Financial Mathematics: Comprehensive Notes

Table of Contents

1. Introduction to Financial Mathematics
2. Simple Interest
3. Compound Interest
4. Present and Future Value
5. Annuities
6. Amortization
7. Bonds and Investments
8. Risk and Return
9. Derivatives Basics
10. Financial Mathematics Quiz

1. Introduction to Financial Mathematics

Financial mathematics is the application of mathematical methods to financial problems. It draws from probability theory, statistics, computational methods, and economic theory to provide powerful tools for analyzing financial markets, developing investment strategies, and managing risk.

Key areas of financial mathematics include:

Interest theory: The study of how money grows over time
Pricing models: Mathematical frameworks for determining the fair value of financial instruments
Risk management: Techniques for measuring and controlling financial risks
Portfolio optimization: Methods for selecting investments to achieve the best returns for a given level of risk

Historical Note: Modern financial mathematics began to flourish in the 1970s with the development of the Black-Scholes model for options pricing, which earned its creators the Nobel Prize in Economics in 1997.

2. Simple Interest

Simple interest is the most basic form of interest calculation, where interest is computed only on the initial principal.

Simple Interest Formula:
I = P × r × t

Where:
I = Interest
P = Principal (initial amount)
r = Interest rate (per period)
t = Time (number of periods)

Future Value with Simple Interest:
FV = P × (1 + r × t)

Example 1: Calculating Simple Interest

Problem: If you invest $5,000 in a savings account with a simple interest rate of 4% per annum, how much interest will you earn after 3 years?

Solution:

Using the simple interest formula: I = P × r × t

I = $5,000 × 0.04 × 3 = $600

Therefore, you will earn $600 in interest after 3 years.

Example 2: Finding the Principal

Problem: If you earned $750 in simple interest over 2.5 years at an interest rate of 6% per annum, what was the initial principal?

Solution:

Using the simple interest formula and solving for P: I = P × r × t

$750 = P × 0.06 × 2.5

P = $750 ÷ (0.06 × 2.5) = $750 ÷ 0.15 = $5,000

Therefore, the initial principal was $5,000.

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3. Compound Interest

Compound interest occurs when interest is added to the principal, so that future interest is calculated on the new total (principal + previously earned interest). This creates exponential growth over time.

Compound Interest Formula:
FV = P × (1 + r)^t

Where:
FV = Future Value
P = Principal (initial amount)
r = Interest rate per period
t = Number of periods

Compound Interest with Multiple Compounding Periods per Year:
FV = P × (1 + r/n)^n×t

Where:
n = Number of compounding periods per unit time

Example 1: Annual Compounding

Problem: If you invest $10,000 at 5% per annum compounded annually, what will be the value of your investment after 10 years?

Solution:

Using the compound interest formula: FV = P × (1 + r)^t

FV = $10,000 × (1 + 0.05)¹⁰ = $10,000 × 1.62889 = $16,288.90

Therefore, after 10 years, your investment will be worth $16,288.90.

Example 2: Quarterly Compounding

Problem: If you invest $5,000 at 6% per annum compounded quarterly, what will be the value of your investment after 5 years?

Solution:

Using the formula for multiple compounding periods: FV = P × (1 + r/n)^n×t

FV = $5,000 × (1 + 0.06/4)^4×5 = $5,000 × (1 + 0.015)²⁰ = $5,000 × 1.3468 = $6,734.00

Therefore, after 5 years, your investment will be worth $6,734.00.

Example 3: Continuous Compounding

Problem: What is the future value of $7,500 invested for 6 years at 4.5% compounded continuously?

Solution:

For continuous compounding, we use the formula: FV = P × e^rt

FV = $7,500 × e^0.045×6 = $7,500 × e^0.27 = $7,500 × 1.3100 = $9,825.00

Therefore, with continuous compounding, your investment will be worth $9,825.00 after 6 years.

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Key Insight: The more frequently interest is compounded, the greater the future value. The mathematical limit of this process is continuous compounding, which uses the formula FV = P × e^rt.

4. Present and Future Value

Present value (PV) and future value (FV) are fundamental concepts in financial mathematics that relate to the time value of money.

Future Value:
FV = PV × (1 + r)^t

Present Value:
PV = FV ÷ (1 + r)^t = FV × (1 + r)^-t

Where:
PV = Present Value
FV = Future Value
r = Discount rate (per period)
t = Number of periods

Example 1: Finding Present Value

Problem: You need $50,000 in 7 years for your child's education. If you can earn 6% per annum compounded annually, how much should you invest today?

Solution:

Using the present value formula: PV = FV × (1 + r)^-t

PV = $50,000 × (1 + 0.06)^-7 = $50,000 × 0.7050 = $35,250

Therefore, you need to invest $35,250 today to have $50,000 in 7 years.

Example 2: Finding the Required Rate

Problem: If you invest $15,000 today and need $25,000 after 5 years, what annual interest rate must you earn?

Solution:

Using the future value formula and solving for r: FV = PV × (1 + r)^t

$25,000 = $15,000 × (1 + r)⁵

(1 + r)⁵ = $25,000 ÷ $15,000 = 1.6667

Taking the 5th root of both sides: (1 + r) = 1.6667^1/5 = 1.1077

r = 1.1077 - 1 = 0.1077 = 10.77%

Therefore, you need an annual interest rate of approximately 10.77%.

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Important Concept: The present value represents the current worth of a future sum of money, given a specified rate of return. It is a crucial concept in investment decisions, as it allows comparison of cash flows occurring at different times.

5. Annuities

An annuity is a series of equal payments made at regular intervals. There are two main types: ordinary annuities (payments at the end of each period) and annuities due (payments at the beginning of each period).

Future Value of an Ordinary Annuity:
FV = PMT × [(1 + r)ⁿ - 1] ÷ r

Present Value of an Ordinary Annuity:
PV = PMT × [1 - (1 + r)^-n] ÷ r

Future Value of an Annuity Due:
FV = PMT × [(1 + r)ⁿ - 1] ÷ r × (1 + r)

Present Value of an Annuity Due:
PV = PMT × [1 - (1 + r)^-n] ÷ r × (1 + r)

Where:
PMT = Payment amount per period
r = Interest rate per period
n = Number of periods

Example 1: Future Value of an Ordinary Annuity

Problem: If you save $2,000 at the end of each year for 10 years and earn 6% per annum, how much will you have at the end of 10 years?

Solution:

Using the formula for the future value of an ordinary annuity: FV = PMT × [(1 + r)ⁿ - 1] ÷ r

FV = $2,000 × [(1 + 0.06)¹⁰ - 1] ÷ 0.06

FV = $2,000 × [1.7908 - 1] ÷ 0.06 = $2,000 × 0.7908 ÷ 0.06 = $2,000 × 13.18 = $26,360

Therefore, you will have $26,360 at the end of 10 years.

Example 2: Present Value of an Ordinary Annuity

Problem: You will receive $5,000 at the end of each year for 5 years. If the discount rate is 7% per annum, what is the present value of this annuity?

Solution:

Using the formula for the present value of an ordinary annuity: PV = PMT × [1 - (1 + r)^-n] ÷ r

PV = $5,000 × [1 - (1 + 0.07)^-5] ÷ 0.07

PV = $5,000 × [1 - 0.7130] ÷ 0.07 = $5,000 × 0.287 ÷ 0.07 = $5,000 × 4.1 = $20,500

Therefore, the present value of the annuity is $20,500.

Example 3: Comparing Ordinary Annuity vs. Annuity Due

Problem: Compare the future value of $3,000 paid annually for 8 years at 5% as an ordinary annuity versus an annuity due.

Solution:

For an ordinary annuity: FV = PMT × [(1 + r)ⁿ - 1] ÷ r

FV = $3,000 × [(1 + 0.05)⁸ - 1] ÷ 0.05 = $3,000 × [1.4775 - 1] ÷ 0.05 = $3,000 × 9.55 = $28,650

For an annuity due: FV = PMT × [(1 + r)ⁿ - 1] ÷ r × (1 + r)

FV = $3,000 × [(1 + 0.05)⁸ - 1] ÷ 0.05 × (1 + 0.05) = $28,650 × 1.05 = $30,082.50

The annuity due provides $1,432.50 more due to the earlier timing of payments, allowing for an extra period of interest.

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6. Amortization

Amortization is the process of paying off a debt (typically a mortgage or loan) through regular payments that cover both principal and interest. With each payment, the principal balance decreases, leading to smaller interest charges on subsequent payments.

Regular Payment Amount Formula:
PMT = P × r × (1 + r)ⁿ ÷ [(1 + r)ⁿ - 1]

Where:
PMT = Payment amount per period
P = Principal (loan amount)
r = Interest rate per period
n = Number of periods

Example 1: Calculating Loan Payments

Problem: You take out a mortgage of $300,000 with a 30-year term at an annual interest rate of 4.5% compounded monthly. What is your monthly payment?

Solution:

For this problem:

P = $300,000

r = 0.045 ÷ 12 = 0.00375 (monthly rate)

n = 30 × 12 = 360 (total number of monthly payments)

Using the payment formula: PMT = P × r × (1 + r)ⁿ ÷ [(1 + r)ⁿ - 1]

PMT = $300,000 × 0.00375 × (1.00375)³⁶⁰ ÷ [(1.00375)³⁶⁰ - 1]

PMT = $300,000 × 0.00375 × 3.8443 ÷ 2.8443 = $300,000 × 0.00507 = $1,521

Therefore, your monthly mortgage payment will be $1,521.

Example 2: Creating an Amortization Schedule

Problem: Create an amortization schedule for the first 3 months of a $50,000 car loan with a 5-year term and an annual interest rate of 6% compounded monthly.

Solution:

For this problem:

P = $50,000

r = 0.06 ÷ 12 = 0.005 (monthly rate)

n = 5 × 12 = 60 (total number of monthly payments)

Using the payment formula: PMT = P × r × (1 + r)ⁿ ÷ [(1 + r)ⁿ - 1]

PMT = $50,000 × 0.005 × (1.005)⁶⁰ ÷ [(1.005)⁶⁰ - 1] = $966.64

Payment #	Beginning Balance	Payment	Interest	Principal	Ending Balance
1	$50,000.00	$966.64	$250.00	$716.64	$49,283.36
2	$49,283.36	$966.64	$246.42	$720.22	$48,563.14
3	$48,563.14	$966.64	$242.82	$723.82	$47,839.32

Note that with each payment, more of the payment goes toward principal and less toward interest.

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Key Point: In the early years of a loan, most of each payment goes toward interest rather than principal. This is why making extra principal payments early in the loan term can significantly reduce the total interest paid over the life of the loan.

7. Bonds and Investments

Bonds are debt securities that represent loans made by investors to borrowers. Understanding bond pricing and yield calculations is essential in financial mathematics.

Bond Price Formula:
P = C × [1 - (1 + r)^-n] ÷ r + F × (1 + r)^-n

Where:
P = Bond price
C = Coupon payment per period
F = Face (par) value
r = Yield rate per period
n = Number of periods until maturity

Current Yield:
Current Yield = Annual Coupon Payment ÷ Bond Price

Yield to Maturity (YTM):
There is no direct formula for YTM; it must be calculated using numerical methods or approximated.

Example 1: Bond Pricing

Problem: A 5-year bond with a face value of $1,000 pays a 6% annual coupon (3% semiannually). If the market yield is 8% compounded semiannually, what is the bond's price?

Solution:

For this problem:

F = $1,000

C = 0.06 × $1,000 ÷ 2 = $30 (semiannual coupon payment)

r = 0.08 ÷ 2 = 0.04 (semiannual yield rate)

n = 5 × 2 = 10 (number of semiannual periods)

Using the bond price formula: P = C × [1 - (1 + r)^-n] ÷ r + F × (1 + r)^-n

P = $30 × [1 - (1 + 0.04)^-10] ÷ 0.04 + $1,000 × (1 + 0.04)^-10

P = $30 × [1 - 0.6756] ÷ 0.04 + $1,000 × 0.6756

P = $30 × 8.11 + $1,000 × 0.6756 = $243.30 + $675.60 = $918.90

Therefore, the bond's price is $918.90, which is less than its face value because the market yield (8%) is higher than the coupon rate (6%).

Example 2: Current Yield and Approximate YTM

Problem: A 10-year, $1,000 face value bond with a 7% annual coupon rate is currently trading at $920. Calculate the current yield and approximate the yield to maturity.

Solution:

Current Yield = Annual Coupon Payment ÷ Bond Price

Current Yield = (0.07 × $1,000) ÷ $920 = $70 ÷ $920 = 0.0761 = 7.61%

For an approximation of the YTM, we can use the formula:

Approximate YTM = [C + (F - P) ÷ n] ÷ [(F + P) ÷ 2]

Where C is the annual coupon payment, F is the face value, P is the price, and n is the number of years to maturity.

Approximate YTM = [$70 + ($1,000 - $920) ÷ 10] ÷ [($1,000 + $920) ÷ 2]

Approximate YTM = [$70 + $8] ÷ [$960] = $78 ÷ $960 = 0.0813 = 8.13%

Therefore, the current yield is 7.61% and the approximate YTM is 8.13%.

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8. Risk and Return

The relationship between risk and return is fundamental in financial mathematics. Investors are compensated for taking on additional risk through higher expected returns.

Expected Return of a Portfolio:
E(R_p) = Σ w_i × E(R_i)

Variance of a Portfolio:
σ_p² = Σ w_i² × σ_i² + Σ Σ w_i × w_j × σ_i × σ_j × ρ_ij

Standard Deviation (Volatility):
σ_p = √(σ_p²)

Sharpe Ratio:
S = (R_p - R_f) ÷ σ_p

Where:
E(R_p) = Expected return of the portfolio
w_i = Weight of asset i in the portfolio
E(R_i) = Expected return of asset i
σ_p² = Variance of the portfolio
σ_i² = Variance of asset i
ρ_ij = Correlation coefficient between assets i and j
R_f = Risk-free rate

Example 1: Portfolio Expected Return and Risk

Problem: A portfolio consists of 60% in Stock A and 40% in Stock B. Stock A has an expected return of 12% with a standard deviation of 20%, while Stock B has an expected return of 8% with a standard deviation of 15%. If the correlation between the stocks is 0.5, calculate the portfolio's expected return and standard deviation.

Solution:

Expected Return of the Portfolio:

E(R_p) = w_A × E(R_A) + w_B × E(R_B)

E(R_p) = 0.6 × 12% + 0.4 × 8% = 7.2% + 3.2% = 10.4%

Portfolio Variance:

σ_p² = w_A² × σ_A² + w_B² × σ_B² + 2 × w_A × w_B × σ_A × σ_B × ρ_AB

σ_p² = 0.6² × 0.2² + 0.4² × 0.15² + 2 × 0.6 × 0.4 × 0.2 × 0.15 × 0.5

σ_p² = 0.36 × 0.04 + 0.16 × 0.0225 + 2 × 0.24 × 0.2 × 0.15 × 0.5

σ_p² = 0.0144 + 0.0036 + 0.0072 = 0.0252

Portfolio Standard Deviation:

σ_p = √(σ_p²) = √0.0252 = 0.1587 = 15.87%

Therefore, the portfolio has an expected return of 10.4% with a standard deviation of 15.87%.

Example 2: Sharpe Ratio Comparison

Problem: Two mutual funds have the following characteristics: Fund X has an expected return of 14% with a standard deviation of 22%, while Fund Y has an expected return of 11% with a standard deviation of 15%. If the risk-free rate is 3%, which fund provides better risk-adjusted returns according to the Sharpe ratio?

Solution:

Sharpe Ratio for Fund X:

S_X = (R_X - R_f) ÷ σ_X = (14% - 3%) ÷ 22% = 11% ÷ 22% = 0.5

Sharpe Ratio for Fund Y:

S_Y = (R_Y - R_f) ÷ σ_Y = (11% - 3%) ÷ 15% = 8% ÷ 15% = 0.53

Therefore, Fund Y provides better risk-adjusted returns with a Sharpe ratio of 0.53 compared to Fund X's Sharpe ratio of 0.5.

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Modern Portfolio Theory Insight: Diversification can reduce portfolio risk without necessarily reducing expected returns. This is because assets that are not perfectly correlated don't move in lockstep, which dampens the overall portfolio volatility.

9. Derivatives Basics

Derivatives are financial contracts whose value depends on underlying assets or variables. Options, futures, forwards, and swaps are common derivatives.

Black-Scholes Option Pricing Model:
C = S × N(d₁) - K × e^-rt × N(d₂)

where:
d₁ = [ln(S/K) + (r + σ²/2) × t] ÷ [σ × √t]
d₂ = d₁ - σ × √t

And:
C = Call option price
S = Current stock price
K = Strike price
r = Risk-free interest rate
t = Time to expiration (in years)
σ = Volatility of the underlying asset
N() = Cumulative normal distribution function

Put-Call Parity:
C + K × e^-rt = P + S

Where:
P = Put option price

Example 1: Binomial Option Pricing (One-Period)

Problem: Using a one-period binomial model, price a European call option with a strike price of $50 on a stock currently trading at $48. Assume the stock can move up by 20% or down by 15% in one period, and the risk-free rate is 5% per period.

Solution:

Current stock price (S) = $48

Up factor (u) = 1.20, so the up-state price (S_u) = $48 × 1.20 = $57.60

Down factor (d) = 0.85, so the down-state price (S_d) = $48 × 0.85 = $40.80

Strike price (K) = $50

Risk-free rate (r) = 5%

Step 1: Calculate the payoffs at expiration:

Call value if stock goes up (C_u) = max(S_u - K, 0) = max($57.60 - $50, 0) = $7.60

Call value if stock goes down (C_d) = max(S_d - K, 0) = max($40.80 - $50, 0) = $0

Step 2: Calculate the risk-neutral probability (p):

p = (e^r - d) ÷ (u - d) = (1.05 - 0.85) ÷ (1.20 - 0.85) = 0.20 ÷ 0.35 = 0.5714

Step 3: Calculate the option price:

C = e^-r × [p × C_u + (1 - p) × C_d]

C = e^-0.05 × [0.5714 × $7.60 + (1 - 0.5714) × $0]

C = 0.9512 × [0.5714 × $7.60] = 0.9512 × $4.34 = $4.13

Therefore, the call option price is $4.13.

Example 2: Put-Call Parity

Problem: If a European call option with a strike price of $45 and 6 months to expiration costs $5, what should be the price of a European put option with the same strike price and expiration? Assume the current stock price is $42 and the risk-free interest rate is 4% per annum compounded continuously.

Solution:

Using the put-call parity formula: C + K × e^-rt = P + S

Given:

C = $5 (call option price)

K = $45 (strike price)

S = $42 (current stock price)

r = 4% = 0.04 (risk-free rate)

t = 0.5 years (time to expiration)

Solving for P (put option price):

P = C + K × e^-rt - S

P = $5 + $45 × e^-0.04×0.5 - $42

P = $5 + $45 × 0.9802 - $42