Lecture 1
Basic interest rate theory
Lecture 1
1 / 22
The time value of money (1)
A fundamental property in financial economics is the time value of money.
Lecture 1
2 / 22
The time value of money (1)
A fundamental property in financial economics is the time value of money.
This means that it is better to have an amount today than the same
amount at any later time.
A dollar today is worth more than a dollar in the future.
Lecture 1
2 / 22
The time value of money (1)
A fundamental property in financial economics is the time value of money.
This means that it is better to have an amount today than the same
amount at any later time.
A dollar today is worth more than a dollar in the future.
The reason for this is that money today can be put into the bank.
Lecture 1
2 / 22
The time value of money (2)
Example
Assume that we have 100 USD today and that there is a bank offering an
interest rate of 5% per year. This means that efter 1 year 100 USD has
grown to
100 · (1 + 0.05) = 105
USD. Hence, it is better to have 100 USD today than 100 USD in a year,
since we can let the money today grow with interest rate.
Lecture 1
3 / 22
The time value of money (2)
Example
Assume that we have 100 USD today and that there is a bank offering an
interest rate of 5% per year. This means that efter 1 year 100 USD has
grown to
100 · (1 + 0.05) = 105
USD. Hence, it is better to have 100 USD today than 100 USD in a year,
since we can let the money today grow with interest rate.
If follows that in this case 100 USD today is equivalent to having 105 USD
in a year.
Lecture 1
3 / 22
The time value of money (3)
We can generalize the previous example.
Lecture 1
4 / 22
The time value of money (3)
We can generalize the previous example.
An amount x put into a bank account with interest rate r grows to
x(1 + r ) = x + xr
after one year.
Lecture 1
4 / 22
The time value of money (3)
We can generalize the previous example.
An amount x put into a bank account with interest rate r grows to
x(1 + r ) = x + xr
after one year.
We call x the principal and xr the interest.
Lecture 1
4 / 22
The time value of money (4)
What happens if we let the money be in the bank for another year?
Lecture 1
5 / 22
The time value of money (4)
What happens if we let the money be in the bank for another year?
After two years we have
x(1 + r ) · (1 + r ) = x(1 + r )2
on our account.
Lecture 1
5 / 22
The time value of money (5)
We can continue in this way, and after n years when the interest rate is r
the amount x has grown to
x(1 + r )n .
Lecture 1
6 / 22
The time value of money (5)
We can continue in this way, and after n years when the interest rate is r
the amount x has grown to
x(1 + r )n .
This exponential behavior of the value is due to the fact that we get
interest on interest.
When n = 2 we have
x(1 + r )2 =
x
+ |{z}
2xr +
xr 2
|{z}
|{z}
principal interest interest on interest
Lecture 1
6 / 22
Compounding (1)
If you have a mortage or a loan on your car or house, then you usually pay
interest every month.
Lecture 1
7 / 22
Compounding (1)
If you have a mortage or a loan on your car or house, then you usually pay
interest every month.
Example
To buy a car you have to take a loan of 10 000 USD. A financial
institution is offering to lend you this amount at an interest rate of 8%. If
the interest payments are to be paid montly, then you have to pay
10 000 ·
0.08
= 67 USD
12
each month as interest rate payment.
Lecture 1
7 / 22
Compounding (1)
If you have a mortage or a loan on your car or house, then you usually pay
interest every month.
Example
To buy a car you have to take a loan of 10 000 USD. A financial
institution is offering to lend you this amount at an interest rate of 8%. If
the interest payments are to be paid montly, then you have to pay
10 000 ·
0.08
= 67 USD
12
each month as interest rate payment.
Note that in this example the interest rate is quoted as a yearly rate. This
is always the case.
Lecture 1
7 / 22
Compounding (2)
In general, if the interest rate r is compounded m times per year, then the
amount x grows to
r
x 1+
m
after 1 period
Lecture 1
8 / 22
Compounding (2)
In general, if the interest rate r is compounded m times per year, then the
amount x grows to
r
x 1+
m
after 1 period, to
r m
x 1+
m
after one year
Lecture 1
8 / 22
Compounding (2)
In general, if the interest rate r is compounded m times per year, then the
amount x grows to
r
x 1+
m
after 1 period, to
r m
x 1+
m
after one year, and to
h
r m in
r mn
x 1+
=x 1+
m
m
after n years.
Lecture 1
8 / 22
Compounding (3)
What happens if m → ∞? Recall that for any y ∈ R
y m
lim 1 +
= ey .
m→∞
m
Lecture 1
9 / 22
Compounding (3)
What happens if m → ∞? Recall that for any y ∈ R
y m
lim 1 +
= ey .
m→∞
m
Hence,
r mn
lim x 1 +
= xe rn .
m→∞
m
This is called continuous compounding.
Lecture 1
9 / 22
Compounding (3)
What happens if m → ∞? Recall that for any y ∈ R
y m
lim 1 +
= ey .
m→∞
m
Hence,
r mn
lim x 1 +
= xe rn .
m→∞
m
This is called continuous compounding.
Continuous compounding is often used in mathematical finance, but in
finance and economics – and in the real world – rates are seldom quoted in
this way.
Lecture 1
9 / 22
Compounding (4)
Observe that
m1 ≤ m2 ⇒
r
1+
m1
m1
r m2
≤ 1+
,
m2
so the compounding factors increase to e r :
r m
1+
↑ e r as m ↑ ∞.
m
Lecture 1
10 / 22
Compounding (5)
Example
Assume that r = 4%. In this case we have
m
1
2
4
12
365
∞
(1 + r /m)m
1.040000
1.040400
1.040604
1.040742
1.040808
1.040811
We se from this example that an interest rate of 4% compounded monthly
represents an efficient rate of 4.07% per year.
Lecture 1
11 / 22
Compounding (6)
The values of m in the previous Example was not selected randomly:
m
1
2
4
12
365
∞
Means
Annually
Semi-annually
Quarterly
Monthly
Daily
Continuously
Lecture 1
12 / 22
Future values (1)
Now assume that we have a stream of cash flows
x = (x0 , x1 , x2 , . . . , xn−1 , xn ),
i.e. a series of payments occuring at the times 0, 1, 2, . . . , n − 1, n.
Each of the individual cash flows xk can be positive or negative.
Lecture 1
13 / 22
Future values (2)
Note that
Positive cash flow = Inflow of money.
Negative cash flow = Outflow of money.
This means that if you put 1 000 000 USD in a bank account, then the
cash flow is negative! You do not lose any money, but you have an outflow
of money (into your own bank account).
Lecture 1
14 / 22
Future values (2)
Note that
Positive cash flow = Inflow of money.
Negative cash flow = Outflow of money.
This means that if you put 1 000 000 USD in a bank account, then the
cash flow is negative! You do not lose any money, but you have an outflow
of money (into your own bank account).
Question
What is the value today of a stream of cash flows at time n if the interest
rate is r ?
Lecture 1
14 / 22
Future values (3)
The cash flow x0 can grow on the bank for n periods, so
x0 → x0 (1 + r )n .
Lecture 1
15 / 22
Future values (3)
The cash flow x0 can grow on the bank for n periods, so
x0 → x0 (1 + r )n .
The cash flow x1 can grow on the bank for n − 1 periods, so
x1 → x1 (1 + r )n−1 .
Lecture 1
15 / 22
Future values (3)
The cash flow x0 can grow on the bank for n periods, so
x0 → x0 (1 + r )n .
The cash flow x1 can grow on the bank for n − 1 periods, so
x1 → x1 (1 + r )n−1 .
..
.
The cash flow xn−1 can grow on the bank for 1 period, so
xn−1 → xn−1 (1 + r ).
Lecture 1
15 / 22
Future values (3)
The cash flow x0 can grow on the bank for n periods, so
x0 → x0 (1 + r )n .
The cash flow x1 can grow on the bank for n − 1 periods, so
x1 → x1 (1 + r )n−1 .
..
.
The cash flow xn−1 can grow on the bank for 1 period, so
xn−1 → xn−1 (1 + r ).
The cash flow xn cannot grow on the bank for any period, so
xn → xn .
Lecture 1
15 / 22
Future values (4)
Summing up, the future value FV of the stream of cash flows
x = (x0 , x1 , . . . , xn−1 , xn ) is given by
FV = x0 (1 + r )n + x1 (1 + r )n−1 + . . . + xn−1 (1 + r ) + xn
n
X
=
xk (1 + r )k .
k=0
Lecture 1
16 / 22
Present values (1)
We have seen that 100 USD today is worth 105 USD in a year if the yearly
interest rate is 5%.
Question
What is 100 USD in a year worth today?
Lecture 1
17 / 22
Present values (1)
We have seen that 100 USD today is worth 105 USD in a year if the yearly
interest rate is 5%.
Question
What is 100 USD in a year worth today?
Answer
It is worth
100
= 95.24 USD.
1.05
Why is this so?
Lecture 1
17 / 22
Present values (1)
We have seen that 100 USD today is worth 105 USD in a year if the yearly
interest rate is 5%.
Question
What is 100 USD in a year worth today?
Answer
It is worth
100
= 95.24 USD.
1.05
Why is this so?If we have 95.24 USD today, then we can put them into a
bank account. After 1 year this is worth
95.24 · 1.05 = 100.
Lecture 1
17 / 22
Present values (1)
We have seen that 100 USD today is worth 105 USD in a year if the yearly
interest rate is 5%.
Question
What is 100 USD in a year worth today?
Answer
It is worth
100
= 95.24 USD.
1.05
Why is this so?If we have 95.24 USD today, then we can put them into a
bank account. After 1 year this is worth
95.24 · 1.05 = 100.
We say that the present value of 100 USD is 95.24 USD.
Lecture 1
17 / 22
Present values (2)
Again, we can consider a general amount and a general interest rate to get
that the one year present value of x when the interest rate is r is
x
.
1+r
Lecture 1
18 / 22
Present values (2)
Again, we can consider a general amount and a general interest rate to get
that the one year present value of x when the interest rate is r is
x
.
1+r
Using the same type of argument we can conclude that the n-year present
value of x is given by
x
.
(1 + r )n
Lecture 1
18 / 22
Present values (2)
Again, we can consider a general amount and a general interest rate to get
that the one year present value of x when the interest rate is r is
x
.
1+r
Using the same type of argument we can conclude that the n-year present
value of x is given by
x
.
(1 + r )n
Now consider a stream of cash flows x = (x0 , . . . , xn ). What is the present
value of this stream of cash flows?
Lecture 1
18 / 22
Present values (3)
The cash flow x0 is given today, so
x0 → x0 .
Lecture 1
19 / 22
Present values (3)
The cash flow x0 is given today, so
x0 → x0 .
The cash flow x1 is given 1 period ahead, so
x1
.
x1 →
1+r
Lecture 1
19 / 22
Present values (3)
The cash flow x0 is given today, so
x0 → x0 .
The cash flow x1 is given 1 period ahead, so
x1
.
x1 →
1+r
..
.
The cash flow xn−1 is given n − 1 periods ahead, so
xn−1
xn−1 →
.
(1 + r )n−1
Lecture 1
19 / 22
Present values (3)
The cash flow x0 is given today, so
x0 → x0 .
The cash flow x1 is given 1 period ahead, so
x1
.
x1 →
1+r
..
.
The cash flow xn−1 is given n − 1 periods ahead, so
xn−1
xn−1 →
.
(1 + r )n−1
The cash flow xn is given n periods ahead, so
xn
xn →
.
(1 + r )n
Lecture 1
19 / 22
Present values (4)
The present value PV of a stream of cash flows is then given by the sum
of all these present values:
PV = x0 +
=
n
X
k=0
xn−1
x1
xn
+ ... +
+
n−1
1+r
(1 + r )
(1 + r )n
xk
.
(1 + r )k
Lecture 1
20 / 22
Present values (5)
With more frequent compound we get
PV =
n
X
k=0
xk
,
(1 + (r /m))k
and with continuous compounding
PV =
n
X
x(tk )e −rtk .
k=0
In the last equation the tk ’s can take on any value in R+ .
Lecture 1
21 / 22
Negative interest rates
Question
What if the interest rate is negative? Can this happen?
Lecture 1
22 / 22