spending_on_plastic by qihao0824

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									    Spending on plastic: The potential for financial
                       distress
1. Introduction                                   particularly those on a mortgage.
1966 might be remembered by many                  Furthermore, when a statement’s
as the year that England won the                  balance is not cleared in full, interest
football world cup. But, it was also the          continues to be charged against all
year when the first credit card was               unpaid transactions.
introduced in the UK. 1987 saw the
arrival of the debit card. Today, we              Indebtedness can undermine economic
take for granted the use of plastic as a          stability. Household financial distress,
means of payment.                                 or the fragility of their financial
                                                  position, has the potential to impact on
Figures from APACS1, the UK payments              consumption. This is significant because
association, show that in 2006 there              household spending constitutes close to
were 142.8 million payment cards in               two-thirds of GDP.      The impact of
issue. The average adult in the UK had            financial distress was readily apparent
2.4 credit cards and 1.6 debit cards.             in the early 1990s when rising interest
The total spend on plastic in 2006 was            rates caused debt repayments to soar.
£320.8 billion, of which £194.9 billion
or 60.8% was on debit cards and                   Indebtedness is firmly on the policy
£125.9 billion or 39.2% was on credit             agenda. The Bank of England publishes
or charge cards.                                  its Financial Stability Report twice a
                                                  year and this includes reference to the
Figures from the Office for National              financial health of the household sector.
Statistics2 show that total consumption           Further, articles on this topic regularly
by households in 2006 was £793.7                  appear in the Bank of England’s
billion. Therefore, approximately 44.5%           publication, Quarterly Bulletin and once
of household spending by value was                a year it presents the results of a
conducted by plastic. Further, 24.6% of           specially commissioned survey on the
household spending by value was on                state of British household finances.3
debit cards and 15.9% on credit cards.
                                                  Meanwhile      the     Department     for
The figures from APACS show an                    Business, Enterprise and Regulatory
amazing 1.9 billion transactions were             Reform (BERR)4 is charged with taking
made on credit or charge cards, though            forward initiatives to address consumer
this figure was unchanged from 2005.              indebtedness. It publishes an Annual
While usage is of interest to policy-             Tackling Over-Indebtedness Report. In
makers, the potential for financial               its 2007 Annual Report BERR reports
stress is a major concern.                        that between 2007Q2 and the same
                                                  quarter a year earlier the write-off rate
While a credit card is a convenient               for credit card debt rose from 5.6% to
vehicle for smoothing consumption by              7.5%. Each write-off rate is calculated
providing immediate credit, it can be             by dividing the amount written-off over
very expensive when balances are not              the latest four quarters by the average
paid off in full. The interest rates on           stock of debt over the same period.
credit cards are typically much higher
than those on other types of loan,                3
                                                    Details of Bank of England publications can be
                                                  found at
                                                  http://www.bankofengland.co.uk/publications/in
1
  The website address for APACS is                dex.htm
                                                  4
http://www.apacs.org.uk/                            The homepage of BERR is
2
  National Statistics Online can be accessed at   http://www.berr.gov.uk/ttp://www.dti.gov.uk/in
http://www.statistics.gov.uk/                     dex.html

                                                                                                1
Write-off rates are increasing. At the        percentage rate, ra. Both interest rates
start of 2000 the write-off rate for          are compound interest rates. This
credit card debt was below 2%.                means that we get ‘interest on the
                                              interest’.
In a submission to the Office of Fair
Trading, the consumer body Which              Our task is to find the monthly rate that
claimed it had identified 12 different        gives us the desired annual percentage
interest calculation methods in use by        rate, the so-called APR. This is achieved
20 of the top credit card providers.5 In      when      the      following condition  is
this case study, we will work through a       satisfied:
simplified example, applying some             (1) (1  rm )12  (1  ra )
techniques in financial mathematics, to
                                              Perhaps the easiest way of thinking
consider the repayment of credit card
                                              about what the equivalence of the
debt and the issue of financial distress.
                                              monthly and annual rate means is to
                                              consider     somebody    investing   an
2. Mathematics of repayments
                                              amount of money x. The individual
We consider a hypothetical individual
                                              would be indifferent between the
with a credit card and no initial
                                              money attracting a monthly interest
outstanding balance. They are issued
                                              rate, rm and interest added annually at
with a statement at the end of each
                                              rate, ra. If we take the twelfth root of
month. When the individual pays the
                                              each side, we obtain
balance off in full by the ‘statement due                                 1
date’, no interest is applied. However, if    (2) (1  rm )  (1  ra )    12

the balance is not settled in full,           We can now solve for the monthly rate,
interest is charged at the rate of 15.9%                           1

p.a. from the date of each transaction        (3) rm  (1  ra )    12
                                                                         1
until the next statement date. The
individual is required to make a              For our individual’s credit card, where
minimum monthly payment of 3% of              the annual interest rate is 15.9%
the statement balance.                        (0.159),    the     equivalent  monthly
                                              interest rate is 1.237%.
Assume our individual purchases a             (4) rm  (1  0.159)
                                                                         1
                                                                          12
                                                                                1  0.01237
holiday costing £2,500 using their card
on the last day of the month. The
                                              In calculating our individual’s new
statement is then issued, which the
                                              credit card balance we are applying a
individual receives shortly afterwards,
                                              monthly interest rate, rm, to an initial
detailing the purchase. It contains a
                                              outstanding         balance, A 1,   and
‘due date’ for payment and informs the
                                              subtracting a payment of proportion p
individual that they will need to make a
minimum         payment       of    £75       of the balance. Hence, in general terms
                                              the amount owing the following month,
(£2,500*(3/100)).
                                              A2, will be
Our individual decides only to make the       (5) A2  A1 (1  rm )  A1 p
minimum payment. Consequently, they           This is equivalent to
are liable to pay interest on the             (6) A2  A1 (1  rm  p)
transaction of £2,500. Given the timing
of the purchase, they will be charged 1       We are now in a position to enter the
month’s     interest.   This    requires      values from our example
calculating a monthly interest rate, rm,      (7) A2  £2,500 (1  0.01237  0.03)
which is equivalent to the annual
                                                   A2  £2,500 (0.98237 )  £2,455 .93
                                              The statement balance the following
5
 Which press release 1 April 2007.            month has decreased to £2,455.93, a
http://www.which.co.uk/press/press_home_569   fall of £44.07. Hence, of the £75 paid,
_64219.jsp

                                                                                               2
£30.93 is payment of interest and             minimum repayment rate, we calculate
£44.07 is repayment of capital.               the next statement balance to be
                                              £2,455.50
In practice an individual will have a         (11) A2  £2,500 [(1.000404 ) 30  0.03]  £2,455 .50
series of transactions on their credit        The small difference in the statement
card statement, not just the one has in       balance is because the calendar year is
our example. A series of separate             not exactly equivalent to 12 months of
interest charges will be applied on all       30 days.
the transactions when the balance is
not paid in full. Where the repayment         To    advance     our examination      of
of capital is sufficient to cover some        servicing credit card debt we continue
transactions it is often the lowest in        with our example, but resume using the
value that are settled first. Typically, as   monthly interest rate. This will simplify
in our example, these charges will be         the calculations and enable us to
dependent on the statement date and           consider the important issue of the
on the date of the transactions.              financial distress of debt.

Consequently, we need to consider the         Consider what happens to the balance
procedure for calculating a daily             owing if the individual, after receiving
interest rate. The daily rate should          the second statement, decides once
satisfy the following condition so as to      again to make only the minimum
have an equivalent APR                        payment. The individual will be liable to
                                              pay interest for another month on the
                     1
(8) rd  (1  ra )       365
                               1
                                              balance owing on the transaction.
Entering our individual’s APR of 15.9%
                                              Hence, at the end of the third month
into this formula, we find that the daily
                                              the balance outstanding, A3, will be
equivalent    rate   is   approximately
                                              (12) A3  A2 (1  rm  p)
0.0404%.
                                              Substituting in for A2 from (6)
We now repeat the calculation of the          (13) A3  A1 (1  rm  p)(1  rm  p)
individual’s second statement balance         This is equivalent to
when only the minimum payment has             (14) A3  A1 (1  rm  p ) 2
been made in response to the first
statement. Assume that the second             If we now enter the numbers from our
statement is issued 30 days after the         example, we find that two months after
first and so 30 days after the                the transaction the amount owing is
transaction of £2,500. Interest is            £2,412.64.
applied to the transaction for 30 days.
                                              (15) A3  £2,500 (1  0.01237  0.03) 2  £2,412 .64
We amend (5) to allow interest to be          Our individual’s second payment of
calculated daily. The amount owing at         £73.68 (3% of £2,455.93) reduces the
the time of the second monthly                amount owing by a further £43.29. But,
statement is obtained by                      after 2 months and payments totalling
                                              £148.68, the balance has only fallen by
(9) A2  A1 (1  rd )  A1 p
                     30
                                              £87.36.
Compound interest is applied daily for
30 days on the initial balance. A             3. Credit card debt in the UK
payment of proportion p of the balance        The potential financial burden from not
is made. After factorising, we see that       being able to meet credit card
this is equivalent to                         payments in full is clear from our
(10) A2  A1[(1  rd )  p ]
                      30
                                              example. So how much unpaid credit
                                              card debt is there? Figures from The
If we now enter our initial statement         Office for National Statistics show that
balance, the daily interest rate and the      the amount of credit card debt
                                              outstanding at the end of 2007 was

                                                                                                 3
£54.88 billion.6 The stock of credit card                                                                          2. Chart 1 shows the amount of
debt rose by £223 million over the year                                                                            credit card debt rising in 2007 but
or 0.4%. Interestingly, the stock of                                                                               the amount of debt as a percentage
credit card debt peaked at £57.90                                                                                  of disposable income falling. How
billion the end of 2005.                                                                                           can these results be reconciled?

Chart 1 displays both the level of credit                                                                          3. Assume an individual uses their
card outstanding and its size relative to                                                                          credit card to purchase a gift for a
household disposable income. 2006 is                                                                               friend. The gift is purchased on the
the first year since the data series                                                                               30th June and costs £500.
began in 1987 when a fall is recorded in
the level of credit card debt. We see a                                                                            The individual’s finances mean they
marked growth in credit card debt                                                                                  can only afford to make the
relative to incomes from the mid-                                                                                  minimum necessary payment of 3%
1990s. In 1995 the amount of debt                                                                                  of the outstanding balance on
outstanding was equivalent to 2.7% of                                                                              receipt of the statement balances for
the household disposable income. By                                                                                June and July. But, they expect to
2005 the figure was 7.2%. However, it                                                                              pay the outstanding balance off in
fell back to 6.5% in 2006 and, despite                                                                             full having received the statement
the small rise in the amount of credit                                                                             balance for August.
card debt, fell further in 2007 to 6.3%.
                                                                                                                   Statements are issued on the last
Chart 1: Credit card debt outstanding                                                                              day of the month and interest is
    8
                      Credit card debt outstanding           Debt to income
                                                                                            £70,000                charged at a rate of 16.9% per
                                                                                            £60,000
                                                                                                                   annum from the transaction date.
    7
                                                                                                                   Interest      is    payable    on     the
                                                                                            £50,000
    6
                                                                                                                   outstanding balance detailed in each
                                                                                            £40,000
                                                                                                                   statement when this balance is not
                                                                                                      £ millions




    5
%




                                                                                            £30,000
                                                                                                                   cleared in full.
                                                                                                                   (i) Calculate the daily rate of
    4
                                                                                            £20,000


    3
                                                                                            £10,000                      interest [assume a calendar
    2                                                                                       £0
                                                                                                                         year of 365 days]
        1987   1989     1991   1993    1995    1997   1999     2001   2003    2005   2007
                                                                                                                   (ii) What payment is made in
Source: EMLR, Office for National Statistics                                                                             response to the statement for
                                                                                                                         June statement?
It is too early to know if the upward                                                                              (iii) What is the balance outstanding
trend in the amount of credit card debt                                                                                  on the statement for July? What
to income has now ceased. Yet the                                                                                        interest     charges    has     the
magnitude of this debt continues to                                                                                      individual     incurred?      What
raise concerns for policy-makers as to                                                                                   payment does the individual
its potential to leave households                                                                                        make       when   receiving     the
financially vulnerable.                                                                                                  statement?
                                                                                                                   (iv) What        payment     does     the
Tasks                                                                                                                    individual make when receiving
  1. Calculate the amount of credit                                                                                      the statement for August so as
  debt written-off over a year if the                                                                                    to clear the balance?
  average amount of debt outstanding                                                                               (v) What is the total amount paid
  in the year is £55 billion and the                                                                                     by the individual for the gift?
  write-off rate is 7.5%




6
 See Table 6.6 of Economic and Labour Market
Review, Office for National Statistics

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