Multiple/Continuous Compounding

Understand Effective Interest Rates Figure out how to use Inflation/Deflation in your decisions

Interest Rate Terms… ● Compounding Period (cp) – the time between points when interest is computed and added to the initial amount. ● Payment Period (pp) – the shortest time between payments. Interest is earned on payment money once per period (cost of money) ● Nominal Rate ( r ) – is a simplified expression of the annual cost of money. It means nothing, unless the compounding period is stated along with it. ● Annual Percentage Rate (APR) – is the nominal interest rate on a yearly basis (credit cards, bank loans, …). It, too, should have a compounding period stated. ● Effective Rate ( i ) – is the rate that is used with the table factors or the closed form equations, and it converts the nominal rate taking into account both the compounding period and the payment period so that the blocks match.

Consider the discrete End-of-Year cash flow tables below: PeriodCash FlowPeriodCash Flow 0- $100,000 3$30, , , , ,000 Determine the Present Worth equivalent if a. the value of money is 12% compounded annually. b. the value of money is 12% compounded monthly. c. the value of money is 12% compounded continuously.

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.12, 5)

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.12, 5) = -100, ,000(3.6048) = $8,144

100,000 30, i r m eff m  11

100,000 30, i r m eff m  11    % 12. (.)..

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.1268, 5)

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.1268, 5) = -100, ,000(3.5449) = $6,346

100,000 30, P = -100, ,000(P/A, i eff, 5) i eff = ????

Now suppose we use an infinite # of compounding periods (continuous). How might we find an answer to our problem of r=12% per year compounded on a continuous basis?

Now suppose we use an infinite # of compounding periods (continuous). How might we find an answer to our problem of r=12% per year compounded on a continuous basis F= P(1+.12/9999) 9999 (one year period) = P(1.1275) = P( )

Continuous Compounding i = e ( r )(# of years) – 1 Examples: r = 12% per year compounded continuously i a = e (.12 )(1) – 1 = 12.75% What would be an effective six month interest rate for r = 12% per year compounded continuously? i 6 month = e (.12 )(.5) – 1 = 6.184%

100,000 30, ie e eff r    % 1..

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.1275, 5)

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.1275, 5) = -100, ,000(3.5388) = $6,164

100,000 30, P = -100, ,000(P/A, i eff, 5) = -100, ,000(P/A,.1275, 5) = -100, ,000(3.5388) = $6,164

Example: Suppose a bank pays interest on a CD account at 6% per annum compounded continuously. What is the effective rate?

Soln: i eff = e =.0618 = 6.18%

Example: Suppose a bank pays interest on a CD account at 6% per annum compounded continuously. What is the effective rate? Soln: i eff = e =.0618 = 6.18% Check: Let r=6%, m=999 i eff = ( 1 + r/m) m - 1 = (1+.06/999) =.0618 = 6.18%

EFFECTIVE INTEREST RATE i e = effective interest rate per payment period = ( 1 + interest rate per cp ) (# of cp per pay period) – 1 = 1 + r m e – 1 m Example: r = 12% APR, compounded monthly, payments quarterly i month = 12% yearly = 1 % compounded monthly 12 months i e = (1 +.01) 3 – 1 =.0303 – or – 3.03% per payment Compounding Period is More Frequent than the Payment Perod

An “APR” or “% per year” statement is a Nominal interest rate – denoted r – unless there is no compounding period stated The Effective Interest rate per period is used with tables & formulas Formulas for Effective Interest Rate: If continuous compounding, use y is length of pp, expressed in decimal years If cp < year, and pp = 1 year, use m is # compounding periods per year If cp < year, and pp = cp, use m is # compounding periods per year If cp cp, use m e is # cp per payment period

CRITICAL POINT When using the factors, n and i must always match! Use the effective interest rate formulas to make sure that i matches the period of interest (sum any payments in-between compounding periods so that n matches i before using formulas or tables)

Note: Interest doesn’t start accumulating until the money has been invested for the full period! periods Shows up here on CFD… (End of Period Convention) X Deposit made here … i Returns interest here!

Problem 1 The local bank branch pays interest on savings accounts at the rate of 6% per year, compounded monthly. What is the effective annual rate of interest paid on accounts? GIVEN: r = 6%/yr m = 12mo/yr FIND i a : DIAGRA M: NONE NEEDED!

Problem 2 What amount must be deposited today in an account paying 6% per year, compounded monthly in order to have $2,000 in the account at the end of 5 years? GIVEN: F 5 = $2 000 r = 6%/yr m = 12 mo/yr FIND P : yrs P? $2 000 DIAGRA M:

Problem 2 – Alternate Soln What amount must be deposited today in an account paying 6% per year, compounded monthly in order to have $2,000 in the account at the end of 5 years? GIVEN: F 5 = $2 000 r = 6%/yr m = 12 mo/yr FIND P : mos P? $2 000 DIAGRA M:

Problem 3 A loan of $5,000 is to be repaid in equal monthly payments over the next 2 years. The first payment is to be made 1 month from now. Determine the payment amount if interest is charged at a nominal interest rate of 12% per year, compounded monthly yrs $5 000 A ? DIAGRA M: GIVEN: P = $5 000 r = 12%/yr m = 12 mo/yr FIND A :

Problem 4 You have decided to begin a savings plan in order to make a down payment on a new house. You will deposit $1000 every 3 months for 4 years into an account that pays interest at the rate of 8% per year, compounded monthly. The first deposit will be made in 3 months. How much will be in the account in 4 years? yrs $1 000 DIAGRAM: F ?F ?

Problem 5 Determine the total amount accumulated in an account paying interest at the rate of 10% per year, compounded continuously if deposits of $1,000 are made at the end of each of the next 5 years yrs $1 000 DIAGRAM:F ?F ?

Problem 6 A firm pays back a $ loan with quarterly payments over the next 5 years. The $ returns 4% APR compounded monthly. What is the quarterly payment amount? yrs = 20 qtrs $A DIAGRAM: $10 000

Suppose the price of copper is $1,000/ton and price rises by 10% per year. In 5 years, Price = 1000(1.1) 5 = $1, But we still only have 1 ton of copper

Suppose the price of copper is $1,000/ton and price rises by 10% per year. In 5 years, Price = 1000(1.1) 5 = $1, But we still only have 1 ton of copper $1,610 5 years from now buys the same as $1,000 now

Suppose the price of copper is $1,000/ton and price rises by 10% per year. In 5 years, Price = 1000(1.1) 5 = $1, But we still only have 1 ton of copper $1,610 5 years from now buys the same as $1,000 now 10% inflation

Suppose the price of copper is $1,000/ton and price rises by 10% per year. In 5 years, Price = 1000(1.1) 5 = $1, But we still only have 1 ton of copper $1,610 5 years from now buys the same as $1,000 now 10% inflation(deflation = neg. inflation)

Suppose inflation equals 5% per year. Then $1 today is the same as $1.05 in 1 year Suppose we earn 10%. Then $1 invested yields $1.10 in 1 year.

Suppose inflation equals 5% per year. Then $1 today is the same as $1.05 in 1 year Suppose we earn 10%. Then $1 invested yields $1.10 in 1 year. In today’s dollars $1.00 $1.10 $1.05 $1.10

That is $1.10 = 1.05 (1+d) 1 1(1+.10) = 1(1+.05)(1+d)

That is $1.10 = 1.05 (1+d) 1 1(1+.10) = 1(1+.05)(1+d) 1+i = (1+j)(1+d) i = d + j + dj

That is $1.10 = 1.05 (1+d) 1 1(1+.10) = 1(1+.05)(1+d) 1+i = (1+j)(1+d) i = d + j + dj i = interest rate (combined) j = inflation rate d = real interest rate (after inflation rate)

Solving for d, the real interest earned after inflation, where i = interest rate (combined) j = inflation rate d = real interest rate (after inflation rate) j ji d    1

Suppose we place $10,000 into a retirement account which earns 10% per year. How much will we have after 20 years?

Solution: F = 10,000(1+.1) 20 = $67,275

How much is $67, years from now worth if the inflation rate is 3%?

Solution: F T = 67,275(P/F,3,20) = 67,275(1.03) -20 = $37,248

Alternate: Recall = ( )/(1+.03) =.068 j ji d    1

Alternate: Recall = ( )/(1+.03) =.068 F T = 10,000(1+d) 20 = 10,000(1.068) 20 = $37,248 j ji d    1

Alternate: Recall = ( )/(1+.03) =.068 F T = 10,000(1+d) 20 = 10,000(1.068) 20 = $37,248 Note: This formula will not work with annuities. j ji d    1

Superwoman wishes to deposit a certain amount of money at the end of each month into a retirement account that earns 6% per annum (1/2% per month). At the end of 30 years, she wishes to have enough money saved so that she can retire and withdraw a monthly stipend of $3,000 per month for 20 years before depleting the retirement account. Assuming there is no inflation and that she will continue to earn 6% throughout the life of the account, how much does Superwoman have to deposit each month? You need only set up the problem with appropriate present worth or annuity factors. You need not solve but all work must be shown.

Take everything to time period 360 FP = A(F/A, 0.5, 360) = 3,000(P/A, 0.5, 240) A 3,000 FP

Take everything to time period 360 FP = A(F/A, 0.5, 360) = 3,000(P/A, 0.5, 240) A(1,004.52) = 3,000(139.58) A 3,000 FP

Take everything to time period 360 FP = A(F/A, 0.5, 360) = 3,000(P/A, 0.5, 240) A(1,004.52) = 3,000(139.58) A = $ A 3,000 FP

Suppose that the solution to the above problem results in monthly deposits of $200 with an amassed savings of $350,000 by the end of the 30th year. For this problem assume that inflation is 3% per annum. Compute the value of the retirement account in year 30 before funds are withdrawn (in today’s dollars)

,000 FP = 350,000 FP T = 350,000(1+j) -n

,000 FP = 350,000 FP T = 350,000(1+j) -n = 350,000(1+0.03) -30 = $144,195

,000 FP = 418,195 FP T = 418,195(1+j) -n = 418,195(1+0.03) -30 = $172,290