Assignment 3 Solutions

Transcription

1 ssignment 3 Solutions Timothy Vis January 30, P 900, r 10%, t 9 months, I?. Given I P rt, we have I (900)(0.10)( 9 12 ) I 40, P 400, t 4 years, r?. Given I P rt, we have 40 (400)r(4), so that r 40 (400)(4) % P 6000, r 6%, t 8 months,? Given, we have 6000(1 + (0.06)( 8 12 )) , r 12%, t 7 months, P? Given P (1 rt), we have 8000 P (1 + (0.12)( 7 12 )), so that P (0.12)( 7 12 ) Solve I P rt for P. I P rt I P rt rt rt I P rt Solve P + P rt for r. P + P rt P P rt P P rt P t P t P r P t Here we are given r 18%, P 835, t 2 months, and asked to find I. I P rt 1

2 (835)(0.18)( 2 12 ) Here we are given P 10, 000, t 6 months, r 6.5% and asked to find. 10, 000(1 + (0.065)( 6 12 )) 10, Here we are given , t 5 months, P 3000 and asked to find r ( r) r r 12 5 ( ) r r 7.8% r Here we are given t 33 days, 1000, P and asked to find r ( r) r r 1 11 ( ) r r 4.205% r 2

3 This problem really takes two steps. First we need to figure out the final value of the note after the 180 days. Then we use this final value to determine the effective rate earned by the third party. For the first part, we have P 10, 000, t 180 days, r 7%, and we must find. (10, 000)(1 + (0.07)( )) 10, 350 With this information we go on to the second part. We now have P 10, 124, t days, 10, 350, and we must find r. 10, , 124( r) 10, , r 10, , r 10, 350 3( 10, 124 1) r r 6.697% r The principal P is the cost of the stock plus the broker s commission on that purchase. The cost of the stock is (450)(21.40) 9630, so the commission, using the commission schedule, is 37 + (0.014)(9630) dding these gives a principal P The final amount is the selling price of the stock minus the broker s commission on that sale. The selling price of the stock is (450)(24.60) 11, 070, so the commission, using the commission schedule, is (0.007)(11, 070) This gives a final amount of 11, , To make things easier, we find the interest earned as P 10, So, we have I , t 26 weeks, P and we need to find r. I P rt ( )r( ) r r 3

4 r % r We have P 10, 000, i 0.08, n 30 and we need to find 10, 000( ) , We have 1000, i 0.015, n 60 and we need to find P P (1 + i) n 1000 ( ) In all cases, we have P 2000, r 0.07, t 5 years. Compounded annually, we get i 0.07, n ( ) lso, I P Compounded quarterly, we get i and n (5)(4) ( ) Then, I P Compounded monthly, we get i and n (5)(12) ( )

5 Then, I P This is just annual percentage yield. In the first case, we have r 0.06, m 12. P Y (1 + r m )m 1 ( ) % In the second case, we have r 0.08, m 2. P Y (1 + r m )m 1 ( ) % This is essentially the same sort of problem; here we have r 0.052, t 8, 160, 000, and we need to find P. Since the compounding is annual, we also have i r and n t. P (1 + i) n 160, 000 ( ) 8 106, Here we have r , 10, P 6, and we need to find t. Since the compounding is annual, we have i r and n t. Note that we can use billions of people rather than people, hence the 10 and 6, with the same outcome. P (1 + i)n ln P ln(1 + i)n ln ln P n ln(1 + i) 5

6 n ln ln P ln(1 + i) ln 10 ln 6 ln( ) To determine the better investment, we consider the effective annual rates of each. Clearly 8.3% compounded annually has an annual percentage yield of 8.3%. It remains to find the PY of 8% compounded quarterly. P Y (1 + r m )m 1 ( ) % The annual yield is better for the annual rate of 8.3% Here we have P 15, 000, 20, 000, r 0.07, m 4, which gives us i , and we need to find n. P (1 + i)n ln ln P n ln(1 + i) n ln ln P ln(1 + i) ln 20, 000 ln 15, 000 ln( ) 17 So 17 quarters are needed or 4.25 years for the desired growth to occur Essentially, we just plug into the various formulas with P 1, r 0.02, t With simple interest, we have: 1(1 + (0.02)(2010)) With annual compounding, we have i 0.02, and n 2010, so that: 1( ) , 350, 474, 163, 852,

7 This is about 193 quadrillion dollars To solve this, we form the appropriate equations for compounding interest on each investment: 4800( )n and 5000( )n. Set these equal to y 1 and y 2 and graph in a window. Find the intersection point of the graphs at n You need to do this graphically. This corresponds to 17 months as the point when the 4800 dollar investment at the higher interest rate becomes more valuable Here we have 20, 000, r , t 10. Since m 1, this gives i , n 10. P (1 + i) n 20, 000 ( ) , Here we have 40, 000, P 32, 000, t 5. Since m 1, this gives n 5 and i r. P (1 + i)n ( P ) 1 n 1 + i i ( P ) 1 n 1 40, 000 ( 32, 000 ) % 7