Lecture 2: Compounding, future value and present value

This lecture contains the single most important tool of the course: moving money through time. A euro today is not the same as a euro next year, because a euro today can be invested and grow. Everything that follows (valuing bonds, stocks, projects, pensions) is an application of two formulas: future value and present value. Take the time to understand these well; the rest of the course is built on them.

Start here if this is new to you

Percentages as decimals. 10% means 10 per hundred, which is 0.10 as a decimal. "10% of 1,000" is 0.10 × 1,000 = 100. To increase something by 10%, multiply by 1 + 0.10 = 1.10. To increase it by 10% twice, multiply by 1.10 twice: 1.10 × 1.10 = 1.10² = 1.21.
Powers. 1.10² means 1.10 × 1.10 = 1.21; 1.10³ = 1.10 × 1.10 × 1.10 = 1.331. A negative power is one divided by the positive power: 1.10⁻² = 1 / 1.21 = 0.8264. On a calculator use the x^y or ^ key.
Timeline notation. T₀ is today, T₁ is one period from now (usually one year), T₂ two periods, and so on. An amount in brackets, like (1,000), is a payment out (you pay); a plain amount is a payment in.
Interest and the rate i. If you put €1,000 in a bank at i = 10% per year, after a year the bank pays you 10% of 1,000 = €100 interest, and you have €1,100. The letter i (or r) always stands for the interest rate per period, as a decimal.
Cost of capital. The interest rate at which you can invest or borrow. It is the rate you use to compare money at different dates. If you can earn 5% on your savings, your cost of capital is 5%.

Future value: what money today becomes

Invest €1,000 today for one year at 10%. In a year you have the €1,000 back plus 10% interest:

1, 000 + (1, 000 \times 0.10) = 1, 000 \times 1.10 = C = 1, 100

This €1,100 is the future value of €1,000 invested for one year at 10%.

Now invest for two years. After the first year you have 1,100. In the second year you earn 10% on the 1,100, not only on the original 1,000, because the interest of year 1 is reinvested. This is compounding: interest on interest.

	T₀	T₁	T₂
Amount	(1,000)	1,100	1,210

1, 000 \times (1 + 0.10) \times (1 + 0.10) = 1, 000 \times (1 + 0.10)^{2} = C = 1, 210

Without compounding (simple interest) you would have 1,000 + 100 + 100 = 1,200. The extra €10 is the interest earned on the first year's interest. Over long periods this effect becomes enormous: €1,000 at 10% for 30 years is 1,000 × 1.10³⁰ = €17,449, not €4,000.

The general formula. For n periods at an interest rate i per period, the future value T of an amount A is:

T = A \times (1 + i)^{n}

Example: at 10% per year, €1,000 today is equivalent to €1,210 in two years. "Equivalent" is the key word: an investor who can earn 10% is indifferent between the two.

Comparing investments with future values

Worked example (from the slides): two investments

	T₀	T₁	T₂	T₃
A	(1,000)	300	400	500
B	(1,000)	400	400	400

Both cost 1,000 today and both pay 1,200 in total. Which is better if the interest rate is 10%? Add up the cash flows at T₃, taking into account that each payment is reinvested at 10% until T₃. A payment at T₁ earns interest for two years, a payment at T₂ for one year, and the payment at T₃ for zero years.

FV (A) = 300 \times 1.1 0^{2} + 400 \times 1.10 + 500 = 363 + 440 + 500 = C = 1, 303

FV (B) = 400 \times 1.1 0^{2} + 400 \times 1.10 + 400 = 484 + 440 + 400 = C = 1, 324

Choose B. B is better than A because it pays sooner, and the money received early can be invested again. Same total, different timing, different value.

Worked example (from the slides): three investments

	T₀	T₁	T₂	T₃
A	(1,000)	400	300	500
B	(1,000)	1,100
C	(1,000)	600	550

The investments end at different dates, so first compute each one's value at its own last date, then bring everything to the same date (T₃) at 10%.

A at T₃: 400 × 1.10² + 300 × 1.10 + 500 = 484 + 330 + 500 = €1,314.
B at T₁: €1,100. Reinvested for two more years: 1,100 × 1.10² = €1,331 at T₃.
C at T₂: 600 × 1.10 + 550 = €1,210. Reinvested one more year: 1,210 × 1.10 = €1,331 at T₃.

B and C have the same value and both are better than A. Note that B and C both equal 1,000 × 1.10³ = 1,331: they simply earn the market rate of 10%, while A earns less.

In plain words

Money has time value: a penny today is not the same as a penny tomorrow. To compare cash flows that arrive at different dates, move them all to one date. Moving forward means multiplying by (1 + i) for every year.

Present value: what future money is worth today

Instead of moving all cash flows to the end, we can bring them all to the present. This is the more common direction, because prices are paid today.

Someone asks to borrow money and will repay €1,000 in a year. You want 10% interest. How much do you lend? Call the amount A. It must grow to 1,000 in one year:

A \times 1.10 = 1, 000 ⟹ A = \frac{1 , 000}{1.10} = C = 909.09

And if the promise is €1,000 in two years?

A \times 1.10 \times 1.10 = 1, 000 ⟹ A = \frac{1 , 000}{1.1 0 ^{2}} = C = 826.45

These amounts are present values. In general, the present value of an amount T received in n years at an interest rate i is:

P V = \frac{T}{( 1 + i ) ^{n}}

Dividing by (1 + i)ⁿ is called discounting, and i is then called the discount rate. Discounting is exactly the reverse of compounding: future value multiplies by (1 + i)ⁿ, present value divides by it.

Step by step: discounting with a calculator

Write the rate as a decimal: 10% is 0.10.
Add 1: 1.10.
Raise to the number of years: 1.10² = 1.21 (for two years).
Divide the future amount by the result: 1,000 / 1.21 = 826.45.

Check: 826.45 × 1.21 = 1,000. If your answer is larger than the future amount, you multiplied where you should have divided.

Worked example (from the slides): the three investments by present value

Same three investments, 10% rate. Discount every cash flow to T₀ and add up.

P V (A) = \frac{400}{1.10} + \frac{300}{1.1 0 ^{2}} + \frac{500}{1.1 0 ^{3}} = 363.64 + 247.93 + 375.66 = C = 987.23

P V (B) = \frac{1 , 100}{1.10} = C = 1, 000

P V (C) = \frac{600}{1.10} + \frac{550}{1.1 0 ^{2}} = 545.45 + 454.55 = C = 1, 000

Each investment costs 1,000. B and C are worth exactly what they cost (they earn exactly 10%); A is worth only 987.23 for a price of 1,000, so it earns less than 10% and is the worst. The ranking is the same as with future values; it always is.

Deciding on a project with present value

Worked example (from the slides): land and offices

You plan to buy land and build offices. Total cost: €200,000, paid today. You can arrange today to sell the whole thing for €220,000 in a year. Do it?

To evaluate the investment you must know the interest rate of alternative, similar investments. Suppose it is 4%. At 4%, your €200,000 would grow in a year to 200,000 × 1.04 = €208,000. The project gives you €220,000, which is more, so the investment is acceptable.

What is the highest cost at which the project is still acceptable? Call it A. It must satisfy A × 1.04 = 220,000, so

A = \frac{220 , 000}{1.04} = C = 211, 538.46

This is the present value of €220,000 received in a year at 4%. At any cost below 211,538.46 the project beats the 4% alternative.

Suppose you pay the €200,000 today and want to sell the project immediately. For how much? For its present value: 220,000 / 1.04 = €211,538.46. This is a fair price for both buyer and seller: the buyer earns exactly 4%, the market rate, and you pocket the difference of 11,538.46 as the value you created.

In plain words

The present value of a project's future cash flows is the most anyone should pay for it. Pay less and you earn more than the market rate; pay more and you earn less. The difference between the present value and the cost is called the net present value (NPV), and a project is worth doing when its NPV is positive.

Perpetuities: the same amount forever

Suppose an investment offers €1,000 per year forever, starting one year from now. Its present value is an infinite sum of discounted payments:

P V = \frac{1 , 000}{1.10} + \frac{1 , 000}{1.1 0 ^{2}} + \frac{1 , 000}{1.1 0 ^{3}} + \dots = \frac{1 , 000}{0.10} = C = 10, 000

The infinite sum has a simple answer because the terms shrink fast. In general, for a perpetuity paying the standard amount T every year forever, with discount rate i:

P V = \frac{T}{i}

Intuition: with €10,000 in the bank at 10% you can withdraw €1,000 of interest every year forever without touching the capital. So €1,000 forever is worth exactly €10,000.

Growing perpetuity. If the first payment is T and each payment grows by d% per period (for example a dividend that grows with the economy), the present value is

P V = \frac{T}{i - d}

which only makes sense if the growth rate d is smaller than the discount rate i. Growth makes the stream more valuable, so the denominator becomes smaller. (In lecture 3 this formula returns as the Gordon growth model for stocks.)

Annuities: the same amount for a fixed number of periods

An annuity pays an equal amount every period for a definite number of periods: a mortgage, a car loan, a pension. The present value of an annuity paying €T every period for n periods at discount rate i is the sum of n discounted payments:

P V = \frac{T}{1 + i} + \frac{T}{( 1 + i ) ^{2}} + \dots + \frac{T}{( 1 + i ) ^{n}} = \frac{T}{i} (1 - \frac{1}{( 1 + i ) ^{n}})

Where does the short formula come from? An annuity for n years is a perpetuity (worth T/i) minus a perpetuity that starts after year n (worth T/i at year n, so T/i divided by (1 + i)ⁿ today). Subtracting gives the formula. The part in brackets is always between 0 and 1; the longer the annuity, the closer it gets to 1 and the closer the annuity gets to a perpetuity.

Future value of an annuity. If you deposit an amount A at the end of every period for n periods, earning i per period, you end up with

T = A \times \frac{( 1 + i ) ^{n} - 1}{i}

where A is the amount invested each period, i the interest earned per period and n the number of periods.

Ordinary annuity versus annuity due

The formulas above assume the first payment comes one period from now (an ordinary annuity, payments at the end of each period). If the first payment is today (an annuity due, payments at the start of each period), every payment is one period earlier and therefore worth more: multiply the ordinary-annuity value by (1 + i), or equivalently take the first payment undiscounted and treat the remaining n − 1 payments as an ordinary annuity. Exercise 2 below uses this.

Worked examples from the slides

1. Which prize?

You win one of the following prizes. You can invest at 5% per year. Which do you choose?

A) €20,000 today. PV = €20,000.
B) €23,000 in two years. PV = 23,000 / 1.05² = 23,000 / 1.1025 = €20,862.
C) €1,000 annually forever. A perpetuity: PV = 1,000 / 0.05 = €20,000.
D) €600 next year, growing at 1.8% annually forever. A growing perpetuity: PV = 600 / (0.05 − 0.018) = 600 / 0.032 = €18,750.

Choose B. Notice that C equals A exactly: €20,000 at 5% produces €1,000 a year forever.

2. Buying a car

A car costs €10,000 cash, or 36 monthly instalments of €400. Your monthly cost of capital is 0.45%. The instalments are an annuity with T = 400, i = 0.0045, n = 36:

P V = \frac{400}{0.0045} (1 - \frac{1}{1.004 5 ^{36}}) = C = 13, 266

The instalment plan costs 13,266 in today's money, more than 10,000. Pay the whole amount today. (36 × 400 = 14,400 in total; even after discounting it is far more expensive, so the plan hides a very high interest rate.)

3. When to pay taxes

You owe €5,000 in taxes. If you pay now you get a 6.8% discount; otherwise you pay the full amount in 3 months. Your cost of capital is 28% per year.

Paying today costs 5,000 − 0.068 × 5,000 = €4,660. If instead you keep the 4,660 invested for 3 months (a quarter of a year, so a quarter of 28% = 7%), it grows to 4,660 × 1.07 = €4,986.20. That is less than the €5,000 you would then owe, so pay today. The logic: if you can turn 4,660 into at least 5,000 within 3 months, waiting is better; here you only reach 4,986.20, which falls €13.80 short. A 6.8% discount for paying 3 months early is the same as earning 5,000 / 4,660 − 1 = 7.3% per quarter, slightly more than your own 7%.

4. The pension fund

A pension fund promises €50,000 in 25 years if you pay €1,000 into it every year until then. You can earn 5% a year on your own. Would you have more by investing on your own?

Future value of an annuity of 1,000 for 25 years at 5%:

T = 1, 000 \times \frac{1.0 5 ^{25} - 1}{0.05} = 1, 000 \times 47.727 = C = 47, 727.10

On your own you reach €47,727, less than the fund's €50,000, so the fund is the better deal.

What interest rate does the fund offer? Solve 50,000 = 1,000 × [(1 + i)²⁵ − 1] / i for i. There is no direct formula; by trial and error (or a financial calculator) i = 5.34%. Check: at 5.34%, (1.0534²⁵ − 1)/0.0534 ≈ 50.0.

Could you beat the fund by saving €500 every 6 months at a nominal 5% per year? A nominal annual rate of 5% paid semiannually means 2.5% per half-year (nominal rate divided by the number of periods). There are 50 half-years in 25 years:

T = 500 \times \frac{1.02 5 ^{50} - 1}{0.025} = C = 48, 742.17

Better than 47,727 (more frequent compounding helps), but still below 50,000. The fund still wins.

Interest rates: nominal versus effective

Quoted interest rates can be simple (nominal) or effective rates. The effective rate takes compounding within the year into account; the nominal rate does not. The distinction matters whenever interest is paid more than once a year.

If an amount A earns interest m times a year at a nominal annual rate i, each payment is i/m, and by the end of the year you have

A (1 + \frac{i}{m})^{m}

so the effective annual rate is

i_{eff} = (1 + \frac{i}{m})^{m} - 1

Worked example (from the slides)

A bank offers 10% per year, compounded semiannually. The rate received every 6 months is i_{6 months} = 10% / 2 = 5%. The effective annual rate is

(1 + \frac{10%}{2})^{2} - 1 = 1.0 5^{2} - 1 = 10.25%

With quarterly compounding (m = 4, 2.5% per quarter):

(1 + \frac{10%}{4})^{4} - 1 = 1.02 5^{4} - 1 = 10.38%

The more often interest is paid, the higher the effective rate, because each interest payment starts earning interest sooner. Monthly compounding would give 10.47%, daily 10.52%.

Rule for exercises. When a problem gives a nominal annual rate and payments every month, quarter or half-year: divide the rate by the number of periods per year (6% per year with monthly payments is 0.5% per month) and count the periods (3 years is 36 months). Use the per-period rate and the number of periods in the annuity formulas. This is what the slides do in every example.

Continuous compounding

What happens if interest is paid every second, or continuously? The effective rate stops rising and settles at a limit. An amount earning interest at a continuous rate r for t years grows by the factor

e^{r t}

where e = 2.71828..., the base of natural logarithms (the e^x key on your calculator). Continuous compounding is used mainly in the pricing of options and in theoretical work, because it makes the mathematics cleaner.

Worked example (from the slides)

A stock generates a continuous return of 10% per year. Its price today is €5. In 18 months (t = 1.5 years):

5 \times e^{0.10 \times 1.5} = 5 \times e^{0.15} = 5 \times 1.1618 = C = 5.81

Exercises from the slides, fully worked out

Exercise 1: furniture

Furniture worth €3,000: pay the whole amount today or 24 monthly instalments of €131. Your cost of capital is 6% per year. How should you pay?

Monthly rate: 6% / 12 = 0.5% = 0.005. Number of periods: 24. Present value of the instalments:

P V = \frac{131}{0.005} (1 - \frac{1}{1.00 5 ^{24}}) = 26, 200 \times (1 - 0.8872) = C = 2, 955.74

The instalment plan is worth €2,955.74 in today's money, less than €3,000. Pay in instalments: the shop is effectively lending you money at less than your own 6%. (Total paid is 24 × 131 = 3,144, but that comparison ignores the time value of money and is wrong.)

Exercise 2: three ways to pay for a car

a) €15,000 cash; b) 20 annual instalments of €1,200, the first due today; c) 15 annual instalments of €1,700 beginning in 5 years. Cost of capital 5%.

Option b is an annuity due: the first 1,200 is paid today (no discounting), the remaining 19 payments form an ordinary annuity.

P V_{b} = 1, 200 + \frac{1 , 200}{0.05} (1 - \frac{1}{1.0 5 ^{19}}) = 1, 200 + 14, 502.39 = C = 15, 702.39

Option c is a deferred annuity. First find its value one period before the first payment, at T₄ (the ordinary-annuity formula values the stream one period before the first payment):

P V_{T_{4}} = \frac{1 , 700}{0.05} (1 - \frac{1}{1.0 5 ^{15}}) = C = 17, 645.42

Then discount this amount four years back to today:

P V_{c} = \frac{17 , 645.42}{1.0 5 ^{4}} = \frac{17 , 645.42}{1.2155} = C = 14, 516.93

Ranking: c (€14,516.93) is cheaper than a (€15,000), which is cheaper than b (€15,702.39). Choose c. Paying late is attractive when the instalments are not increased enough to compensate for the delay.

Exercise 3a: saving for retirement

You are 30 and want to retire in 35 years, at 65. You want a pension of €18,000 a year until you are 85 (20 yearly payments after retirement). The interest rate is always 6%. How much must you put into your retirement fund each year?

Step 1: how much money do you need at 65? The present value, at age 65, of an annuity of 18,000 for 20 years at 6%:

P V_{65} = \frac{18 , 000}{0.06} (1 - \frac{1}{1.0 6 ^{20}}) = 300, 000 \times 0.6882 = C = 206, 458.58

Step 2: which yearly deposit grows to that amount in 35 years? Use the future value of an annuity and solve for A:

206, 458.58 = A \times \frac{1.0 6 ^{35} - 1}{0.06} = A \times 111.43 ⟹ A = C = 1, 852.73

Saving about €1,853 a year for 35 years at 6% funds a pension of 18,000 a year for 20 years. Compounding does most of the work: you deposit 35 × 1,853 = €64,846 in total and withdraw 20 × 18,000 = €360,000.

Exercise 3b: saving twice a year

You can save every 6 months and receive 3% every 6 months. How much should you save per half-year?

The target at 65 stays €206,458.58 (the pension side of the problem is unchanged). Deposits: 35 years × 2 = 70 half-years at 3% per half-year:

206, 458.58 = A \times \frac{1.0 3 ^{70} - 1}{0.03} = A \times 230.59 ⟹ A = C = 895.33

Per year that is 2 × 895.33 = €1,790.66, less than the €1,852.73 of part (a), because the half-yearly deposits start earning interest earlier and 3% per half-year is slightly more than 6% per year (effective 6.09%).

Exercise 4a: saving for your studies

Your parents want €20,000 in 4 years and will deposit an equal amount at the end of every year. The bank offers a nominal annual rate of 5.913% compounded semiannually. How much per year?

The deposits are yearly but the interest is credited every half-year, so first convert the nominal rate to an effective annual rate:

i_{eff} = (1 + \frac{0.05913}{2})^{2} - 1 = 1.02956 5^{2} - 1 = 6.00%

(The odd-looking 5.913% is chosen so that the effective rate is a round 6%.) Now solve the future value of an annuity for A with n = 4 and i = 6%:

20, 000 = A \times \frac{1.0 6 ^{4} - 1}{0.06} = A \times 4.3746 ⟹ A = C = 4, 571.80

Exercise 4b: living off the account

Your studies last 18 months. You deposit the €20,000 in an account earning 3% every 6 months and withdraw an equal amount at the end of every 6 months (three withdrawals), leaving €1,850 at the end of the 18 months. How much can you withdraw each time?

Today's €20,000 must equal the present value of three withdrawals W plus the present value of the €1,850 left after the third period:

20, 000 = \frac{W}{0.03} (1 - \frac{1}{1.0 3 ^{3}}) + \frac{1 , 850}{1.0 3 ^{3}} = 2.8286 W + 1, 693.01

W = \frac{20 , 000 - 1 , 693.01}{2.8286} = C = 6, 472.08

Check: 20,000 × 1.03 − 6,472.08 = 14,127.92; × 1.03 − 6,472.08 = 8,079.68; × 1.03 − 6,472.08 = 1,850.0. It works.

Common mistakes to avoid

Adding up cash flows from different dates without discounting ("the plan costs 3,144 in total"). Always move everything to one date first.
Mixing an annual rate with monthly periods. Convert the rate to the period of the payments (6% a year with monthly payments: 0.5% per month, 36 periods for 3 years).
Forgetting that the annuity formula values the stream one period before the first payment. A first payment today needs the annuity-due adjustment; a first payment in year 5 needs a further discounting of 4 years.
Using the nominal rate when the effective rate is asked, or the other way round. Nominal 10% semiannual is effective 10.25%.
For a growing perpetuity, forgetting that T is the first payment (one period from now), not today's payment.

In plain words

Lecture 2 in one breath: multiply by (1 + i) per period to go forward, divide to go back. A constant stream forever is worth T / i; a constant stream for n periods is worth T / i times (1 − 1 / (1 + i)ⁿ); growth lowers the denominator to i − d. Convert nominal rates to per-period rates before using any of these.

Andere hoofdstukken van Money and Capital Markets

Test jezelf met oefenvragen