In advertisements that offer bank loans or savings accounts, interest is usually quoted as an annual rate, also called anominal rate, even if the actual interest period is different. This interest periodis the time which elapses between successive dates when interest is added to the account. For some bank accounts the interest period is one year, but recently it has become increasingly common for financial institutions to offer other interest schemes. For instance, many US bank accounts now add interest daily, some others at least monthly. If a bank offers 9% annual rate of interest with interest payments each month, then(1/12)9%= 0.75% of the capital accrues at the end of each month. The annual rate must be divided by the number of interest periods to get theperiodic rate—that is, the interest per period.
Suppose a principal (or capital) ofS0yields interest at the ratep% per period (for example one year). As explained in Section 1.2, aftertperiods it will have increased to the amount
S(t )=S0(1+r)t where r=p/100
Each period the principal increases by the factor1+r. Note thatp% meansp/100, and we say that theinterest rateisp% orr.
1 Claimed to be Newton’s reaction to the outcome of the “South Sea Bubble”, a serious financial crisis in 1720, in which Newton lost money.
The formula assumes that the interest is added to the principal at the end of each period.
Suppose that the annual interest rate isp%, but that interest is paid biannually (i.e. twice a year) at the ratep/2%. Then the principal after half a year will have increased to
S0+S0
p/2 100 =S0
1+r
2
Each half year the principal increases by the factor1+r/2. After 2 periods (= one year) it will have increased toS0(1+r/2)2, and aftert years to
S0
1+r 2
2t
Note that a biannual interest payment at the rate12ris better for a lender than an annual interest payment at the rater. This follows from the fact that(1+r/2)2=1+r+r2/4>1+r.
More generally, suppose that interest at the rate p/n% is added to the principal atn different times distributed more or less evenly over the year. For example,n=4 if interest is added quarterly,n=12 if it is added monthly, etc. Then theprincipal will be multiplied by a factor(1+r/n)neach year. Aftertyears, the principal will have increased to
S0
1+r n
nt
(1) The greater isn, the faster interest accrues to the lender. (See Problem 10.2.6.)
E X A M P L E 1 A deposit of £5000 is put into an account earning interest at the annual rate of 9%, with interest paid quarterly. How much will there be in the account after 8 years?
Solution: The periodic rate r/n is 0.09/4 = 0.0225 and the number of periods nt is 4ã8=32. So formula (1) gives:
5000(1+0.0225)32≈10 190.52
E X A M P L E 2 How long will it take for the £5000 in Example 1 (with annual interest rate 9% and interest paid quarterly) to increase to £15 000?
Solution: Aftert quarterly payments the account will grow to 5000(1+0.0225)t. So 5000(1+0.0225)t =15 000 or 1.0225t =3
To findtwe take the natural logarithm of each side:
tln 1.0225=ln 3 (because lnap=plna) t= ln 3
ln 1.0225 ≈49.37
Thus it takes approximately 49.37 quarterly periods, that is approximately 12 years and four months, before the account has increased to £15 000.
S E C T I O N 1 0 . 1 / I N T E R E S T P E R I O D S A N D E F F E C T I V E R A T E S 347
Effective Rate of Interest
A consumer who needs a loan may receive different offers from several competing financial institutions. It is therefore important to know how to compare various offers. The concept ofeffective interest rateis often used in making such comparisons.
Consider a loan which implies an annual interest rate of 9% with interest at the rate 9/12=0.75% added 12 times a year. If no interest is paid in the meantime, after one year an initial principal ofS0 will have grown to a debt ofS0(1+0.09/12)12 ≈S0ã1.094. In fact, as long as no interest is paid, the debt will grow at a constant proportional rate that is (approximately) 9.4% per year. For this reason, we call 9.4% the effective yearly rate.
More generally:
E F F E C T I V E Y E A R L Y R A T E
When interest is addedntimes during the year at the rater/nper period, then the effective yearly rateRis defined as
R= 1+r
n n
−1
(2)
The effective yearly rate is independent of the amountS0. For a given value ofr >0, it is increasing inn. (See Problem 10.2.6.)
E X A M P L E 3 What is the effective yearly rateRcorresponding to an annual interest rate of 9% with interest compounded: (i) each quarter; (ii) each month?
Solution:
(i) Applying formula (2) withr =0.09 andn=4, the effective rate is R=
1+0.09/44
−1=(1+0.0225)4−1≈0.0931 or 9.31%
(ii) In this caser=0.09 andn=12, so the effective rate is R =
1+0.09/1212
−1=(1+0.0075)12−1≈0.0938 or 9.38%
A typical case in which we can use the effective rate of interest to compare different financial offers is the following.
E X A M P L E 4 When investing in a savings account, which of the following offers are better: 5.9% with interest paid quarterly; or 6% with interest paid twice a year?
Solution: According to (2), the effective rates for the two offers are R=
1+0.059/44
−1≈0.0603, R=
1+0.06/22
−1=0.0609 The second offer is therefore better for the saver.
NOTE 1 In many countries there is an official legal definition of effective interest rate which takes into account different forms of fixed or “closing” costs incurred when initiating a loan.
Theeffective rate of interestis then defined as the rate which implies that the combined present value of all the costs is equal to the size of the loan. This is the internal rate of return, as defined in Section 10.7. (Present values are discussed in Section 10.3.)
P R O B L E M S F O R S E C T I O N 1 0 . 1
1. (a) What will be the size of an account after 5 years if $8000 is invested at an annual interest rate of 5% compounded (i) monthly; (ii) daily (with 365 days in a year)?
(b) How long does it take for the $8000 to double with monthly compounding?
2. (a) An amount $5000 earns interest at 3% per year. What will this amount have grown to after 10 years?
(b) How long does it take for the $5000 to triple?
3. What annual percentage rate of growth is needed for a country’s GDP to become 100 times as large after 100 years? (100√
100≈1.047.)
4. (a) An amount of 2000 euros is invested at 7% per year. What is the balance in the account after (i) 2 years; (ii) 10 years?
(b) How long does it take (approximately) for the balance to reach 6000 euros?
5. Calculate the effective yearly interest if the nominal rate is 17% and interest is added:
(i) biannually; (ii) quarterly; (iii) monthly.
6. Which terms are preferable for a borrower: (i) an annual interest rate of 21.5%, with interest paid yearly; or (ii) an annual interest rate of 20%, with interest paid quarterly?
7. (a) A sum of $12 000 is invested at 4% annual interest. What will this amount have grown to after 15 years?
(b) How much should you have deposited in a bank account 5 years ago in order to have $50 000 today, given that the interest rate has been 5% per year over the period?
(c) A credit card is offered with interest on the outstanding balance charged at 2% per month.
What is the effective annual rate of interest?
8. What is the nominal yearly interest rate if the effective yearly rate is 28% and interest is com- pounded quarterly?
S E C T I O N 1 0 . 2 / C O N T I N U O U S C O M P O U N D I N G 349