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

value at t=0 on a timeline

Question 2

What is a future value (FV)?

Answer 2

the amount an investment is worth after one or more periods
‘later’ money on a timeline

Question 3

What is interest rate (r)?

Answer 3

the ‘exchange rate’ between earlier and later money
discount rate/ cost of capital/ opportunity cost of capital/ required return
terminology depends on usage

Question 4

What is the future value formula?

Answer 4

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 5

What does the future value formula mean?

Answer 5

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

Question 6

An example of future value (one period)

Answer 6

suppose you invest £100 for one year at 10% per year

what is the future value in one year?
interest = 100 × 0.10 = 10

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

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

Question 7

An example of future value (two periods)

Answer 7

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

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 8

What is simple interest?

Answer 8

interest earned only on the original principal

Question 9

What is compound interest?

Answer 9

interest earned on principal and interest received
‘interest on interest’ - interest earned on reinvestment of previous interest payments

Question 10

Future value example

Answer 10

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

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

How much will you have in 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 11

eg Future value in 200 years

Answer 11

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?

FV = PV × (1 + r)^t = 5 × (1 + 0.06)^200 = 575,629.52

How much can you get if the investment only earns 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 12

What are 2 important relationships in future value?

Answer 12

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

Question 13

What is a dividend?

Answer 13

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 14

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 14

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

Question 15

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

Answer 15

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

Question 16

What does discounting mean

Answer 16

finding the present value of one or more future amounts

Question 17

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 17

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 18

What are the 2 important relationships of the present value?

Answer 18

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 19

How do we find the implied interest rate in an invesmtnet?

Answer 19

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

Question 20

How do we find the number of periods/ years?

Answer 20

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 21

What typically contains multiple cash flows?

Answer 21

an asset or project

Question 22

How do we value an asset or project (since money has time value)?

Answer 22

we need to transform all cash flows to one point in time before they can be meaningfully added together

this transformation is either done through compounding or discounting

Question 23

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 23

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 24

What is annuity?

Answer 24

a finite series of equal payments that occur at regular intervals

Question 25

What is ordinary annuity?

Answer 25

when the first payment occurs at the end of the period

Question 26

What is annuity due?

Answer 26

when the first payment occurs at the beginning of the period

Question 27

What is perpetuity?

Answer 27

infinite series of equal payments that occur at regular intervals

Question 28

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

Answer 28

C x [1- 1/(1+r)^T]/r = C/r x [1- 1/(1+r)^T]

Question 29

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

Answer 29

C x [(1+r)^T - 1]/ r

Question 30

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

Answer 30

C/r

Question 31

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 31

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

Question 32

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 32

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 33

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 33

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 34

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 34

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 35

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 35

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 36

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 36

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