple += simple_interest
    compound += compound_interest
    print(
        f"{year:<6}{inr(simple_interest, '0.01'):>17}{inr(simple, '0.01'):>16}"
        f"{inr(compound_interest, '0.01'):>19}{inr(compound, '0.01'):>18}"
    )
print(f"After 3 years compound interest is ahead by {inr(compound - simple, '0.01')}: interest earned on interest.")

print()
print("Longer horizons")
print(f"{'Years':<7}{'Simple':>12}{'Compound':>12}{'Gap':>12}{'Interest on interest':>22}")
for years in (1, 5, 10, 20, 30):
    s = PRINCIPAL * (1 + RATE * years)  # P(1 + rt)
    c = PRINCIPAL * (1 + RATE) ** years  # P(1 + r)^t
    share = (c - s) / (c - PRINCIPAL) * 100  # part of all compound interest that is interest on interest
    print(f"{years:<7}{inr(s):>12}{inr(c):>12}{inr(c - s):>12}{share.quantize(Decimal('0.1'), rounding=ROUND_HALF_UP):>20} %")

₹1,00,000 at 7.5 % a year, interest added once a year

The first three years
Year    Simple interest  Simple balance  Compound interest  Compound balance
1             ₹7,500.00    ₹1,07,500.00          ₹7,500.00      ₹1,07,500.00
2             ₹7,500.00    ₹1,15,000.00          ₹8,062.50      ₹1,15,562.50
3             ₹7,500.00    ₹1,22,500.00          ₹8,667.19      ₹1,24,229.69
After 3 years compound interest is ahead by ₹1,729.69: interest earned on interest.

Longer horizons
Years        Simple    Compound         Gap  Interest on interest
1         ₹1,07,500   ₹1,07,500          ₹0                 0.0 %
5         ₹1,37,500   ₹1,43,563      ₹6,063                13.9 %
10        ₹1,75,000   ₹2,06,103     ₹31,103                29.3 %
20        ₹2,50,000   ₹4,24,785   ₹1,74,785                53.8 %
30        ₹3,25,000   ₹8,75,496   ₹5,50,496                71.0 %

The gap is ₹1,729.69 after three years and ₹5,50,496 after 30, when interest on interest makes up 71 % of all the compound interest earned. Time does the work: the gap after 30 years is more than 300 times the gap after three, because every year’s interest earns in every later year.

A quoted rate earns more when interest is added more often

Deposits quote a nominal ratenominal rate: The yearly rate a deposit or loan quotes before compounding is counted. 7 % compounded quarterly is a nominal rate: 1.75 % is added four times a year., a yearly figure, together with a compounding frequencycompounding frequency: How often interest is added to the balance in a year: yearly, half-yearly, quarterly, monthly or daily. The more often, the more a quoted yearly rate earns.: how often a share of it is added to the balance. “7 % compounded quarterly” means 7 % ÷ 4 = 1.75 % is added four times a year, and each addition earns in the quarters after it. India Post states the frequency for several of its schemes: its recurring deposit at 6.7 % is compounded quarterly, while Kisan Vikas Patra, the National Savings Certificate and the Public Provident Fund are compounded yearly.

In words: with m additions a year, each adds the rate divided by m, and there are m × t of them in t years. The effective annual rateeffective annual rate: What a quoted rate earns in one year once compounding is counted: (1 + rate ÷ m)^m − 1 for m additions a year. 7 % compounded quarterly is 7.1859 % a year. says what that earns in one year, (1 + r ÷ m)^m − 1. For 7 % compounded quarterly, 1.0175 × 1.0175 × 1.0175 × 1.0175 − 1 = 7.1859 %. The model starts there, ₹1,00,000 for five years with interest added quarterly. Switch “Interest added” to yearly, then monthly and daily, and read the effective yearly rate:

The program prints the same five frequencies side by side, and turns the effective rate back into a quoted one:

"""One quoted rate, five ways of adding the interest.

A nominal rate of 7 % a year is split into m equal parts, and one part is added m times a year:
amount = P(1 + r/m)^(m*t), and the effective annual rate is (1 + r/m)^m - 1. The last part turns the effective rate
back into the nominal rate that gives it, and the RD line uses the 6.7 % compounded quarterly that India Post lists
for its recurring deposit. Every rate here is an example for the arithmetic. Decimal module, rounded half up.
"""

from decimal import ROUND_HALF_UP, Decimal, getcontext

getcontext().prec = 40


def inr(amount):
    """Whole rupees with Indian digit grouping: 141478.3 -> "₹1,41,478"."""
    whole = str(Decimal(amount).quantize(Decimal(1), rounding=ROUND_HALF_UP))
    head, tail = whole[:-3], whole[-3:]
    groups = []
    while len(head) > 2:
        groups.insert(0, head[-2:])
        head = head[:-2]
    if head:
        groups.insert(0, head)
    return "₹" + ",".join(groups + [tail]) if groups else "₹" + tail


def pct(fraction, places="0.0001"):
    """A fraction as a percentage: 0.0718590 -> "7.1859 %"."""
    return f"{(fraction * 100).quantize(Decimal(places), rounding=ROUND_HALF_UP)} %"


PRINCIPAL = Decimal(100000)
NOMINAL = Decimal("0.07")
YEARS = 5

print(f"{inr(PRINCIPAL)} at 7 % a year (nominal), for {YEARS} years")
print(f"{'Interest added':<16}{'Times a year':>13}{'Effective rate':>16}{'After 5 years':>15}")
for name, m in (("yearly", 1), ("half-yearly", 2), ("quarterly", 4), ("monthly", 12), ("daily", 365)):
    per_period = NOMINAL / m
    effective = (1 + per_period) ** m - 1
    amount = PRINCIPAL * (1 + per_period) ** (m * YEARS)
    print(f"{name:<16}{m:>13}{pct(effective):>16}{inr(amount):>15}")

print()
quarterly_effective = (1 + NOMINAL / 4) ** 4 - 1
print(f"7 % compounded quarterly is {pct(quarterly_effective)} a year.")
back = 4 * ((1 + Decimal("0.07")) ** (Decimal(1) / 4) - 1)
print(f"The quarterly-compounded rate that is exactly 7 % a year: {pct(back)}.")
rd = (1 + Decimal("0.067") / 4) ** 4 - 1
print(f"6.7 % compounded quarterly is {pct(rd, '0.01')} a year.")

₹1,00,000 at 7 % a year (nominal), for 5 years
Interest added   Times a year  Effective rate  After 5 years
yearly                      1        7.0000 %      ₹1,40,255
half-yearly                 2        7.1225 %      ₹1,41,060
quarterly                   4        7.1859 %      ₹1,41,478
monthly                    12        7.2290 %      ₹1,41,763
daily                     365        7.2501 %      ₹1,41,902

7 % compounded quarterly is 7.1859 % a year.
The quarterly-compounded rate that is exactly 7 % a year: 6.8234 %.
6.7 % compounded quarterly is 6.87 % a year.

Yearly, the five years end at ₹1,40,255; quarterly at ₹1,41,478; daily at ₹1,41,902. Each step adds less than the one before, so daily compounding is worth only 0.0211 percentage points a year more than monthly. India Post’s 6.7 % compounded quarterly works out to 6.87 % a year. That is what the effective rate is for: two quotes can be compared only once both are turned into what they earn in a year.

How long money takes to double: the rule of 72, then the exact answer

India Post states that money in Kisan Vikas Patra, at 7.5 % compounded yearly, doubles in 115 months, which is 9 years and 7 months. The rule of 72rule of 72: A shortcut for the years money takes to double: 72 divided by the yearly rate in per cent. At 7.5 % it gives 9.6 years, close to the exact 9.584. gives a quick estimate: divide 72 by the rate in per cent. 72 ÷ 7.5 = 9.6 years, or 115.2 months. The exact time comes from solving (1 + r)^t = 2 for t:

In words: the natural logarithm of 2 divided by the natural logarithm of one plus the rate. At 7.5 %, 0.693147 ÷ 0.072321 = 9.584 years, which is 115.01 months: the scheme’s 115 months. The program checks the rule at six rates, then each rate and term in the National Savings Institute’s table of KVP rates since the scheme was relaunched in 2014:

"""How long money takes to double: the rule of 72 against the exact answer.

With interest added once a year at rate r, money doubles when (1 + r)^t = 2, so t = ln 2 / ln(1 + r) years exactly.
The rule of 72 estimates t as 72 / (r in per cent). Part 2 checks the rows of the National Savings Institute's table of
Kisan Vikas Patra rates after the 2014 relaunch, each a rate with the months in which the deposit doubles, and works
out the rate those months imply, 2^(12 / months) - 1. Decimal module, 40 digits, rounded half up.
"""

from decimal import ROUND_HALF_UP, Decimal, getcontext

getcontext().prec = 40
TWO = Decimal(2)


def q(value, places):
    """Round half up to the given places, for printing."""
    return value.quantize(Decimal(places), rounding=ROUND_HALF_UP)


def inr(amount):
    """Rupees and paise with Indian digit grouping: 1999.849 -> "₹1,999.85"."""
    whole, _, paise = str(q(Decimal(amount), "0.01")).partition(".")
    head, tail = whole[:-3], whole[-3:]
    groups = []
    while len(head) > 2:
        groups.insert(0, head[-2:])
        head = head[:-2]
    if head:
        groups.insert(0, head)
    return "₹" + ",".join(groups + [tail]) + "." + paise if groups else "₹" + tail + "." + paise


def exact_years(rate_pct):
    """Years to double at rate_pct per cent, compounded yearly: ln 2 / ln(1 + r)."""
    return TWO.ln() / (1 + Decimal(rate_pct) / 100).ln()


print("Part 1: the rule of 72 and the exact doubling time, interest added yearly")
print(f"{'Rate':<8}{'Rule of 72':>12}{'Exact':>10}{'Rule is off by':>16}")
for rate in ("4", "6", "7.5", "9", "12", "18"):
    rule = Decimal(72) / Decimal(rate)
    exact = exact_years(rate)
    print(f"{rate + ' %':<8}{str(q(rule, '0.01')) + ' y':>12}{str(q(exact, '0.01')) + ' y':>10}{str(q((rule - exact) * 12, '0.1')) + ' months':>16}")

print()
print("Part 2: Kisan Vikas Patra rows since the 2014 relaunch (rate, months to double)")
NSI_ROWS = [("8.7", 100), ("7.8", 110), ("7.7", 112), ("7.6", 113), ("7.5", 115), ("7.3", 118), ("6.9", 124), ("7.0", 123), ("7.2", 120)]
print(f"{'Rate':<7}{'Months':>8}{'Rule of 72':>12}{'Exact':>10}{'Rate the months imply':>23}")
for rate, months in NSI_ROWS:
    rule_months = Decimal(72) / Decimal(rate) * 12
    exact_months = exact_years(rate) * 12
    implied = (TWO ** (Decimal(12) / months) - 1) * 100
    print(f"{rate + ' %':<7}{months:>8}{q(rule_months, '0.1'):>12}{q(exact_months, '0.1'):>10}{str(q(implied, '0.001')) + ' %':>23}")

print()
years = exact_years("7.5")
print(f"At 7.5 % compounded yearly the exact doubling time is {q(years, '0.001')} years = {q(years * 12, '0.01')} months.")
grown = 1000 * (1 + Decimal("0.075")) ** (Decimal(115) / 12)
short = (2000 - grown) * 100
print(f"₹1,000 grown at 7.5 % a year for 115 months: {inr(grown)}, {q(short, '1')} paise short of the ₹2,000 a doubling pays.")

Part 1: the rule of 72 and the exact doubling time, interest added yearly
Rate      Rule of 72     Exact  Rule is off by
4 %          18.00 y   17.67 y      3.9 months
6 %          12.00 y   11.90 y      1.3 months
7.5 %         9.60 y    9.58 y      0.2 months
9 %           8.00 y    8.04 y     -0.5 months
12 %          6.00 y    6.12 y     -1.4 months
18 %          4.00 y    4.19 y     -2.3 months

Part 2: Kisan Vikas Patra rows since the 2014 relaunch (rate, months to double)
Rate     Months  Rule of 72     Exact  Rate the months imply
8.7 %       100        99.3      99.7                8.673 %
7.8 %       110       110.8     110.7                7.855 %
7.7 %       112       112.2     112.1                7.709 %
7.6 %       113       113.7     113.6                7.639 %
7.5 %       115       115.2     115.0                7.501 %
7.3 %       118       118.4     118.1                7.303 %
6.9 %       124       125.2     124.7                6.938 %
7.0 %       123       123.4     122.9                6.996 %
7.2 %       120       120.0     119.6                7.177 %

At 7.5 % compounded yearly the exact doubling time is 9.584 years = 115.01 months.
₹1,000 grown at 7.5 % a year for 115 months: ₹1,999.85, 15 paise short of the ₹2,000 a doubling pays.

The rule is nearly exact around 7.5 to 9 % and drifts away on both sides: at 4 % it says 18 years where 17.67 is right, and at 18 % it says 4 years where 4.19 is right. In the scheme’s table the months, not the rounded rate, decide: 124 months at “6.9 %” means a deposit that doubles at 6.938 % a year. Walk through the rates in order:

At every rate in the table, from 6.9 % to 8.7 %, the rule of 72 lands less than 0.6 of a month from the exact doubling time, and the scheme’s months within a month of it.

Interest that is paid out never compounds

India Post’s Monthly Income SchemeA five-year post office deposit that pays its interest out to the depositor every month. pays 7.4 % a year, and pays the interest out every month. On a deposit of ₹1,00,000 that is 7.4 % ÷ 12 of it, ₹616.67, every month, because the deposit itself never grows. Spent as it arrives, that interest never earns interest: it is simple interest. Compare it with the same interest left in at the same rate, which is an assumption made only to see the difference:

The program follows both for five years, and then a second case where simple interest is the rule itself:

"""Interest taken out every month, or left in: the same rate, two different growths.

Part 1: Rs 1,00,000 for 5 years at 7.4 % a year, the rate India Post lists for its Monthly Income Scheme, whose
interest is paid out every month. Taken out and spent, the interest never earns interest, which is simple interest.
Left in at the same 7.4 % a year, added monthly (an assumption for the comparison), it compounds.
Part 2: a one-year time deposit closed after 8 months. The National Savings Institute's rule for closing between six
months and a year is simple interest at the Post Office Savings Account rate, 4 % a year.
Decimal module, rounded half up.
"""

from decimal import ROUND_HALF_UP, Decimal


def inr(amount, places="1"):
    """Rupees with Indian digit grouping: 144604.4 -> "₹1,44,604"; places="0.01" keeps paise."""
    value = Decimal(amount).quantize(Decimal(places), rounding=ROUND_HALF_UP)
    whole, _, paise = str(abs(value)).partition(".")
    head, tail = whole[:-3], whole[-3:]
    groups = []
    while len(head) > 2:
        groups.insert(0, head[-2:])
        head = head[:-2]
    if head:
        groups.insert(0, head)
    text = ",".join(groups + [tail]) if groups else tail
    return "₹" + text + ("." + paise if paise else "")


PRINCIPAL = Decimal(100000)
RATE = Decimal("0.074")
MONTHLY = RATE / 12

print(f"Part 1: {inr(PRINCIPAL)} at 7.4 % a year for 5 years")
print(f"Interest for one month: 7.4 % / 12 of {inr(PRINCIPAL)} = {inr(PRINCIPAL * MONTHLY, '0.01')}")
print(f"{'After year':<12}{'Paid out: received':>20}{'Left in: balance':>18}{'Difference':>12}")
received = Decimal(0)
balance = PRINCIPAL
for month in range(1, 61):
    received += PRINCIPAL * MONTHLY  # always on the first Rs 1,00,000
    balance += balance * MONTHLY  # on the whole balance
    if month % 12 == 0:
        print(f"{month // 12:<12}{inr(received):>20}{inr(balance - PRINCIPAL):>18}{inr(balance - PRINCIPAL - received):>12}")
print(f"Paid out, the 5 years earn {inr(received)}: exactly 7.4 % × 5 = 37 % of the deposit, simple interest.")
print(f"Left in, they earn {inr(balance - PRINCIPAL)}: an effective {((1 + MONTHLY) ** 12 - 1) * 100:.2f} % a year.")

print()
print(f"Part 2: a one-year time deposit of {inr(PRINCIPAL)} closed after 8 months")
early = PRINCIPAL * Decimal("0.04") * 8 / 12
print(f"Simple interest at 4 % a year for 8 months: {inr(early, '0.01')}")

Part 1: ₹1,00,000 at 7.4 % a year for 5 years
Interest for one month: 7.4 % / 12 of ₹1,00,000 = ₹616.67
After year    Paid out: received  Left in: balance  Difference
1                         ₹7,400            ₹7,656        ₹256
2                        ₹14,800           ₹15,899      ₹1,099
3                        ₹22,200           ₹24,772      ₹2,572
4                        ₹29,600           ₹34,325      ₹4,725
5                        ₹37,000           ₹44,609      ₹7,609
Paid out, the 5 years earn ₹37,000: exactly 7.4 % × 5 = 37 % of the deposit, simple interest.
Left in, they earn ₹44,609: an effective 7.66 % a year.

Part 2: a one-year time deposit of ₹1,00,000 closed after 8 months
Simple interest at 4 % a year for 8 months: ₹2,666.67

Paid out, the five years bring ₹37,000, exactly 7.4 % × 5 of the deposit. Left in, they earn ₹44,609, because the balance grows every month, an effective 7.66 % a year. A monthly payout is not a worse deal: it is what someone who lives on the interest needs. It just cannot compound, because nothing stays behind to earn.

Closing a deposit early can switch compounding off by rule. The National Savings Institute’s rule for a post office time deposit closed after six months but before a year is simple interest at the Post Office Savings Account rate, 4 % a year: ₹1,00,000 closed after eight months earns ₹2,666.67, whatever rate the deposit was opened at.

Practise: interest, rates and doubling

Practice picks its first question from the idea your answers have found hardest so far, from simple interest to payouts. Work rupee amounts to the nearest rupee and rates to two decimal places, unless a question says otherwise:

Interview questions

Warm-up (fresher): What is the difference between simple and compound interest?

Simple interest is worked out on the principal alone, so ₹1,00,000 at 7.5 % earns ₹7,500 every year, ₹2,25,000 in 30 years. Compound interest is worked out on the whole balance, interest already added included, so the same deposit earns ₹7,500 in the first year, ₹8,062.50 in the second, and grows to ₹8,75,496 in 30 years. The difference is interest on interest, and it grows with time: 13.9 % of all the compound interest after 5 years, 71 % after 30. Compounding needs the interest to stay invested; interest paid out and spent grows like simple interest.

Warm-up (fresher): What does an effective annual rate tell you that a quoted rate does not?

A quoted, or nominal, rate is split into equal parts that are added through the year. The effective annual rate says what those parts earn in one year once each has compounded: (1 + r ÷ m)^m − 1 for m additions a year. 7 % added quarterly is 7.1859 % a year, and added monthly, 7.2290 %. Two deposits can be compared only on effective rates, or on the same frequency: 7 % compounded quarterly earns more than 7.1 % compounded yearly, although its headline rate is lower.

Key takeaways

• Simple interest is principal × rate × time: the same amount every year, because interest already earned is never put to work.
• Compound interest is worked out on a growing balance, P × (1 + r)^t: on ₹1,00,000 at 7.5 % the gap from simple interest is ₹1,729.69 after three years and ₹5,50,496 after 30.
• A quoted rate earns more when interest is added more often: 7 % compounded quarterly is 7.1859 % a year, and only effective rates can be compared with each other.
• The rule of 72 estimates the years to double; ln 2 ÷ ln(1 + r) gives them exactly, 115.01 months at 7.5 %, which matches the 115 months stated for Kisan Vikas Patra.
• Interest paid out and spent never compounds: ₹1,00,000 at 7.4 % paid monthly brings ₹37,000 in five years, against ₹44,609 when the interest stays in.