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## Chapter 5

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**Chapter 5**Mathematics of Finance**Compound Interest**• In the last class session, we covered the basics regarding compound interest • Tonight, we build on these results to get some help for working with financial investments. • Let P = principal invested, let S = the compound amount (i.e., the amount of money we have after n time periods) and let r = interest rate.**Compound Interest (cont.)**• Then S = P(1 + r)n • Example: Suppose that we purchase a certificate of deposit for $1000 to mature in 4 years with an interest rate of 4.5% • S = $1000(1 + 0.045)n =1000(1.1925) • S = $1,192.50 • Note that in this problem, the idea is to find S, given P, r, and n**Quarterly Compounding**• Suppose now compounding is quarterly. How long does it take for $1500 to become $2100? • $2100 = $1500(1 + 0.0125)n • ln(2100/1500)/ln(1.0125) = n • So, n = 27.088 quarters or 6.77 years • Shows that quarterly compounding decreases the length of time it takes for $1500 to become $2100 at 5% interest.**Power of Compound Interest**• Suppose a person at age 23 puts $1000 aside at 6% interest for retirement. How much money will be available at age 65? • S = $1000(1 + 0.06)42 = $11,560 • Suppose a person at age 60 puts $1000 aside at 6% interest for retirement. How much money will be available at age 65? • S = $1000(1 + 0.06)5 = $1,340**Power of Compound Interest**• Suppose that you have outstanding credit card debt of $4700. The interest rate on the card is 23% per year. How much would be owed at the end of one year if there was no need to make the minimum payments? • S = 4700(1 + 0.24) = $5828.00 • After 5 years it would be: • S = 4700(1 + 0.24)5 = $13,778.63 • After 5 years with monthly compounding: • S = 4700(1 + 0.02)60 = $15,420.84**Two Variations on Compound Interest**• Suppose we know P, S, and n and want to find r. • Example: Assume that we buy a CD for $600 and after 5 years it is worth $900. What interest rate was earned? • Then, $900 = $600(1 + r)5 • So, (900/600)1/5 = (1 + r) • Or, 1.0845 = 1 + r • This means that r = 0.0845**Two Variations (cont.)**• Another use for the compound interest formula comes up if we know S, P, and r and want to find n. • Suppose we know that a CD purchased for $1500 will eventually be valued at $2100. At 5% interest, how long will it take for this to happen. • $2100 = $1500(1.05)n • ln(2100/1500) = n(ln(1.05)) • n = ln(2100/1500)/(ln(1.05)) = 6.897 years**Doubling Time of Money**• Given that S = P(1+ r)n then the principal invested, P, doubles when S = 2P • Example: How long does it take for $1500 to double at 5% • $3000 = $1500(1 +0.05)n • ln(3000/1500) = ln(2) = n(ln(1.05)) • n = ln(2)/ln(1.05) = 14.21 • n = 14.21 years**Doubling Time of Money**• With quarterly compounding, how long does it take for the $1500 to double? • $3000 = $1500(1 + 0.0125)n • n = ln(2)/ln(1.0125) • Daily compounding? • n = ln(2)/ln(1.0001) • Continuous compounding: Use rule of 70, so at 5% interest it takes 14 years(i.e., 5x14 = 70**Effective Rate of Interest**• Definition: The effective rate re that is equivalent to a nominal (annual) rate of interest r compounded n times per year is • re = (1 + (r/n))n – 1 • Example: suppose that compounding occurs 4 times per year • re = (1 + (r/4))4 - 1**Effective Interest Rates**Suppose compounding is quarterly and we want to know the equivalent annual rate of interest • Again, let P = $1500, let the given (nominal) interest rate equal 5% and suppose money is invested for one year • Annual compounding: S = $1500(1.05) = $1575 • Quarterly compounding: S = $1500(1.0125)4 = $1500(1.0509) = $1576.41**EffectiveRate (cont.)**• So, the effective interest rate calculation asks: What annual rate would bring $1500 to $1576.41? This would be re = 0.0509 or 5.09% • Another way to say this is that 5% interest compounded quarterly is the same as 5.09% compounded annually.**Another example**• What effective rate is equivalent to a nominal rate of 6% compounded monthly • re = (1 + (0.06/12))12 = 1.0617 • What if compounding occurs daily? • re = (1 + (0.06/365))365 = 1.0618 • What if compounding occurs continuously? • In this case, as we saw in Chapter 4, the effective rate of interest is e0.06 = 1.0618**#27 p. 222**• A zero-coupon bond is a bond that is sold for less than its face value and has no periodic interest payments; instead the bond is redeemed at face value at maturity. Suppose that such a bond sells today for $420 and can be redeemed in 14 years for $1000. The bond earns what nominal rate compounded semiannually?**Solution**• 1000 = 420(1 + (r/2))28 • Let x = r/2 • 2.381 = (1 + x)28 • ln(2.381) = 28ln(1 + x) • Ln(1 + x ) = ln(2.381)/28 = 0.031 • So, (1 + x) = e0.031 = 1.0315 • This means that x = 0.0315; r = 0.0630**#28, p. 222**• Suppose that $1000 is hidden under a mattress for safekeeping. Each year the purchasing power of money is 96.5% of what it was the previous year. After 6 years, what is the purchasing power of the $1000? • S = $1000(1 + (-0.035))6 • S = $1000(1 – 0.035)6 = $807.54**Homework**• Pp. 221-222: 7,9,11,15,21,25**Present Value**• We know that the future sum, S, that is obtained by investing the principal, P, at 100r% for n years is given by • S = P(1 + r)n • So, if we want to find the value today of a future sum to be paid in n years when the interest rate is 100r%, we would calculate • P = S/(1 + r)n • Or, P = S(1 + r)-n**Example #1**• Find the present value of $1000 due after 3 years if the interest rate is 9% • P = 1000(1.09)-3 = 1000/1.295=772.18 • Same problem but with monthly compounding • P = 1000(1.0075)-36 =1000/1.309=764.15**Example #2**• Suppose that a trust fund for a child’s education is set up by a single payment today so that at the end of 15 years $50,000 will be available for college tuition payments. If the interest rate is 7% compounded semiannually, how much should be paid into the fund today? • P = 50,000(1.035)-30 = $17,813.92**Equations of Value**• Suppose that Mr. Smith owes Mr. Jones two sums of money: $600 due in 5 years and $1000 due in 2 years. The interest rate is 8% compounded quarterly. How much should Mr. Smith pay today to retire both of the debts? • x = 1000(1.02)-8 + 600(1.02)-20 • x = 1000/1.17 + 600/1.49 = 854.70 + 403.78 = $1,258.48**Equations of Value**• A debt of $3000 due in 6 years is to be paid off by three payments, $500 now, $1500 in 3 years and a final payment after 5 years. How large should the final payment be? The interest rate is 6% compounded annually. • 3000 = 500(1.06)6 + 1500(1.06)3 + x(1.06) • Divide both sides by (1.06) • x = 3000(1.06)-1 – 500(1.06)5 – 1500(1.060)2 • x = 2830.19 – 669.11 – 1685.40 = $475.68**Net Present Value**• Suppose that you can invest $20,000 in a business that will guarantee you a cash flow of $10,000 in year 2, $8,000 in year 3, and $6,000 in year 4. the interest rate 7% compounded annually. What is the net present value of the business venture? This is the value today of the cash flow less investment cost. • NPV = 10,000(1.07)-2 + 8,000(1.07)-3 + 6,000(1.07)-4 – 20,000 = -$471.31 • With a negative NPV, the investment should not be undertaken.**Homework**• P. 226-227: 1, 13, 15, 19, 21**Geometric Sequence**• Definition: The sequence of n numbers a, ar, ar2, ar3, … ,arn-1 where a ≠ 0 is called a geometric sequence with first term a and common ratio r Example: Suppose $100 is invested at 6% for 4 years. Then the compound amounts at the end of each year is a geometric sequence and can be written as: 100(1.06), 100(1.06)2, 100(1.06)3, 100(1.06)4**Geometric Series**• Definition: A geometric series is defined as the sum of terms in a geometric sequence. • Example: 100(1.06) + 100(1.06)2 + 100(1.06)3 + 100(1.06)4 • In general, a geometric series can be written as: a + ar + ar2 + ar3 + …. + arn-1**Geometric Series**• Suppose we wish to compute the sum of terms in a geometric series • s = a + ar + ar2 + ar3 + …. + arn-1 • rs = ar + ar2 + ar3 + ar4 + …. + arn • s – rs = a – arn • s(1 – r) = a(1 – rn) • s = a(1 – rn)/(1 – r)**Examples**• Suppose a = 1, r = ½, and n = 6. Find s. • s = a(1 – rn)/(1 – r) • s = (1 – 0.57)/(1 – 0.5) = .9922/0.5 = 1.9844 ≈ 127/64 (see p. 230). • Suppose we wish to evaluate 35 + 36 + 37 + 38 + 39 + 310 + 311 • a = 35 = 243, r = 3, n = 7 • s = 243(1 – 37)/(1 – 3) = 243(-2,186)/(-2) = 265,599**Annuities**• An annuity pays a fixed amount of money per period (year, quarter, month) for a fixed number of years. • Suppose that the payment per period is $R, a period is one year long, the interest rate is r, and there are n payments. In general, how much would we pay today for this income stream? • A = R(1 + r)-1 + R(1 + r)-2 + … + R(1 + r)-n • A = R(1 + r)-1 [1 - (1 + r)-n]/[1 - (1 + r)-1]**Annuities (cont.)**• With a little algebra, The annuity value can be written as • A = R[1 – (1 + r)-n]/r • This formula gives the present value A of an annuity of R dollars per payment for n periods at the interest rate of r per period. • Example: Find the present value of an annuity of $100 per month for 3.5 years at interest rate 6% compounded monthly • Using Appendix B , p. 954, this is $100(37.7983) = $3779.83**Annuities (cont.)**• Or, this value can be found approximately using a calculator. Let anr = [1 – (1 + r)-n]/r • For r = 0.005 and n = 42, anR = 37.8, so A = $3780 • Value of a consol (promise to pay in perpetuity). In this case, anr = 1/r. So, a promise to pay $100 each year in perpetuity would be valued at $2000 if the interest rate is 5%. If the interest rate is 10%, A= $1000**Example 6, p. 231**• Given an interest rate of 5% compounded annually, find the present value of an annuity of $2000 due at the end of each year for three years and $5000 due at thereafter at the end of each year for four years. • A = $5000(5.786) - $3000(2.723) = $2885.91 • where a7,0.05 = 5.786 and a3,0.05 = 2.723**Example 7, p. 232**• If $10,000 is used to purchase an annuity consisting of equal annual payments for four years and the interest rate is 6% compounded annually, find the amount of each payment. • Know that A = Ranr so, R = A/anr • R = $10,000/a4,0.06 = $10,000/3.465 = $2885.91**Another Example**• Suppose that a home mortgage of $150,000 is to be retired by making 48 monthly payments. The interest rate is 6%. Determine the monthly payment. • The way to look at this is that the bank is purchasing an annuity valued today at $150,000 with monthly payment for 4 years at 6% compounded monthly. • R = A/anr = 150,000/a48,0.005 = 150,000/42.580 = $3522.78**Bond Values**• Suppose that you buy a 20-year corporate bond for $10,000. The coupon rate is 5% paid semiannually. How much would you pay for the bond if the interest rate is now 6%? • S = Ranr + 10,000/(1 + r)40 • R = $250, n = 40, r = 0.03, anr = 23.114772 • R = $5778.69 S = $5778.69 + $3065.57 = $8844.26**Bond Prices (cont.)**• Suppose now that interest rates are 4%. What would you pay for the same bond? • S = Ranr + 10,000/(1 + r)40 • R = $250, n = 40, r = 0.02, anr=27.355479 • S = $6838.87 + 4528.90 = $11,367.77 • Two conclusions • When interest rates are above (below) the coupon rate, bonds trade at a discount (premium) • Bond prices vary inversely with the interest rate**Amount of an annuity**• Thus far, we have considered the value of an annuity. This is what should be paid today for a stream of future periodic payments • Now we wish to consider the amount of an annuity: The amount of money available at the end of the period of a stream of payments compounded forward • The formula for the amount of an annuity is • S = R[(1 + r)n – 1]/r = Rsnr**Amount of an Annuity**• Find the amount of an annuity consisting of payments of $50 at the end of every 3 months for three years at 6% compounded quarterly. Also, find the compound interest that would accrue. • S = Rsnr • S = 50s12,0.015 = 50(13.041) = $652.06 • Compound interest accruals are $52.06. • $52.06 = $652.06 - $600**Sinking Fund**• A sinking fund is a fund into which periodic payments are made to satisfy a future obligation. Suppose that a machine costing $7000 is to be replaced at the end of 8 years, when it will have a salvage value of $700. How much must be set aside each quarter to replace the machine if the interest rate is 8% compounded quarterly?**Solution**• The amount needed after 8 years is $7000 - $700 = $6300 • The required quarterly payment is • R = S/snr • R = $6300/s32,0.02 = 6300/44.227 = $142.45 • So, a quarterly payment of $142.45 will accumulate to $6300 after 8 years at 8% interest compounded quarterly**Retirement Planning**• A woman at age 25 has $17,000 in her retirement plan and wishes to contribute $2000 each year until she retires at age 65. Assume an interest rate of 6% compounded annually. How much will she have available for retirement • S = Rsnr + 17000(1 + r)40 • S = 2000(154.762) + 17000(10.29) • S = 309,254 + 174,857.21 = $484,381.21**Homework**• P. 236-238: 1, 5, 7, 13, 19, 23, 25, 27, 31,33, 41, 43