Single paymentsCompound interestAnnuitiesFuture valuePresent valuesBasic Mathematics of FinanceL.N. StoutIllinois Wesleyan UniversityFebruary 22, 2008L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 1 / 23Symbol meaning and mnemonicP Principler annual interest ratet number of years, timem number of compounding periods per yearmt number of compounding periods in t yearsn number of compounding or payment periods n = mti Interest rate per compounding period i =rmPnfuture value after n periodsR payment amount made at a regular intervalA present value of an annuityS future value of an annuitySavings plan or Sinking fundL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 2 / 23Single payments Compound interestWe find the amount which accumulates when a principle P is put in anaccount with an annual interest rate r compounded m times a year for tyears is given byPn= P1 +rmmtL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 3 / 23Single payments Compound interest 1 2 3 · · · n − 2 n − 1 n = mtP-P1 +rmmtL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 4 / 23Single payments Compound interestExampleSuppose I deposit $2000 now at 5% annual interest compounded monthlyand ask how much I will have in that account 15 years from now. HereP = $2000, r = .05, m = 12, and t = 15. I will end up with$20001 +.051212·15= $4227.41Doing the same calculation for someone who needs the money in 45 yearsinstead gives$20001 +.051212·45= $18, 886.98With compound interest it pays to wait longer!L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 5 / 23Single payments Compound interestExampleA related calculation lets me calculate how much I would need to depositnow to have $1,000,000 in 15 years, again assuming 5% annual interestcompounded monthly. Here I take1000000 = P1 +.051212·15and solve for P:P =10000001 +.051212·15= 473103.16L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 6 / 23Single payments Compound interestExampleWe can also see how long it takes my $2000 to grow into $10000 bysolving10000 = 20001 +.051212·tfor t using logarithms:5 =100002000=1 +.051212·tlog(5) = 12t log1 +.0512log(5)12 log1 +.0512= tt = 32.26 yearsL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 7 / 23AnnuitiesAn annuity is a regular sequence of payments. We usually assume that thecompounding period and the payment period coincide, so monthlypayments go with monthly compounding. We will consider an annuity due:regular payments of size R are made at the end of each period for nperiods. We assume an interest rate of i per period.L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 8 / 23Annuities 1 2 3 · · · n − 2 n − 1 nR R R R R R RL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 9 / 23AnnuitiesWhen working with annuities we will need the formula for the sum of ageometric series:IfS = a0+ a0r + a0r2+ . . . a0rnthenrS = a0r + a0r2+ . . . + a0rn+ a0rn+1Subtracting gives(1 − r )S = a0(1 − rn+1) so S =a0(1 − rn+1)1 − rL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 10 / 23Annuities Future valueTo find the value of this sequence of payments right after the nth paymentis made we find the future value of each of the payments:L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 11 / 23Annuities Future value 1 2 3 · · · n − 2 n − 1 nR-R(1 + i)n−1R-R(1 + i)...RL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 12 / 23Annuities Future value 1 2 3 · · · n − 2 n − 1 nR-R(1 + i)n−1R-R(1 + i)...RL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 12 / 23Annuities Future value 1 2 3 · · · n − 2 n − 1 nR-R(1 + i)n−1R-R(1 + i)...RL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 12 / 23Annuities Future value 1 2 3 · · · n − 2 n − 1 nR-R(1 + i)n−1R-R(1 + i)...RL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 12 / 23Annuities Future valueSo the sum of the present values of the payments isR + R(1 + i) + R(1 + i)2+ R(1 + i)3+ · · · + R(1 + i)n−1a geometric series with n terms, common ratio 1 + i, and first term R.The formula for the sum of a geometric series givesS = R1 − (1 + i)n1 − (1 + i)= R1 − (1 + i)n−i= R(1 + i)n− 1iL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 13 / 23Annuities Future valueExampleLet us consider the future value of $1000 paid at the end of each monthinto an account paying 8% annual interest for 30 years. How much willaccumulate? This is a future value calculation with R = 1000, n = 360,and i =.0812. This account will accumulate$1000(1 +.0812)360− 1.0812≈ $1, 490, 359.45Note that this is much larger than the sum of the payments, since many ofthose payments are earning interest for many years.L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 14 / 23Annuities Future valueWe can also figure out what payment would need to be made each monthto achieve a financial goal.ExampleHow much do you have to resolve to save at the end of each month inorder to accumulate $24000 in 4 years if you can get 6% annual interestcompounded monthly? Here we know S = 24000, n = 48 and i =.0612, andwe solve for R:$24000 = R(1.005)48− 1.005R = $24000.005(1.005)48− 1= $443.64Here the total of the payments is $443.64 · 48 = $21294.72 the rest comesfrom the interest earned.L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 15 / 23Annuities Present valuesSometimes we want to know how much the annuity is worth at thebeginning of the picture. Here we move each of the payments back in time:L.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 16 / 23Annuities Present values 1 2 3 · · · n − 2 n − 1 nRR1 + iRR(1 + i)2RR(1 + i)3R...R(1 + i)nL.N. Stout (Illinois Wesleyan University) Basic Mathematics of Finance February 22, 2008 17 /