The Time Value of Money Flashcards
Bonds and stocks
buy bond today and will get face value back and interest payments later
have to convert all benefits and costs to PV to compare
purchase stocks today and receive dividends in future
Timelines
linear representations of the timing of potential cash floes
- inflows are positive
- outflows are negative
3 rules of time travel
- comparing and combining values
- Moving cash flows forward in time
- moving cash flows back in time
- comparing and combining values
- a dollar today and a dollar in a year aren’t equivalent
- need to calculate PV of values to compare
- Moving cash flows forward in time
to move a cash flow forward in time you must compound it
if you believe you can earn 10% on $1000 today then:
it’s worth 1000 x 1.1 x 1.1 = $1210 in 2 years
- this is called compounding
- moving cash flows back in time
to move a cash flow back in time, we must discount it
e.g savings bond pays $15000 in 10 years
if competitive market interest rate is fixed at 6% per year, what is the bond value worth today?
PV = 15000/1.06^10 = $8375.92 today
= C/(1+r)^n
Applying rules of time trade
if we plan too save $1000 today and $1000 at the end of each year for 3 years at 10% fixed interest rate what will we end up with?
Compound each amount adding 1000 each year
use timeline to help
Valuing a stream of cash flows
- If we want to find the PV of a stream of cash flows, we simply add the PV’s of each
- if you have to pay out or lose money at some point then it’s a negative cash flow
thus, PV = Co + C1/(1+r) + C2/(1+r)^2 etc etc
if you pay back a loan over 4 years, first year you pay 5000, then 8000 each year from that, how much did you borrow in the first place with a 6% fixed interest rate?
PV = 5000/1.06 + 8000/1.06^2 + 8000/1.06^3 + 8000/1.06^4
= $24,890.65
Future value of cash flows
FV = PV x (1+r)^n
Calculating Net Present Value
- for evaluating investment decisions
= sum of all PV of cash flows
e.g if you invest $1000 today you will receive $500 at the end of next 3 years. You could otherwise earn 10% on your money
PV = -1000 + 500/1.1 + 500/1.1^2 + 500/1.1^3 = $243.43
should take investment as is equivalent to receiving $243.43 today
Perpetuities
when a constant cash flow will occur at regular intervals forever it is called a perpetuity
the value of the perpetuity is simply the cash flow divided by the interest rate
PV = C/r
e.g if you need $30,000 of funds per year forever starting next year and get 8% fixed interest rate what will you need to invest now?
PV = C/r = 30,000/0.08 = $375,000
Annuities
when a constant cash flow will occur at regular intervals for a finite no. of N period
PV = C/(1+r) + C/(1+r)^2 ….
= C/r x (1-(1/(1+r)^N)). N on the bottom
e.g you’ve won $30 million. You can taker it either as 30 payments of a million a year or $15 mill today. Interest rate is 8% what should you take?
first one is an annuity as first payment today and is finite
PV = 1/0.08 x (1-1/1.08^29)
= $11.16 million today
adding 1 mill upfront PV = $12.16 mill
therefore more valuable to receive $15 mill upfront
Future value of an annuity
FV = PV x (1+r)^N
=
FV=C/r (1-1/(1+r)^N ) (1+r)^N
e.g Ellen is 35 and wants to plan her retirement. At the end of each year until she is 65 she will save $10,000 in a retirement account. Gets 10%
FV = 10,000/0.1 x (1.1^30 -1) = $1.645 mill
Growing perpetuity
assume you expect the amount of your perpetual payment to increase at a constant rate, g
PV = C/r-g
e.g cost of $30,000 party grows by 4% per year
PV = 30,000/(0.08-0.04) = $750,000 today
