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Exponential Growth Exponential Growth Discrete Compounding Suppose that you were going to invest $5000 in an IRA earning interest at an annual rate of 5.5%. How much interest would you earn during the 1st year? How much is in the account after 1 year? Exponential Growth Interest after 1 year: i1 5000 0.055 $275 P r Account value after 1 year: F1 5000 0.055 5000 1.055 $5275 5000 P i1 $275 What would happen during the 2nd year? Exponential Growth Interest made during the 2nd year: i2 5275 0.055 $290.13 F 1 r Value of account after 2nd year: F2 5275 0.055 5275 1.055 $5,565.13 5275 F 1 i2 $290.13 What about for the 3rd year? Exponential Growth Interest made during 3rd year: i3 $5,565.13 0 . 055 $ 306 . 08 F2 r Value of the account after 3rd year: F3 5565 . 13 5565 .130.055 5565 .131.055 $5,871.21 F2 i3 $306.08 Exponential Growth Summarizing our calculations: P 5000 F1 50001.055 $5,275 F2 5275 1.055 50001.055 $5,565.13 2 5000 1.055 F3 5565 . 131.055 50001.055 $5,871.21 3 5000 1.055 2 Exponential Growth From our calculations, a $5,000 investment into an account with an annual interest rate of 5.5% will have a value of F after t years according to the formula: Ft 50001.055 t Exponential Growth In general, P dollars invested at an annual rate r, has a value of F dollars after t years according to: Ft P1 r t Notice that the interest was paid on a yearly basis, while our money remained in the account. This is called compounding annually or one time per year. Exponential Growth What would happen if the interest was paid more times during the year? Suppose interest is collected at the end of each quarter, (interest is paid four times each year). What would happen to our investment? Exponential Growth Since the annual interest rate is 5.5% this rate needs to be adjusted so that interest is paid on a quarterly basis. The quarterly rate is: r 5.5 r1 1.375 % 4 n 4 Exponential Growth Interest made during 1st quarter: 0.055 i 5000 $68.75 4 P 1 4 r1 4 Value of account after 1st quarter: 0.055 0.055 F1 5000 5000 50001 $5,068.75 4 4 4 P i1 4 Exponential Growth Interest made during the 2nd quarter: 0.055 i 5068.75 4 $69.70 F 2 4 1 4 r1 4 Value of account after 2nd quarter: 0.055 0.055 F2 5068.75 5068.75 5068.751 $5,138.45 4 4 4 F1 4 i 2 $69.70 4 Exponential Growth Interest made during the 3rd quarter: 0.055 i 5138.45 4 $70.65 F 3 4 2 4 r1 4 Account value after 3rd quarter: 0.055 0.055 F3 5138.45 5138.45 5,138.451 $5,209.10 4 4 4 F2 4 i 3 $70.65 4 Exponential Growth Interest made during the 4th quarter: 0.055 i 5209.10 4 $71.63 F 4 4 3 4 r1 4 Account value after 4th quarter: 0.055 0.055 F4 5209.10 5209.10 5209.101 $5,280.72 4 4 4 F3 4 i 4 $71.63 4 Exponential Growth Summarizing our results for 1 year: P 5000 0.055 F1 50001 $5,068.75 4 4 2 0.055 0.055 F2 5068 . 75 1 5000 1 $5,138.45 4 4 4 F1 4 3 0.055 0.055 F3 5138.45 1 5000 1 $5,209.10 4 4 4 F2 4 4 0.055 0.055 F4 5209.10 1 4 50001 4 $5,280.72 4 F3 4 Exponential Growth Notice that the exponent corresponds to the number of quarters in a year: So So So So So for for for for for 1 year there are 4 quarters 2 years there are 8 quarters 3 years there are 12 quarters 4 years there are 16 quarters t years there are 4t quarters Exponential Growth So the value of a $5,000 investment with an annual interest rate of 5.5% compounded quarterly after t years is given by: 0.055 Ft 50001 4 4t Exponential Growth In general, P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years according to: r F P 1 n nt Exponential Growth From the last slide, we can also say: r P F 1 n nt In other words, we can find the present value (P) by knowing the future value (F). Exponential Growth Notice for each of the 3 years the account that is compounded quarterly is worth more than the one compounded annually t 1 2 3 n=1 n=4 F F $5,275.00 $5,280.72 $5,565.13 $5,577.21 $5,871.21 $5,890.34 Exponential Growth It would seem the larger n is the more an investment is worth, but consider: t 1 2 3 n=52 n=365 F F $5,282.55 $5,282.68 $5,581.07 $5,581.34 $5,896.45 $5,896.89 Exponential Growth Notice value of the investment is leveling off when P, r, and t are fixed, but n is allowed to get really big. n r This suggests that 1 n is leveling off to some special number Exponential Growth There is a clever technique that allows us to find this value. We let m = n/r, so that n = mr. For any value of r, m gets larger as n increases. We rewrite the expression: r F P1 rm mrt 1 P 1 m m rt Exponential Growth m As m gets big, 1 1 2.7182818284 5905 e m 10 100 1000 10000 100000 1000000 10000000 100000000 m 1 1 m m 2.59374246 2.704813829 2.716923932 2.718145927 2.718268237 2.718280469 2.718281694 2.718281786 Exponential Growth So as m gets large, r F P1 rm mrt 1 P 1 m Pe rt This is for continuous compounding In Excel, use the function EXP(x) m rt Exponential Growth So P dollars will grow to F dollars after t years compounded continuously at r % by the equation: F Pe rt We can also find P by knowing F as follows: rt P Fe Exponential Growth How do we compare investments with different interest rates and different frequencies of compounding? Look at the values of P dollars at the end of one year Compute annual rates that would produce these amounts without compounding. Annual rates represent the effective annual yield Exponential Growth In our current example when we compounded quarterly, after one year 41 we had: 0.055 F1 50001 $5280.72 4 Notice we gained $280.72 on interest after a year. That interest represents a gain of 5.61% on $5000: Effective Annual 280 .72 y 0.0561 Yield (y) 5000 Exponential Growth Effective annual yield (Discrete): find the difference between our money after one year and our initial investment and divide by the initial investment. Therefore, interest at an annual rate r, compounded n times per year has n yield y: r P 1 P n n r y 1 1 P n Exponential Growth You may need to find the annual rate that would produce a given yield. n r y 1 1 Need to solve for r : This tells you the annual interest rate r that will produce a given yield when compounding n times a year. Note: This is only for Discrete Compounding n r y 1 1 n n y 11/ n 1 r n r 1/ n y 1 1 n 1/ n n y 1 1 r Exponential Growth Effective Annual Yield (Continuous): Pe P r y e 1 P r Annual interest rate: r ln y 1 Exponential Growth Ex. Find the final amount if $10,000 is invested with interest calculated monthly at 4.7% for 6 years. Soln. F P1 0.047 126 10,0001 12 r nt n $13,250.49 Exponential Growth Ex. Find the annual yield on an investment that computes interest at 4.7% compounded monthly. r n Soln. y 1 n 1 1 0.047 12 12 1 1.048025794 1 0.0480 About 4.80% Exponential Growth • Ex. Find the rate, compounded weekly, that has a yield of 9.1% 521 0.091 1 • Soln. r n 1 y 1 / n 1 1 / 52 0.087167685 About 8.72% Exponential Growth Examples that use the word continuous to describe compounding period mean you use: F Pert Ex. How much would you have after 3 years if an investment of $15,000 was placed into an account that earned 10.3% interest compounded continuously? Exponential Growth Soln. F Pe rt 15,000e 0.103 3 $20,430.94 Exponential Growth Ex. Find the annual rate of an investment that has an annual yield of 9% when compounded continuously. Soln. r ln y 1 ln0.09 1 0.0862 Approx 8.62% Exponential Growth Where else can compound interest be used? Financing a home Financing a car Anything where you make monthly payments (with interest) on money borrowed Exponential Growth The average cost of a home in Tucson is roughly around $225,000. Suppose you were planning to put down $25,000 now and finance the rest on a 30 year mortgage at 7% compounded monthly. How much would your monthly payments be? Exponential Growth For a 30 year mortgage, you’ll be making 360 monthly payments. At the end of the 360 months we want the present value (P) of all the monthly payments to add up to the amount you plan to finance, e.g. $200,000 The $200,000 is called the principal Exponential Growth Let’s say that Pk represents the present monthly value k months ago. Then after 360 months, we want: 200,000 P1 P2 P3 P360 Exponential Growth Since we’re borrowing money here, each Pk can be expressed as r Pk F 1 n k But where F represents the future value for Pk. In other words, F is your monthly payment. Exponential Growth Remember we want: 200,000 P1 P2 P3 P360 So if we insert: r Pk F 1 n k We have instead: 1 2 r r r 200,000 F 1 F 1 F 1 n n n 360 Exponential Growth Now for a little algebra (factor out F): 1 2 360 r r r 200,000 F 1 1 1 n n n Divide both sides by the stuff in [ ] 200,000 r r r 1 1 1 n n n 1 2 360 F Exponential Growth The last result will tell us our monthly payment F: 200,000 r r r 1 1 1 n n n 1 2 360 F Notice that all we need to is figure out how to add up the numbers in the bottom. This is where we use Excel. Exponential Growth Since we’re compounding monthly at 7%, r = 0.07 and n = 12 So: 200,000 0.07 0.07 0.07 1 1 1 12 12 12 1 2 360 F Exponential Growth We’ll do the rest of our calculation in Excel So our monthly payments F: F 200,000 360 0.07 1 0.07 2 0.07 1 1 1 12 12 12 200,000 1330.60 150.3076 Exponential Growth Now that we know what F is we can figure out what each Pk is. Again, each Pk will tell us what F dollars was worth k months ago We’ll again use Excel to answer this question. Exponential Growth In Excel: End This number tells us that our monthly payment of $1330.60 was worth $1322.89 one month ago. Notice that as we descend down the table the values get smaller because we’re going farther back in time. This number tells us how much of the monthly payment is for interest. Notice that as we descend the table the interest goes up. This tells us that in the beginning of a payment plan a lot of the monthly payment is toward interest and only a small portion is going toward principal while the reverse is true at the end. Start Exponential Growth What your outstanding balance looks like with each monthly payment? Balance $ Balance 250000 200000 150000 100000 50000 0 -50000 0 Balance 100 200 Months 300 400 Exponential Growth Things to notice: After 360 months of payments of 1330.61, you’re really paying $479,019.60 on $200,000 borrowed. The mortgage company has made 139% profit on your borrowing $200,000.