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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

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