How to Calculate Present Values

Present Value of a single CF

• The present value of a single cash flow C expected n years from now is given by:-

• You can also use the PVIF table

𝐶(1+𝑟 )𝑛

PVIF Table

Future Value of a single CF

• The future value of a single cash flow C at a time n years from now is given by:-

C *

• You can also use the FVIF table

FVIF Table

Perpetuity

• Perpetuity is when the same cash flow C is paid every year for an infinite period

• C is the periodic cash-flow • r is the discount rate• PV is the present value of this cash flow

stream Example C=100, r=10% PV=1000

Growing Perpetuity

• When the cash flows paid out every year grow at constant rate of growth g

• • C1 is the cash-flow at the end of year 1• r is the discount rate• g is the growth rate• PV is the present value of this cash flow streamExample C1=100, r=10%, g=5% PV=2000

PV of Annuity

• When the same cash flow C is paid out for n years

• C is the periodic cash-flow• r is the discount rate• n is the number of periods for which the cash

flow will last• PV is the present value of this cash flow stream• If n is in months, this is the EMI (Equated

Monthly Installments) formula

PVIFA Table• Alternatively, the PVIFA table can also be used

to calculate the PV of an annuity

FVIFA Table• To calculate future value of an annuity, you

can use the FVIFA table

2-14

• Factory costs 800,000• Produces 170,000 for 10 years• Find

– A. Its NPV if r = 14%• (86,739.66)

– B. Value of factory at the end of 5 years• (583,623.76)

2-25

• If r= 8%, what amount needs to be set aside for – A perpetuity of 1 bn

• 12.5 bn– Perpetuity of 1 bn growing at 4%

• 25 bn– 1 bn for 20 yrs

• 9.82 bn– 1 bn spread evenly over the year for 20 yrs

• 10.20 bn

2-34

• Couple retires in 3 yrs• Needs 15k per month, gets 9k from other

sources• Has a fund of 1,000,000• Fund grows at 3.5%• For how many years can withdrawals be

made?– 21.38 yrs

2-39

• Pipeline generates 2 mn in year 1• Cash flows declining by 4% p.a. • If r= 10%, find:-

– PV of pipeline if cash flows continue forever• 14.29 mn

– PV of pipeline if cash flows last for 20 years• 13.35 mn• Can we use Value perpetuity minus TV in yr 21?

Growing Annuity

• When the cash flows paid out every year grow at constant growth rate g and are paid for n years

• C1 is the cash-flow at the end of year 1• r is the discount rate• g is the growth rate• n is the number of periods for which the cash

flow will last• PV is the present value of this cash flow stream

Cash flows starting at beginning of year

• All the present value formulae above assume that cash flows start at the end of year 1. If cash flows start at the beginning of year (first cash flow is at time 0), then the present values can be calculating by multiplying all the above PV formulae by an additional (1 + r) factor

• The rationale is that is cash flows start at time 0, then every cash flow is effectively being discounted for one less year than assumed in the formulae in the previous slides

Calculating Future Values

• To calculate the future value of a stream of cash flows (the value of all cash flows at the end of year n instead of their value at time 0), all the present value formulae must be multiplied by the factor

Compounding at higher frequencies

• To compound at higher frequencies, divide the rate by number of periods per year and multiply the time (in years) by number of periods per year

• For continuous compounding, use exp(rt) as the compounding factor

Different rates of borrowing and lending

• We need to consider each cash flow individually• Example:-

– Rate of borrowing is 12%– Rate of lending is 10%– Calculate the value of the project at the end of its life

cycle

Year CF0 -551 602 803 -85

Different rates of borrowing and lending

Year CFAmount Left after repayment in Yr 2

Amount invested at year 3

Amount left at the end of the project

0 -55  

1 60 -1.60  

2 80 78.21  

3 -85     1.0288