Financial Math: Formulas, Calculations, and Applications

Posted on Dec 13, 2024 in Mathematics

Financial Mathematics: Formulas and Calculations

Tutorial 1: Decimal Shifting and Interest Formulas

• If you get a positive value times a number, you need to shift the decimal to the right as many times as the number specified.
• If the number is negative, move it to the left.

Simple Interest Formula:

S = FV = P(1 + i)^k

Compound Interest Formula:

S_k = P(1 + i)^k

S_N = P(1 + I/T)ⁿ

Where:

• I = Interest
• T = Frequency of compounding
• K = Number of years
• N = Total number of periods (K * T or T * K)

Depreciation Formula:

V₀ or P = Initial value

V_k = P(1 – d)^k

Tutorial 2: Interest Rate Calculations

1. Five-Year Investment Returns:

1 + r = (^FV⁄_PV)^1/5

(i) r = 10.38%

(ii) r = 10.47%

• r = 10.51%
• r = 10.52%
• r = 10.52%

2. Variable Interest Rate Calculation:

1 + r = (1 + 0.06/12)⁸ ⋅ (1 + 0.072/12)⁴

1 + r = (1.005)⁸ ⋅ (1.006)⁴

1 + r = (1.0407) ⋅ (1.0242) = 1.06591

r = 6.59%

For an initial outlay of $1000, the net return is 1,000(1.067) – 10 = 1,057.

Rate of return: 5.7%

For larger outlays, e.g., 10,000: 10,000(1.067) – 10 = 10,660.

Rate of return: 6.6%

3. Bond Investment Analysis:

2500 = 97(1 + r)⁴⁰ Take logs of both sides.

Ln(2500/97) = 40Ln(1 + r), or 3.249335 = 40Ln(1 + r), or Ln(1 + r) = 0.0812334

Take the exponential of both sides: 1 + r = 1.084624 and r = 8.4624%

97(1.0867)⁴⁰ = 97(27.822) = 2698.72

Either (i) the rate of return is less than the bond rate or (ii) the $97 would have grown to more than $2,500; hence the purchase wasn’t a good investment.

4. Present Value Calculations:

(i) 10,000

(ii) 10,000(1.08)^-2 = 10,000(0.8573) = 8573.39

• 10,000(1.08)^-10 = 10,000(0.4632) = 4631.93

5. Future Value Calculations:

(i) 1,050(1.05)^-1 = 1000

(ii) 1,108(1.05)^-2 = 1004.99 (*)

• 1,160(1.05)^-3 = 1002.05

6. Present Value of Multiple Cash Flows:

PV = 10,000(1.07)^-2 + 5,000(1.07)^-3 + 15,000(1.07)^-5

PV = 8,734.39 + 4,081.49 + 10,694.79

PV = 23,510.67

7. Interest Rate Calculation with Quarterly Compounding:

100,000(1 + ⁱ⁄₄)¹⁶ = 125,000

(1 + ⁱ⁄₄)¹⁶ = 1.25 → 1 + ⁱ⁄₄ = (1.25)^1/16 = 1.014044

i = 0.0562 or 5.62%

OR use logarithms:

Ln[(1 + i/4)¹⁶] = Ln 1.25 and 16Ln(1 + i/4) = 0.22314

Ln(1 + i/4) = 0.0139465 and 1 + i/4 = 1.014044.

8. Time to Double Investment:

15,000(1 + ^0.055⁄₁₂)^12k = 30,000

(1 + ^0.055⁄₁₂)^12k = 2

12k Ln(1 + ^0.055⁄₁₂) = Ln 2

12k ⋅ 0.0045728 = 0.69315

k = 12.63 years. About 12 years and 7 ½ months.

Tutorial 3: Net Present Value and Loan Calculations

1. Net Present Value (NPV) Comparison:

Add up PV to get NPV

2. Finding the Internal Rate of Return (IRR):

Find i such that NPV(i) = 0

NPV(10%) = -15,000 + 909.09 + 4,793.39 + 10,518.41

NPV(10%) = 1,220.89 > 0

NPV(12%) = -15,000 + 892.86 + 4,623.72 + 9,964.92

NPV(12%) = 481.51 > 0

NPV(13%) = -15,000 + 884.96 + 4,542.25 + 9,702.70

NPV(13%) = 129.91 > 0

NPV(14%) = -15,000 + 877.19 + 4,462.91 + 9,449.60

NPV(14%) = -210.29 < 0

Say i is approximately i = 13.38%

3. Present Value of an Annuity:

PV = ^{150[1 – (1 + 0.052/52)-156]}⁄_0.052/52

PV = ^{150[1 – 0.8556]}⁄_0.001 = 21,656.12

4. Future Value of an Annuity:

FV = ^{150[(1.001)156 – 1]}⁄_0.001

FV = ^{150[1.16873 – 1]}⁄_0.001 = 25,310.26

FV = PV(1.001)¹⁵⁶

25,310.26 = 21,656.12(1.16873) = 25,310.27

Almost perfect match.

5. Loan Repayment Calculations:

(a) R = ^{120,000(0.05/12)}⁄_{[1 – (1 + 0.05/12)^-120]} = ⁵⁰⁰⁄_{[1 – 0.60716]}

R = 1272.79

• Outstanding Balance: B = 1272.79[1 – (1 + ^0.05⁄₁₂)^-96]⁄_(0.05/12)

B = ^1272.79⁄_0.05/12[1 – 0.6709] = 100,536.97

• New R = ^{100,536.97(0.09/12)}⁄_{[1 – (1 + 0.09/12)^-96]}

New R = ^{100,536.97(0.0075)}⁄_{[1 – 0.48806]} = 1472.89