FM213 MT Notes
Anya Shah
2023
, 1 Present Values
Assuming interest rate r and investment period T :
Future Value FV ( r , T )=X ( 1+r )T
X
Present Value PV ( r , T )=
( 1+ r )T
Net Present Value
K
ci c1 c ck
NPV =∑ =c 0+ + 2 2 +…+
i=0 ( 1+ r ) i
(1+ r ) ( 1+ r ) ( 1+r )k
1.1 Present Value of Perpetuity
c
P V P=
r
Proof:
c c c
P= + + +…
1+r ( 1+r ) ( 1+ r )3
2
1
Let X =
1+r
2 3
P=cX +c X +c X + …
Multiply by X
2 3 4
XP =c X + c X + c X +…
Subtract the two
P− XP=cX
cX c
P= =
1− X r
1.2 Present Value of Annuity
For a T year annuity paying c per year:
c 1
P V A= (1− )
r ( 1+r )T
Proof:
The value of a perpetuity paying C per year for the first year is:
c
P V 1=
r
The value of a perpetuity paying C per year and with a first payment
in T + 1years is:
Anya Shah
2023
, 1 Present Values
Assuming interest rate r and investment period T :
Future Value FV ( r , T )=X ( 1+r )T
X
Present Value PV ( r , T )=
( 1+ r )T
Net Present Value
K
ci c1 c ck
NPV =∑ =c 0+ + 2 2 +…+
i=0 ( 1+ r ) i
(1+ r ) ( 1+ r ) ( 1+r )k
1.1 Present Value of Perpetuity
c
P V P=
r
Proof:
c c c
P= + + +…
1+r ( 1+r ) ( 1+ r )3
2
1
Let X =
1+r
2 3
P=cX +c X +c X + …
Multiply by X
2 3 4
XP =c X + c X + c X +…
Subtract the two
P− XP=cX
cX c
P= =
1− X r
1.2 Present Value of Annuity
For a T year annuity paying c per year:
c 1
P V A= (1− )
r ( 1+r )T
Proof:
The value of a perpetuity paying C per year for the first year is:
c
P V 1=
r
The value of a perpetuity paying C per year and with a first payment
in T + 1years is:
