Time value of money: future value, present value, NPV and IRR
Money received sooner is worth more than money received later, because it can be invested. All the formulas below use one idea: moving a dollar amount across time at a rate of return.
Future value
FV = PV x (1 + r)^n
Worked example. $10,000 invested at 6% a year for 8 years: $10,000 x 1.06^8 = $10,000 x 1.5938 = $15,938.48.
If interest compounds quarterly the rate per period is 6% / 4 = 1.5% and there are 8 x 4 = 32 periods: $10,000 x 1.015^32 = $16,103.24. More frequent compounding gives a larger amount.
Rule of 72. Years to double is about 72 divided by the annual rate in percent. At 6%, 72 / 6 = 12 years. Check: $10,000 x 1.06^12 = $20,121.96. The rule is an approximation that works best between 6% and 10%.
Present value
PV = FV / (1 + r)^n
Worked example. A client wants $20,000 in 5 years and can earn 7%. PV = $20,000 / 1.07^5 = $20,000 / 1.4026 = $14,259.72. The client needs $14,259.72 today.
Present value of a series. To receive $1,000 at the end of each year for 5 years with a 6% discount rate, discount each payment: $943.40 + $890.00 + $839.62 + $792.09 + $747.26 = $4,212.36.
A higher discount rate gives a lower present value; a longer time gives a lower present value.
Net present value
NPV is the sum of the present values of all cash flows, including the initial outlay, discounted at the required rate of return.
Worked example. A project costs $10,000 now and returns $3,000, $4,000 and $5,000 at the end of years 1, 2 and 3. The required return is 8%.
| Year | Cash flow | Discount factor | Present value |
|---|---|---|---|
| 0 | -$10,000 | 1.0000 | -$10,000.00 |
| 1 | $3,000 | 1/1.08 | $2,777.78 |
| 2 | $4,000 | 1/1.08^2 | $3,429.36 |
| 3 | $5,000 | 1/1.08^3 | $3,969.16 |
| NPV | +$176.29 |
NPV is positive, so the project earns more than 8% and adds value. Decision rule: accept if NPV is greater than zero; reject if negative. The profitability index (present value of inflows / outlay) is 10,176.29 / 10,000 = 1.018.
Internal rate of return
IRR is the discount rate at which NPV equals zero. There is no direct formula. Solve by trying rates.
Worked example (same project).
| Trial rate | NPV |
|---|---|
| 8% | +$176.29 |
| 9% | -$20.07 |
| 10% | -$210.37 |
The sign changes between 8% and 9%, so the IRR is just under 9%. Linear interpolation between 8% and 10%: 8% + 2% x 176.29 / (176.29 + 210.37) = 8% + 0.91% = 8.91%. Refining gives IRR = 8.90%. At 8.90%, NPV is almost exactly zero.
Decision rule: accept if IRR is greater than the required return (8.90% > 8%: accept). The NPV and IRR rules agree here.
When NPV and IRR disagree. For mutually exclusive projects of different size or cash-flow timing, IRR can rank the projects differently than NPV. NPV assumes cash flows are reinvested at the required rate; IRR assumes reinvestment at the IRR itself, which is less realistic. A project with cash flows that change sign more than once can have more than one IRR. In these cases NPV is the better guide because it measures the dollar value added.
Where it appears in advice. IRR is used to compare investments with irregular cash flows, such as private equity. The money-weighted return of a portfolio with deposits and withdrawals is an IRR.
Key facts — Time value of money: future value, present value, NPV and IRR - FV = PV x (1 + r)^n; PV = FV / (1 + r)^n. - Rule of 72: doubling time = 72 / rate. - NPV > 0: accept. IRR > required return: accept. - IRR is the rate at which NPV = 0. - NPV is preferred when NPV and IRR conflict.
Exam traps
- Using the annual rate with monthly periods.
- Forgetting the initial outlay in NPV.
- Assuming a higher IRR project is always better than a larger project with a higher NPV.
Descriptive statistics: mean, median, mode, range and standard deviation
Mean, median, mode and range
Worked example. A fund's monthly returns (in percent) over seven months: 4, 8, 6, 5, 3, 8, 9.
- Sort: 3, 4, 5, 6, 8, 8, 9.
- Mean = (4 + 8 + 6 + 5 + 3 + 8 + 9) / 7 = 43 / 7 = 6.14.
- Median = the middle value of the sorted list = 6. With an even count, the median is the average of the two middle values; for 2, 4, 7, 9 the median is (4 + 7) / 2 = 5.5.
- Mode = the most frequent value = 8.
- Range = highest minus lowest = 9 - 3 = 6.
The median is not changed by an outlier; the mean is. If the last value were 90, the mean would jump while the median would stay at 6.
Arithmetic and geometric mean. The arithmetic mean averages the returns. The geometric mean gives the true compound rate. A fund that gains 50% one year and loses 50% the next has an arithmetic mean of (50% + (-50%)) / 2 = 0%, but $100 becomes $150 then $75. The geometric mean is the square root of (1.50 x 0.50) minus 1 = square root of 0.75 minus 1 = -13.40% a year. The geometric mean is always at most the arithmetic mean, and the gap grows with volatility.
Standard deviation, calculated by hand
Standard deviation (SD) measures how far returns scatter around their mean. It is the square root of the variance.
Worked example. Four annual returns: 10%, 4%, -2%, 8%.
- Mean = (10 + 4 - 2 + 8) / 4 = 5.
- Deviations from the mean: 5, -1, -7, 3.
- Squared deviations: 25, 1, 49, 9. Sum = 84.
- Population variance (divide by n = 4) = 84 / 4 = 21. SD = square root of 21 = 4.58%.
- Sample variance (divide by n - 1 = 3) = 84 / 3 = 28. SD = square root of 28 = 5.29%.
Use the sample SD when the returns are a sample of a longer record (the usual case for a fund's history); use the population SD when the data are the whole group.
Interpretation. For a roughly normal distribution, about 68% of outcomes fall within one SD of the mean, 95% within two, and 99.7% within three. A fund with a mean return of 8% and an SD of 12% has a 68% range of -4% to 20% and a 95% range of -16% to 32%.
Coefficient of variation = SD / mean, the risk per unit of return. Fund A has mean 8%, SD 12%: 1.50. Fund B has mean 6%, SD 6%: 1.00. Fund B has less risk per unit of return.
Key facts — Descriptive statistics: mean, median, mode, range and standard deviation - Mean is sensitive to outliers; median is not. - Range = highest - lowest. - Population SD divides by n; sample SD divides by n - 1. - About 68%, 95%, 99.7% within 1, 2, 3 SDs for normally distributed returns. - Geometric mean is at most the arithmetic mean.
Exam traps
- Reading SD as a loss measure. It measures dispersion in both directions.
- Using the arithmetic mean to describe multi-year growth.
- A data set can have no mode, or more than one mode.
Covariance, correlation and combined risk
Covariance measures whether two assets tend to move together. Correlation rescales covariance to a number between -1 and +1:
Correlation = covariance(X, Y) / (SD of X x SD of Y)
Worked example. Annual returns of two funds over four years (in percent):
| Year | Fund X | Fund Y |
|---|---|---|
| 1 | 10 | 6 |
| 2 | 4 | 3 |
| 3 | -2 | 5 |
| 4 | 8 | -2 |
| Mean | 5 | 3 |
- Deviations of X: 5, -1, -7, 3. Deviations of Y: 3, 0, 2, -5.
- Products: 5 x 3 = 15; -1 x 0 = 0; -7 x 2 = -14; 3 x -5 = -15. Sum = -14.
- Sample covariance = -14 / 3 = -4.667.
- SD of X (sample) = 5.29 (from the earlier example). Squared deviations of Y: 9, 0, 4, 25 = 38; variance = 38 / 3 = 12.667; SD of Y = 3.56.
- Correlation = -4.667 / (5.29 x 3.56) = -0.248.
The correlation is weakly negative. Combining the funds should reduce portfolio risk.
Why correlation matters: a two-asset portfolio. Portfolio SD is not the weighted average of the SDs. For weights w1 and w2:
SD(portfolio) = square root of [ (w1 x SD1)^2 + (w2 x SD2)^2 + 2 x w1 x w2 x SD1 x SD2 x correlation ]
With 60% in a stock fund (SD 20%) and 40% in a bond fund (SD 10%), correlation 0.3:
- (0.60 x 20)^2 = 144
- (0.40 x 10)^2 = 16
- 2 x 0.60 x 0.40 x 20 x 10 x 0.3 = 28.8
- Variance = 144 + 16 + 28.8 = 188.8. SD = 13.74%.
The weighted average of the SDs is 0.6 x 20 + 0.4 x 10 = 16%. Diversification cut the risk to 13.74%. If the correlation were +1, the portfolio SD would equal 16%, so there would be no diversification benefit. The lower the correlation, the larger the benefit; at -1 risk can be eliminated with the right weights. Diversification reduces unsystematic risk only; it cannot reduce market risk.
R-squared is the square of the correlation with a benchmark: the share of a fund's movement explained by the benchmark. A correlation of 0.9 gives R-squared of 0.81, meaning 81%.
R-squared is usually read together with tracking error, the amount by which a fund's returns deviate from its benchmark's. A high R-squared says the fund moves with the benchmark. A low tracking error says its returns stay close to the benchmark's. An adviser who wants to know how closely a fund follows its index looks at both: R-squared close to 1 (100%) and a small tracking error describe an index-like fund. Sharpe ratio and standard deviation describe risk-adjusted return and volatility, and beta describes sensitivity to the market; none of them measures how closely a fund follows a benchmark.
Key facts — Covariance, correlation and combined risk - Correlation ranges from -1 to +1. Covariance has no fixed range. - Portfolio SD is less than the weighted average SD unless correlation is +1. - R-squared = correlation squared. - R-squared plus tracking error measure how closely a fund follows its benchmark.
Exam traps
- Correlation of zero means no linear relationship, not no risk.
- A negative covariance does not mean negative returns.
- Diversification does not remove systematic risk.