# Mathematics. Rosella Castellano. Rome, University of Tor Vergata

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1 and Loans Mathematics Rome, University of Tor Vergata

2 and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings account. The bank pays you 5% interest rate, in the form of a check, each year. How much interest will you earn each year? Because the bank pays you 5% interest each year, your annual (or yearly) interest will be: , 000 = E.50 Then, the annual interest paid to you, is given by: I = PV r.

3 and Loans Future Value for Simple Interest Present Value for Simple Interest If the investment is made for a period of t years, then the total interest accumulated is t times this amount, which gives us the following: Simple Interest The simple interest on an investment (or loan) of PV Euro at an annual interest rate of r for a period of t years is I = PV r t. Example The simple interest over a period of 4 years on a E. 5,000 investment earning 8% per year is I = PV r t: I = (5, 000)(0.08)(4) = E.1, 600.

4 and Loans Future Value for Simple Interest Present Value for Simple Interest Given your E. 1,000 investment at 5% simple interest, how much money will you have after 2 years? To nd the answer, we need to add the accumulated interest to the principal to get the future value (FV ) of your deposit. FV = PV + I = 1, (1, 000)(0.05)(2) = E. 1, 100 FV = PV + I = PV + PVrt = PV (1 + rt)

5 and Loans Future Value for Simple Interest Present Value for Simple Interest The future value of an investment of PV Euro at an annual simple interest rate of r for a period of t years is FV = PV (1 + rt). Example The value, at the end of 4 years, of a E. 5,000 investment earning 8% simple interest per year is: FV = PV (1 + rt) = 5, 000[1 + (0.08)(4)] = E. 6, 600. To nd the interest paid, we could also have computed: I = FV PV = E. 1, 600

6 and Loans Future Value for Simple Interest Present Value for Simple Interest We often want to turn an interest calculation around: rather than starting with the present value and nding the future value, there are times when we know the future value and need to determine the present value. Solving the future value formula for PV gives us the following: PV = FV [1 + rt] 1 Example If an investment earns 5% simple interest and will be worth E. 1,000 in 4 years, then its present value (its initial value) is: PV = 1, 000 [ ] 1 = E. 833, 33

7 and Loans Future Value for The time value of money Present Value for You deposit E. 1,000 into a savings account. The bank pays you 5% interest, which it deposits into your account, or reinvests, at the end of each year. At the end of 5 years, how much money will you have accumulated? At the end of the rst year: PV (1 + rt) = 1, 000( ) = E. 1, 050.

8 and Loans Future Value for The time value of money Present Value for At the end of the second year, the bank will pay you another 5% interest, but this time computed on the total in your account, which is E. 1,050. Thus, you will have a total of: 1, 050 ( ) = E.1, Notice that if you were being paid simple interest on your original E. 1,000, you would have only E. 1,100 at the end of the second year. The extra E is the interest earned on the E. 50 interest added to your account at the end of the rst year.

9 and Loans Future Value for The time value of money Present Value for Having interest earn interest is called compounding the interest. We could continue like this until the end of the fth year, but notice what we are doing: each year we are multiplying by ( ). So, at the end of 5 years, you will have: 1, 000 ( ) 5 E. 1, It is interesting to compare this to the amount you would have if the bank paid you simple interest: FV SI = 1, 000 ( ) = E.1, The extra E is again the e ect of compounding the interest.

10 and Loans Future Value for The time value of money Present Value for Banks often pay interest more often than once a year. Paying interest quarterly (four times per year) or monthly is common. If your bank pays interest monthly, how much will your E. 1,000 deposit be worth after 5 years? The bank will not pay you 5% interest every month, but will give you 1/12 of that each month. Thus, instead of multiplying by ( ) every year, we should multiply by each month. Because there are 5 12 = 60 months in 5 years, the total amount you will have at the end of 5 years is FV = = E.1,

11 and Loans Future Value for The time value of money Present Value for Compare this to the E. 1, you would get if the bank paid the interest every year. You earn an extra E if the interest is paid monthly because interest gets into your account and starts earning interest earlier. The amount of time between interest payments is called the compounding period.

12 and Loans Future Value for The time value of money Present Value for The preceding calculations generalize easily to give the general formula for future value when interest is compounded. or where: FV = PV 1 + r mt m FV = PV (1 + i) n i = r m is the interest paid each compounding period n = m t is the total number of compounding periods.

13 and Loans Future Value for The time value of money Present Value for Example In November 2011, the Bank of Montreal was paying 1.30% interest on savings accounts. If the interest is compounded quarterly, nd the future value of a E. 2,000 deposit in 6 years. What is the total interest paid over the period? We use the future value formula with m = 4: FV = PV 1 + r mt = 2, = m 4 2, The total interest paid is: I = FV PV = 2, , 000 = E

14 and Loans Future Value for The time value of money Present Value for The examples illustrate the concept of the time value of money: a given amount of money received now will usually be worth a di erent amount to us than the same amount received some time in the future. In the example above, we can say that E.2, 000 received now is worth the same as E.2, received 6 years from now, because if we receive E.2, 000 now, we can turn it into E.2, by the end of 6 years.

15 and Loans Future Value for The time value of money Present Value for As we did for simple interest, we can solve the future value formula for the present value and obtain the following formula. The present value of an investment earning interest at an annual rate of r compounded m times per year for a period of t years, with future value FV, is: where: or PV = FV 1 + r mt m PV = FV (1 + i) n i = r/m is the interest paid each compounding period n = mt is the total number of compounding periods.

16 and Loans Future Value for The time value of money Present Value for Example To nd the amount we need to invest in an investment earning 12% per year, compounded annually, so that we will have E. 1 million in 20 years: PV = 1million [ ] 20 = E.103, Put another way, E.1, 000, years from now is worth only E.103, to us now, if we have a 12% investment available.

17 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund A typical de ned-contribution pension fund works as follows: Every month while you work, you and your employer deposit a certain amount of money in an account. This money earns (compound) interest from the time it is deposited. When you retire, the account continues to earn interest, but you may then start withdrawing money at a rate calculated to reduce the account to zero after some number of years. This account is an example of an annuity, that is an account earning interest into which you make periodic deposits or from which you make periodic withdrawals.

18 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund There are various terms used for accounts into which you make payments, based on their purpose. Examples include savings account, pension fund, and sinking fund. A sinking fund is generally used by businesses or governments to accumulate money to pay o an anticipated debt, but we ll use the term to refer to any account into which you make periodic payments.

19 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund Suppose you make a payment of E. 100 at the end of every month into an account earning 3.6% interest per year, compounded monthly. This means that your investment is earning 3.6%/12 = 0.3% (i = 0.036/12 = 0.003) per month. What will be the value of the investment at the end of 2 years (24 months)? Think of the deposits separately. Each earns interest from the time it is deposited, and the total accumulated after 2 years is the sum of these deposits and the interest they earn.

20 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund In other words, the accumulated value is the sum of the future values of each single deposit, taking into account how long each deposit sits in the account. Figure 1 shows a timeline with the deposits and the contribution of each to the nal value.

21 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund For example, the very last deposit (at the end of month 24) has no time to earn interest, so it contributes only E The very rst deposit, which earns interest for 23 months, by the future value formula for compound interest contributes E. 100( ) 23 to the total. Adding together all of the future values gives us the total future value: FV = ( ) + 100( ) ( ) 23 = = 100[1 + ( ) + ( ) 2 ++( ) 23 ]

22 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund This sum is well-known and there is a convenient formula for its value. This formula allows us to calculate the future value as: 1 + x + x x n 1 = x n 1 x 1 In our case, with (x = ), this formula allows us to calculate the future value: FV = 100 ( )24 1 ( ) 1 = 100( )24 1 = 2,

23 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund A sinking fund is an account earning compound interest into which you make periodic deposits. Suppose that the account has an annual rate of r compounded m times per year, so that i = r/m is the interest rate per compounding period. If you make a payment of R at the end of each period, then the future value after t years, or n = mt periods, will be FV = R (1 + i)n 1 i

24 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund Example At the end of each month you deposit E. 50 into an account earning 2% annual interest compounded monthly. To nd the future value after 5 years, we use i = 0.02/12 and n = 12 5 = 60 compounding periods, so: FV = 50 ( /12) /12 = 3,

25 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund Example Your retirement account has E 5,000 in it and earns 5% interest per year compounded monthly. Every month for the next 10 years you will deposit E 100 into the account. How much money will there be in the account at the end of those 10 years? Solution (part 1) This is a sinking fund with R = 100, r = 0.05, m = 12, so i = 0.05/12, and n = = 120. Ignoring for the moment the E 5,000 already in the account, your payments have the following future value: FV = 100 ( /12) /12 = 15,

26 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund What about the E 5,000 that was already in the account? Solution (part 2) That sits there and earns interest, so we need to nd its future value as well, using the compound interest formula: FV = PV (1 + i) n = 5, = 8, 235, Hence, the total amount in the account at the end of 10 years will be: E.15, , = E.23,

27 and Loans Sinking Funds Future Value of a Sinking Fund Payment Formula for a Sinking Fund Suppose that an account has an annual rate of r compounded m times per year, so that i = r/m is the interest rate per compounding period. If you want to accumulate a total of FV in the account after t years, or n = mt periods, by making payments of R at the end of each period, then each payment must be: R = FV i (1 + i) n 1

28 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity Suppose we deposit an amount PV now in an account earning 3.6% interest per year, compounded monthly. Starting 1 month from now, the bank will send us monthly payments of E What must PV be so that the account will be drawn down to E. 0 in exactly 2 years? As before, we write i = r/m = 0.036/12 = 0.003, and we have R = 100. The rst payment of E. 100 will be made 1 month from now, so its present value is. 1 R (1 + i) = 100 ( ) = 100 ( ) 1 = E.99.70

29 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity The second payment, 2 months from now, has a present value of: 1 R (1 + i) 2 = 100 ( ) 2 = 100 ( ) 2 = E That much of the original PV funds the second payment.

30 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity This continues for 2 years, at which point we receive the last payment, which has a present value of and that exhausts the account. 1 R (1 + i) n = 100 ( ) 24 = 100 ( ) 24 = E.93.06

31 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity Figure 2 shows a timeline with the payments and the present value of each.

32 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity Because PV must be the sum of these present values, we get: PV = 100( ) ( ) ( ) 24 = = 100[( ) 1 + ( ) 2 ++( ) 24 ]. We can again nd a simpler formula for this sum: x 1 + x x n = x x x n 1 = = 1 x n x 1

33 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity So, in our case 24 1 ( ) PV = 100 ( ) ( ) = = = 2, If we deposit E. 2, initially and the bank sends us E. 100 per month for 2 years, our account will be exhausted at the end of that time.

34 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity An annuity is an account earning compound interest from which periodic withdrawals are made. Suppose: 1 the account has an annual rate of r compounded m times per year, so that i = r/m is the interest rate per compounding period. 2 the account starts with a balance of PV. If you receive a payment of R at the end of each compounding period, and the account is down to E. 0 after t years, or n = mt periods, then: PV = R 1 n (1 + i) i

35 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity Example At the end of each month you want to withdraw E. 50 from an account earning 2% annual interest compounded monthly. If you want the account to last for 5 years (60 compounding periods), it must have the following amount to begin with: PV = = 2, Notice that: if you make your withdrawals at the end of each compounding period, you have an ordinary annuity. if, instead, you make withdrawals at the beginning of each compounding period, you have an annuity due.

36 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity Because each payment occurs one period earlier, there is one less period in which to earn interest, hence the present value must be larger by a factor of (1 + i) to fund each payment. So, the present value formula for an annuity due is: PV = R (1 + i) 1 (1 + i)n i

37 and Loans Present Value of an Annuity Annuity due Payment Formula for an Ordinary Annuity Suppose: 1 an account has an annual rate of r compounded m times per year, so that i = r/m is the interest rate per compounding period; 2 the account starts with a balance of PV. If you want to receive a payment of R at the end of each compounding period, and the account is down to E.0 after t years, or n = mt periods, then: R = PV i 1 (1 + i) n

38 and Loans In a typical installment loan, such as a car loan or a home mortgage, we borrow an amount of money and then pay it back with interest by making xed payments (usually every month) over some number of years. From the point of view of the lender, this is an annuity. Thus, loan calculations are identical to annuity calculations. The process of paying o a loan is called amortizing the loan, meaning to kill the debt owed

39 and Loans Example Jonny and So a are buying a house, and have taken out a 30-year, E 90,000 mortgage at 8% interest per year. What will their monthly payments be? Solution: From the bank s point of view, a mortgage is an annuity. In this case, the present value is PV = E.90, 000, r = 0.08, m = 12, and n = = 360. To nd the payments, we use the payment formula: 0.08/12 R = 90, ( /12) 360 =

40 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond A bond pays interest until it reaches its maturity, at which point it pays you back an amount called its maturity value or par value. The two parts, the interest and the maturity value, can be separated and sold and traded by themselves.

41 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond A zero coupon bond is a form of bond (government or corporate) that pays no interest during its life but promises to pay you the maturity value when it reaches maturity. Zero coupon bonds are often created sintetically, that is by removing or stripping the interest coupons from an ordinary bond (coupon bond), so are also known as strips. Zero coupon bonds sell for less than their maturity value, and the return on the investment is the di erence between what the investor pays and the maturity value. Although no interest is actually paid, we measure the return on investment by thinking of the interest rate that would make the selling price (the present value) grow to become the maturity value (the future value).

42 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond The e ective interest rate r e of an investment paying a nominal interest rate of r nom compounded m times per year is: r e = 1 + r nom m 1. m Hint: to compare rates of investments with di erent compounding periods, always compare the e ective interest rates rather than the nominal rates.

43 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond Example To calculate the e ective interest rate of an investment that pays 8% per year, with interest reinvested monthly, set r nom = 0.08 and m = 12, to obtain: r e = =

44 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond Here is some additional terminology on ZCB: Treasury Bills (T-Bills), Italian BOT, Maturity Value, Discount Rate, and Yield The maturity value of a ZCB is the amount of money it will pay at the end of its life, that is, upon maturity. Example A 1 year E. 10,000 ZCB has a maturity value of E. 10,000, and so will pay you E. 10,000 after one year.

45 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond The cost of a ZCB is generally less than its maturity value. In other words, a ZCB will generally sell at a discount, and the discount rate is the annualized percentage of this discount; that is, the percentage is adjusted to give an annual percentage. Example A 1 year E. 10,000 ZCB with a discount rate of 5% will sell for 5% less than its maturity value of E. 10,000, that is, for E. 9,500.

46 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond Suppose that a corporation o ers a 10-year bond paying 6.5% with payments every 6 months. If we pay E. 10,000 for bonds with a maturity value of E. 10,000, we will receive 6.5/2 = 3.25% of E. 10,000, or E. 325, every 6 months for 10 years, at the end of which time the corporation will give us the original E. 10,000 back. But bonds are rarely sold at their maturity value. Rather, they are auctioned o and sold at a price the bond market determines they are worth.

47 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond Separately, we determine the present value of an investment worth E. 10,000 ten years from now, if it earned 7% compounded semiannually. For the rst calculation, we use the annuity present value formula, with i = 0.07/2 and n = 2 10 = 20. n 1 (1 + i) PV = R = i 20 1 ( /2) = /2 = 4,

48 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond For the second calculation, we use the present value formula for compound interest: PV = 10, 000( /2) 20 = E.5, Thus, an investor looking for a 7% return will be willing to pay E. 4, for the semiannual payments of E 325 and E. 5, for the E. 10,000 payment at the end of 10 years, for a total of for the 10,000 bond. E.4, , = E.9,

49 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond Example Suppose that bond traders are looking for only a 6% yield on their investment. How much would they pay per E. 10,000 for the 10-year bonds above, which have a coupon interest rate of 6.5% and pay interest every six months? Solution (part 1) We redo the calculation with r = For the annuity calculation we now get: 20 1 ( /2) PV = /2 = 4,

50 and Loans Zero-Coupon Bond E ective Interest Rate Coupon Bond Solution (part 2): For the compound interest calculation we get PV = 10, 000( /2) 20 = E.5, Thus, traders would be willing to pay a total of E.4, E.5, = E.10, for bonds with a maturity value of E. 10,000.

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