Equivalence Factors and Formulas Moving Money Around

Moving Money from Present to Future Finding F given P, interest rate i, and N periods (F/P, i,N)= (1+ i ) N “spcaf” F = P (F/P, i,N) = P (1+ i ) N F = 100*(1.1) 10 F = 100*2.594 = P F

Moving Money from Future to Present Finding P given F, interest rate i, and N periods (P/F, i,N)= 1/(1+ i ) N “sppwf” P = F (P/F, i,N) = F/ (1+ i ) N P = (P/F, 0.1,10) = 100 P F

Moving a Uniform Series to the Future Finding F given A, interest rate i, and N periods (F/A, i,N)= ((1+ i ) N –1)/ i “uscaf” F = A (F/A, i,N) =A ((1+ i ) N –1)/ i F = * (F/A, 0.1,10) = A= F

Spreading a Future amount into a Uniform Series Finding A given F, interest rate i, and N periods (A/F, i,N)= i /((1+ i ) N –1) “ussff” A = F (A/F, i,N) = F (i /((1+ i ) N –1)) A = * (A/F, 0.1,10) = A= F

Moving an Annuity to the Present Finding P given A, interest rate i, and N periods (P/A, i,N)= ((1+ i ) N –1)/( i(i+1) N ) “uspwf” P = A (P/A, i,N) P = (P/A, 0.1,10) =100 A= P

Spreading a Present amount into a Uniform Series Finding A given P, interest rate i, and N periods (A/P, i,N)= ( i(i+1) N )/((1+ i ) N –1) “crf” A = P (A/P, i,N) A = 100(A/P, 0.1,10) = P A=

Moving an Arithmetic Gradient to the Present P Finding P given G, interest rate i, and N periods (P/G, i,N)= ((1+ i ) N – i N - 1) / ( i 2 (1+ i ) N ) P = G (P/G, i,N) P = 4.37(P/G, 0.1,10) = 100

Moving an Arithmetic Gradient to a Uniform Series Finding P given G, interest rate i, and N period (A/G, i,N)= ((1+ i ) N – i N - 1 ) / ( i (1+ i ) N - i ) A = G (A/G, i,N) P = 4.37(A/G, 0.1,10) = A=

The Geometric Gradient Finding P given A, g, i, and N (P/A, g, i,N)= (1-(1+ g ) N (1+ i ) -N )/(i-g) P = A (P/A, g, i,N) P = 7.31(P/A, 0.2, 0.1,10)=100

Equivalence does not mean Equal TimeSingle Payment Uniform Series Geometric Gradient Arithmetic Gradient Single Payment 0$100 1$16.27$7.21$0.00 2$16.27$8.65$4.37 3$16.27$10.38$8.74 4$16.27$12.46$ $16.27$14.95$ $16.27$17.94$ $16.27$21.53$ $16.27$25.83$ $16.27$31.00$ $16.27$37.20$39.32$259.37

Most Common Interest Factors So far, we have seen the interest factors most commonly used in determining equivalences –(F/P, i, N) –(P/F, i, N) –(F/A, i, N) –(A/F, i, N) –(P/A, i, N) –(A/P, i, N)

Simple Question 1 You are 20 yrs old and want to retire at age 60 with $1,000,000. How much do you have to put away each year if you earn 10% interest on it? –We want the value of equal payments that are equivalent to a future million of dollars. –The interest is compounded, and rate is 10%/yr –The number of periods is 40 years. –The answer comes from the equivalence between an future worth (F) and an annuity (A)

Simple Question 1 (cont’d) N =40 years, i =10%, (A/F, i,N) = i /((1+ i ) N –1) = 0.1/( ) 40 –1) = Or alternatively, you can find that (A/F, i,N) = at the end of the Workbook, on page 169 F =$1,000,000 Therefore, A = $1,000,000 (0.0023) = about $2,300.

Simple Question 2 At age 5, you were left $10,000 from an aunt. Your parents put the money in a trust fund earning 10%/year. You are now 25 years old and may draw from the fund. How much money is there? –We want the future worth of an investment. –The interest is compounded, and rate is 10%/yr. –The number of periods is 20 years. –The answer comes from the equivalence between present worth (P) and future worth (F) –F = $10,000 (F/P,0.1,20) »10,000 (6.7275) = $67,275

Simple Question 2 (cont’d) N =20 years, i =10%, (F/P, i,N) = (1+ i ) N = (1+0.1) 20 = Or alternatively, you can find that (F/P, i,N) = at the end of the Workbook, on page 169 P =$10,000 Therefore, F = $10,000 (6.7275) = $67,275

Simple Question 3 You buy a house for $100,000. You finance the full amount with a 30 years mortgage. The interest rate is 9%/yr and your payments are monthly. What is the total of all the payments made? –We want the present worth of the value of equal payments. –The interest is compounded, and rate is 0.75%/month. –The number of periods is 360 months. –The answer comes from the equivalence between an annuity (A) and a present worth (P).

Simple Question 3 (cont’d) N = 360 months i = 0.75% (A/P, i, N) = (0.0075(1.0075) 360 )/((1.0075) 360 –1) = Therefore, A = P (A/P, i,N) = 100,00 ( ) = $804.62/month Total of all payments = 360 (804.62) = $289,664.

Simple Question 4 You can afford $300/month on a car. If the dealer’s interest rate is 6%/yr and the loan is for 60 months, how much can you finance? –We want the present worth of the value of equal payments. –The interest is compounded, and rate is 0.5%/mo. –The number of periods is 60 months. –The answer comes from the equivalence between an an annuity (A) and a present worth (P).

Simple Question 4 (cont’d) N =60 months, i =0.5%, (P/A, 0.5, 60 )= , A = $300 Therefore, P= $300 ( ) = $15,518

Simple Question 5 You win the lottery and the government promises to pay $1,000,000 in ten years. Banks currently pay 10%/yr on savings. What is the prize worth to you right now? –We want the present worth of a future cash flow. –The interest is compounded, and rate is 10%/yr. –The number of periods is 10. –The answer comes from the equivalence between future worth (F) and present worth (P)

Simple Question 5 (cont’d) N =10 years, i =10%, (P/F, i,N) = 1/(1+ i ) N = (1+0.1) -10 = Or alternatively, you can find that (P/F, i,N) = at the end of the Workbook, on page 169 F =$1,000,000 Therefore, P = $1,000,000 (0.3855) = $385,500.

Simple Question 7 Your daughter starts college in 18 years. If you put $100/month in a CD returning 6%/year (compounded monthly), how much will you have then? –We want the future worth of a series of equal payments. –The interest is compounded, and rate is 0.5%/mo –The number of periods is 216 months. –The answer comes from the equivalence between an annuity (A) and future worth (F)

Simple Question 7 (cont’d) N =216 months, i =0.5%, (F/A, i,N) = ((1+ i ) N –1)/ i = (( ) )/0.005 = Unfortunately, your Workbook (page 161) can’t help here. A =$100 Therefore, F = $100 ( ) = about $38,735.