Week 13 - Introduction to Valuation: Time value of money

Question 1

What is the present value (PV)?

Answer 1

the current value of future cash flows discounted at the appropriate discount rate

PV is the amount you would need to invest today to reach a certain future value, given the discount rate

value at t=0 on a timeline

Question 2

What is a future value (FV)?

Answer 2

The amount an investment will grow to at a specified time in the future, considering compound interest

Question 3

What is interest rate (r)?

Answer 3

the ‘exchange rate’ between earlier (PV) and later money (FV)

It’s the percentage by which the value of money increases or decreases over a specific period

Question 4

What terms refer to the rate used to discount future cash flows to their present value

Answer 4

discount rate (generally used when calculating present value)

Cost of capital (often used in investment or business decisions, reflecting the return needed to justify an investment)

Opportunity cost of capital (reflects the potential return on the next best alternative investment)

Required return (the return that an investor expects to earn on an investment)

Question 5

What is the time value of money concept?

Answer 5

Money today is worth more than the same amount of money in the future because of the opportunity to earn interest.

The further into the future a payment is, the less valuable it is today, due to inflation and the opportunity cost of capital

Question 6

What is the future value formula?

Answer 6

FV = PV x (1+r)^t

FV - future value
PV - present value
r - period interest rate, expressed as a decimal
t - number of periods

Question 7

What does the future value formula mean?

Answer 7

the amount of money an investment will grow to over some period of time at a given rate of interest

Question 8

An example of future value (one period)

suppose you invest £100 for one year at 10% per year
what is the future value in one year?

Answer 8

interest = 100 × 0.10 = 10

value in one year = principal + interest = 100 + 10 = 110

future value (FV) = 100 × (1 + 0.10) = 110

Question 9

An example of future value (two periods)

at the end of the first period, you have £110. How much you can get at the end of the second period depends on what you do with the £10 interest at the end of the first period (2 options)

Answer 9

1. withdraw £10 interest and leave £100 in the bank
   payoff: 10 + 100 × (1 + 0.10) = 120
   (simple interest)
2. leave the entire £110 in the bank to earn interest in the second year
   payoff: 110 × (1 + 0.10) = 121
   (compound interest)

Question 10

What is simple interest?

Answer 10

interest earned only on the original principal

Question 11

What is compound interest?

Answer 11

interest earned on the original principal and any accumulated interest

ie compounding (earning interest on interest) - interest earned on reinvestment of previous interest payments

Question 12

What is the simple interest formula?

Answer 12

FV with simple interest
FV = PV × (1 + r × t)

FV = Future Value
PV = Present Value
r = Periodic interest rate (expressed as a decimal)
t = Number of periods

In simple interest, you’re just adding a fixed amount of interest each period, without compounding, grows linearly

Question 13

What is the compound interest formula?

Answer 13

FV with compound interest
FV = PV × (1 + r)ˆt

FV = Future Value
PV = Present Value
r = Periodic interest rate (expressed as a decimal)
t = Number of periods

The formula shows how the money grows exponentially with each compounding period

Question 14

Future value example

Deposit £5,000 today in an account paying 12%. How much will you have in 6 years with compound interest?

How much will you have in 6 years with simple interest?

Answer 14

FV = PV × (1 + r)^t = 5,000 × (1 +0.12)^6 = 5,000 × 1.974 = 9,869

6 years with simple interest:
FV = PV × (1 + r × t) = 5,000 × (1 + 0.12 × 6) = 8,600

Compound interest = 9,869 – 5,000 = 4,869
The interest on interest = 4,869 – (8,600 – 5,000) = 1,269

Question 15

eg Future value in 200 years

Suppose you had a relative deposit £5 for you at 6% interest 200 years ago. How much would the investment be worth today by compounding interest?

How much can you get if the investment only earns simple interest?

Answer 15

investment worth today by compounding interest:
FV = PV × (1 + r)^t = 5 × (1 + 0.06)^200 = 575,629.52

simple interest:
FV = PV × (1 + r × t) = 5 × (1 + 0.06 × 200) = 65

The effect of compounding is small for a small number of periods but increases as the number of periods increases
Simple interest is constant each year. The size of the compound interest keeps increasing because more and more interest builds up and there is thus more to compound.

Question 16

What are 2 important relationships in future value?

Answer 16

1. the longer the time period, the higher the future value
2. the higher the interest rate, the larger the future value

Question 17

Why does the longer the time period the higher the FV?

Answer 17

Longer time periods increase the future value: As time passes, compound interest has more periods to accumulate, which leads to exponential growth in the investment

Question 18

Why does the higher interest rates increase the FV?

Answer 18

Higher interest rates increase the future value: With a higher interest rate, the growth per period is larger, leading to a higher future value over time

Question 19

What is a dividend?

Answer 19

a payment made by firms to stockholders. It is usually cash but may also be stock. A dividend represents part of the investor’s return for buying the stock (the other part of the return is any capital gain made when the stock is sold)

Question 20

What is the dividend growth formula?

Answer 20

FV = D_0 x (1+r)ˆt

D_0 = the current dividend
r = growth rate
t = time period

Question 21

Suppose an investor buys 1 share in BT plc. The company pays a current
dividend of £1.10, which is expected to grow at 40% per year for the next
five years. What will the dividend be in five years?

Answer 21

FV = D_0 x (1+r)^t
FV = 1.10 x (1+0.4)^5 = 5.92

Question 22

Why is the present value worth less than the future value?

Answer 22

because of opportunity cost, risk and uncertainty (discount rate)

Question 23

Why does opportunity cost affect the PV?

Answer 23

The money you receive in the future could have been used for investment today, so there’s an opportunity cost to waiting for a payment in the future instead of using that money now

Question 24

Why does risk and uncertainty affect the PV?

Answer 24

The future is uncertain, and there is always a risk that the expected future payment may not materialise (due to factors like inflation, economic conditions, etc.).

The higher the perceived risk, the higher the discount rate used to calculate present value, and the lower the present value

Question 25

Why does time value of money also affect PV?

Answer 25

The value of money changes over time. The general principle is that a sum of money today is worth more than the same sum in the future, because money today can be invested to earn a return

Question 26

What is the process of discounting?

Answer 26

involves applying a discount rate to the future value (FV) to find the present value (PV)

Question 27

What is the discounting formula to find the PV?

Answer 27

PV = FV/ (1+r)ˆt PV = Present value FV = Future value r = Discount rate (which accounts for the time, opportunity cost, and risk) t = Time period

Question 28

What questions does the PV answer?

Answer 28

How much money do I need to invest today to reach a certain amount in the future? What is the current worth of a sum of money or future cash flows?

Question 29

When we talk about the 'value' of something what does this mean

Answer 29

we are talking about the present value unless we specifically indicate that we want the future value

Question 30

How do you answer the questions: * how much do I have to invest today to have some amount in the future? * what is the current value of an amount to be received in the future?

Answer 30

Rearrange FV = PV × (1 + r)^t to solve for PV: PV = FV / (1 + r)^t when we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value

Question 31

What are the 2 important relationships of the present value?

Answer 31

1. for a given interest rate, the longer the time period, the lower the present value 2. for a given time period, the higher the interest rate, the smaller the present value

Question 32

Why for a given interest rate, the longer the time period, the lower the present value?

Answer 32

The further in the future a cash flow occurs, the less it is worth today because of the time value of money—meaning the opportunity cost of not being able to use that money for other investments or purposes. The higher the interest rate, the more discounted future amounts are

Question 33

Why for a given time period, the higher the interest rate, the smaller the present value?

Answer 33

Discounting Effect: When you use a higher interest rate, you're discounting future cash flows more heavily. The interest rate represents the "opportunity cost" of having money today versus in the future. A higher rate means you could earn more if you invested the money now, so you would need less today to achieve the same future value. Time Value of Money: As the interest rate increases, the amount you need today (PV) to get the same amount in the future (FV) decreases, because the higher interest rate allows you to earn more on your investment over time

Question 34

What is the discount rate?

Answer 34

essentially the interest rate that we would use to discount future cash flows to their present value. If you know the future value (FV), the present value (PV), and the time period (t), you can solve for the discount rate (r)

Question 35

How do we find the implied interest rate (discount rate) in an investment?

Answer 35

rearrange FV = PV x (1+r)^t so r = (FV/PV)^1/t -1

Question 36

How do we find the number of periods/ years?

Answer 36

rearrange FV = PV x (1+r)^t FV/PV = (1+r)^t ln(FV/PV) = tln(1+r) ln(FV/PV) / ln(1+r) = t

Question 37

What is the rule of 72?

Answer 37

a quick way to estimate how long it will take for an investment to double, given a fixed annual interest rate. You divide 72 by the interest rate to get the approximate number of years it will take for the investment to double. For your example: Interest rate = 8% Rule of 72: 72 / 8 = 9 years This gives a good approximation

Question 38

What typically contains multiple cash flows?

Answer 38

an asset or project, it's essential to account for the time value of money (TVM). This is because a pound received today is worth more than a pound received in the future, due to its potential earning power

Question 39

How do we handle multiple cash flows?

Answer 39

For a series of cash flows, you typically need to calculate the Net Present Value (NPV) or Future Value (FV) of all those flows by applying either discounting or compounding. Discounting is used to calculate the present value (PV) of future cash flows. Compounding is used to calculate the future value (FV) of present cash flows.

Question 40

Eg Future value: You think you will be able to deposit £4,000 at the end of each of the next three years in a bank account paying 8% interest. You currently have £7,000 in the account. How much will you have in 3 years?

Answer 40

Cash flow in Y0: FV3 = 7,000 × (1 + 0.08)^3 = 8,817.93 Cash flow in Y1: FV3 = 4,000 × (1 + 0.08)^2 = 4,665.60 Cash flow in Y2: FV3 = 4,000 × (1 + 0.08)^1 = 4,320.00 Cash flow in Y3: FV3 = 4,000 Total FV in three years = 8,817.98 + 4,665.60 + 4,320.00 + 4,000.00 = 21,803.58

Question 41

Eg Present value You are offered an investment that will pay you £200 at the end of one year, £400 the year after, £600 in year 3, and £800 at the end of year 4. You can earn 12% on very similar investments. What is the most you should pay for this one?

Answer 41

Find the present value of each cash flow and add them together: Cash flow in year 1: PV0 = 200 / (1 + 0.12)1 = 178.57 Cash flow in year 2: PV0 = 400 / (1 + 0.12)2 = 318.88 Cash flow in year 3: PV0 = 600 / (1 + 0.12)3 = 427.07 Cash flow in year 4: PV0 = 800 / (1 + 0.12)4 = 508.41 Total present value = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93

Question 42

Eg Investment decision Your broker calls you and tells you that he has this great investment opportunity. If you invest £100 today, you will receive £40 in one year and £75 in two years. If you require a 15% return on investments of this risk, should you take the investment?

Answer 42

PV_0 = 40 / (1 + 0.15)ˆ1 + 75/(1 + 0.15)ˆ2 = 91.49 Your broker is asking you to invest £100 – you should reject the investment.

Question 43

Why is the timing of cash flows is crucial when calculating the present or future value of a project or asset?

Answer 43

If cash flows occur at the end of a period, that's typically the assumption unless stated otherwise. This is important because it directly impacts how you discount or compound those cash flows

Question 44

What are the cash flow timing implications?

Answer 44

For the NPV calculation, if we discount back to the present: The cash flow in Year 1 would be discounted by (1+r)ˆ1 The cash flow in Year 2 would be discounted by (1+r)ˆ2 The cash flow in Year 3 would be discounted by (1+r)ˆ3 For the FV calculation, if we compound forward: The cash flow in Year 1 would be compounded by (1+r)ˆ2 to get its value at the end of Year 3 The cash flow in Year 2 would be compounded by (1+r)ˆ1 The cash flow in Year 3 would remain as is, since it's already at the end of the third period

Question 45

What is annuity?

Answer 45

a finite series of equal payments that occur at regular intervals over a fixed period of time

Question 46

What are the two main types of annuities?

Answer 46

ordinary annuity and annuity due

Question 47

What is ordinary annuity?

Answer 47

when the first payment occurs at the end of the period This is the most common type of annuity. For example, if you make monthly payments into a retirement account that starts at the end of each month, this would be an ordinary annuity

Question 48

What is annuity due?

Answer 48

when the first payment occurs at the beginning of the period An example would be a rent payment made at the start of each month (instead of at the end). Annuity due payments are slightly more valuable than ordinary annuities because each payment is invested for one additional period

Question 49

What is perpetuity?

Answer 49

infinite series of equal payments that occur at regular intervals Since perpetuities last forever, they don’t have a fixed term (i.e., no end date). The most common example is the British Government’s Consol Bonds, which pay interest indefinitely

Question 50

What is the PV of annuity equation when annuity (t=T)

Answer 50

PV of annuity = C x [1- 1/(1+r)^T]/r = C/r x [1- 1/(1+r)^T] C = Cash flow (payment) per period r = Interest rate per period T = Total number of periods

Question 51

What is the FV of annuity equation when (t=T)

Answer 51

C x [(1+r)^T - 1]/ r C = Cash flow (payment) per period r = Interest rate per period T = Total number of periods

Question 52

What is PV of perpetuity equation when perpetuity (t->∞)

Answer 52

C/r

Question 53

Annuity - Future value example: Suppose you begin saving for your retirement by depositing £2,000 per year in a savings account. If the interest rate is 7.5%, how much will you have in 40 years?

Answer 53

FV of annuity = C x [(1+r)^T - 1]/ r = 2000 x [(1+0.075)^40 - 1]/ 0.075

Question 54

Annuity Present value example You can afford to pay £150 per month towards a car. The bank can lend you the money at 1% per month for 48 months. How much can you borrow? You are borrowing money TODAY, so you need to compute the present value

Answer 54

PV of annuity = C/r x (1- 1/(1+r)^T) = 150/0.01 x [1- 1/(1+0.01)^48] = 5,696.09

Question 55

Annuity example finding the payment: Suppose you want to borrow £10,000 for a new car. You can borrow at 0.6667% per month. If you take a 4-year loan, what is your monthly payment?

Answer 55

10,000 = C/ 0.00667 x [1 - 1/(1+0.00667)^48] 10,000 = C x 0.2731/0.00667 C = 10,000/40.962 C = 244.13

Question 56

Annuity example finding the number of payments: Suppose you borrow £2,000 at 5% and you are going to make annual payments of £734.42. How long before you pay off the loan?

Answer 56

2000 = 734.42/0.05 x [1 - 1/(1+0.05)^t] 2000x0.05/734.42 = 1- 1/1.05^t 1/1.05^t = 1- 0.1362 1.05^t = 1.1576 tln(1.05) = ln(1.1576) t= 3 years

Question 57

Annuity example finding the rate: Suppose you borrow £10,000 from your parents to buy a car. You agree to pay £207.58 per month for 60 months. What is the monthly interest rate?

Answer 57

trial and error process 10,000 = 207.58/r x [1- 1/(1+r)^60] try r=1% PV = 207.58/0.01 x [1- 1/1+0.01)^60] = 9,331.77<10,000 implies r=1% is larger than a true r try r=0.5% PV = 207.58/0.005 x [1- 1/1+0.005)^60] = 10,737,19>10,000 implies r=0.5% is smaller than a true r try r=0.75% PV = 207.58/0.0075 x [1- 1/1+0.0075)^60] = 9,999.83≈10,000 implies r=0.75% the rate we borrow at

Question 58

Annuity due example: You have won a competition. You will receive £100,000 a year for 20 years, starting today. If you can earn 12 percent on your investments, what are your winnings worth today?

Answer 58

PV of annuity due = PV of ordinary annuity x (1+r) PVA due = C/r x [1 - 1/(1+r)^T] x (1+r) = 100/0.12 x [1 - 1/(1+0.12)^20] x (1+0.12) = 836.57769

Question 59

What is the trial and error method to find the interest rate (discount rate) that makes the PV of the annuity equal to the loan amount

Answer 59

PV of annuity = C x [1- 1/(1+r)^T]/r = C/r x [1- 1/(1+r)^T] Choose an initial interest rate guess (r₀). Start with a reasonable guess, say 5% (0.05) or 10% (0.10). Calculate the present value (PV) of the annuity using the guessed rate. Plug the guessed rate into the formula and calculate the resulting PV. Compare the computed PV with the actual loan amount: If the computed PV is greater than the loan amount, the guessed interest rate is too low. Increase the rate and repeat the calculation. If the computed PV is less than the loan amount, the guessed interest rate is too high. Decrease the rate and repeat the calculation. Repeat the process: Adjust the interest rate based on the outcome, recalculating the PV each time, until the computed PV is very close to the actual loan amount. The rate that makes the computed PV match the loan amount is the correct interest rate for the loan.