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Learning Objectives
Differences between simple and compound interest
Calculation and comparison of effective and nominal interest rates
Discounted cashflow models representing real world applications
Price and yield of simple bonds
Key financial statistics used to analyze range of financial market scenarios
1. Simple and Compound Interest
1.1 Simple Interest
Simple interest is not commonly used in finance. It is calculated by multiplying the interest rate by the principal amount. Both the principal and the interest payment remain unchanged over time.
1.2 Compound Interest
Compound interest accrues on both the principal and the accumulated interest, meaning that each period's interest increases without any payments being made.
1.3 Simple vs. Compound Interest
The simple interest rate remains constant regardless of the time period, while compound interest grows with each period. The longer the investment term, the greater the disparity between simple and compound interest. This difference is even more significant when the interest rate is high.
2. Nominal and Effective Interest Rates
The nominal annual rate is the stated interest rate that does not consider compounding; it is also known as the Annual Percentage Rate (APR).
The effective rate accounts for compounding and is referred to as the Effective Annual Rate (EAR) or Annual Percentage Yield (APY). The more frequently interest is compounded, the higher the effective rate will be.
2.1 Understanding Types of Rates
When dealing with interest rates, it is crucial to know whether the rate is nominal or effective. If the rate type is not explicitly stated, follow this guidance:
If the compounding period is not given: Assume the rate is effective with the compounding frequency equal to the stated period (e.g., a 5% per year rate indicates effective annual compounding).
If the compounding period is given but the rate type is not stated: Assume the rate is nominal (e.g., a 5% per year rate compounded semi-annually).
3. Discounted Cash Flows
Discounted cash flow (DCF) is a method used for valuing an asset based on future cash flows, which are discounted to determine their present value.
3.1 Time Value of Money
The value of a dollar today is not the same as the value of a dollar in the future due to several factors:
- Opportunity Cost: The potential gains from alternative investments.
- Inflation: The decrease in purchasing power over time.
- Default Risk: The risk that future cash flows may not be received.
Thus, money received today is worth more than the same amount received in the future.
Using Simple Interest:
FV = PV x (1 + r x n) or PV = FV / (1 + r x n)
Using Compound Interest:
FV = PV x (1+r)^n or PV = FV/(1+r)^n
In Excel:
=FV(rate, nper, pmt, [pv], [type])
PV(rate, nper, pmt, [FV],[type])
Where 'rate' is the effective interest rate per period, 'nper' is the number of payments, and 'pmt' is any periodic payments made.
3.2 Annuities
The Discount Factor (DF) can be used to calculate present value (PV) or future value (FV). Simply multiply FV by the discount factor to find PV, or divide PV by DF to find FV.
DF=1/(1+r)^n.
Annuities can also be discounted to present value using the discount factor.
Example:
For a car priced at $12,000, with a 20% down payment, the loan amount is $9,600 at an 8% APR compounded monthly for 3 years. The payment will be a 36-month annuity.
Using Excel's PMT function:
Rate = 8% / 12
nper = 36
pv = -9,600
This results in a monthly payment of $300.83, as the annual rate is converted to a monthly rate to match the compounding frequency.
Amortization Schedule:
This schedule outlines the breakdown of principal repayment and the interest component of each monthly payment.
3.3 Net Present Value
Net Present Value (NPV) is the value of all future cash flows over the life of an investment, discounted to present value, minus the initial investment.
Using Excel, the NPV formula can be applied. Remember to include only future cash flows and then add back the original investment. An alternative method involves using discount factors.
3.4 Internal Rate of Return (IRR)
The IRR is the rate of return that results in a zero NPV. For instance, if the required rate of return is 5% and the IRR is 8%, the investment remains worthwhile even if the required rate increases to 8%.
NPV Decision Rule:
1. Positive NPV: Proceed with the investment.
2. IRR exceeds the required rate (cost of capital): Proceed.
4. Bond Pricing
Bonds are debt instruments issued by governments or organizations to raise capital. They have a maturity date, a coupon rate (interest rate as a percentage of the par value), and a face/nominal/par value that will be repaid upon maturity. To price a bond, the NPV of all future cash flows is calculated, discounted using the Yield to Maturity (YTM), which represents the return investors would earn if they purchased the bond at its NPV.
4.1 Relationship between Bond Price and YTM
As bond prices increase, yields decrease, and vice versa. If YTM equals the coupon rate, the investor pays par value for the bond. If YTM is lower than the coupon rate, the bond is sold at a premium; if higher, it is sold at a discount. YTM is the IRR of the cash flows associated with this investment. In Excel, the RATE function can be used to calculate YTM.
5. Statistics for Financial Markets
5.1 Measuring Returns
Returns on market securities can be calculated using the logarithmic return (continuously compounded return) with the Excel function:
=LN(Closing price / Opening price)
5.2 Measuring Volatility
Volatility measures the spread of returns of a security, indicating how much returns vary from the average. This is calculated using the Excel function `STDEV.S` (for a sample; use `STDEV.P` for a whole population). Volatility must be compared over the same periods; for example, standard deviations of daily returns cannot be compared with those of monthly returns. Therefore, volatility is typically quoted on an annual basis in financial markets.
Annualized Standard Deviation = period standard deviation x scaling factor
In Excel, periodic standard deviation is calculated using the STDEV (continuously compounding return), and the scaling factor SQRT(number of periods in a year) (use 52 for weekly returns and 360 for daily returns).
5.3 Comparing Securities
Correlation (r) measures the strength of the relationship between two variables, represented by the correlation coefficient, where 1 indicates perfect linear correlation and -1 indicates a perfect inverse relationship. In Excel, use the function CORREL(array1, array2) to calculate r.
5.4 The Normal Distribution (Bell Curve)
In a normal distribution, half the values lie below the mean and half above, with fewer values located far from the mean. A higher standard deviation indicates greater dispersion from the mean.
In a normal distribution:
68% of values lie within 1 standard deviation,
95% within 2 standard deviations,
99.7% within 3 standard deviations.
5.5 Finding Probabilities
Higher stock volatility corresponds to a greater standard deviation, which implies a higher potential for loss. The probabilities of gains or losses for any given period, along with the distances from the mean, can be calculated using Excel's statistical functions.