module 2: time value of money (chapter 4) Flashcards
what are the 3 fundamental rules to time value of money?
- only cash flows at the same point in time can be compared or combined
- to move a cash flow forward in time, compound it (multiply by (1+r)^n)
- to move a cash flow backward in time, discount it (divide by (1+r)^n)
consider a cash flow to be received 6 years from today of 1200. if r = 4% per year, what is the pv of this cash flow?
pv = 1200/1.04^6
suppose you invest 1k today at an interest rate of 8% per year. how much will this grow to after 15 years if you receive compound interest
fv = 1.08^15 * 1000
to get the pv of a stream of cash flows…
add up values
pv0 = c0 + c1/(1+r) + c2/(1+r)^2 +…
suppose r = 4%, c0 = 250, c1 = -175, c2 = -225 and c5 = 500. calculate pv0
250 - 175/1.04 - 225/(1.04^2) + 500/1.04^5
suppose r = 4%, c0 = 250, c1 = -175, c2 = -225 and c5 = 500. calculate fv8
1.04^8 (250 - 175/1.04 - 225/(1.04^2) + 500/1.04^5)
an investment that costs 10k today will produce cash flows of 4k one year from today and 7.5k 2 years from today. assume the int rate is 5% and that you have 10k today. if you want to spend as much as you can 2 years from today, should you make this investment?
npv = -10k + 4k/1.05 + 7.5k/1.05^2
npv > 0 , make the investment
an investment that costs 10k today will produce cash flows of 4k one year from today and 7.5k 2 years from today. assume the int rate is 5% and that you have 10k today. if you want to spend the same amount today, after 1 year and after 2 years. show how the spending changes if you make the investment.
max spending after 2 years = 11700
4000/1.05 + 7500/1.05^2 = x(1 + 1/1.05 + 1/1.05^2)
x = 3711
what is a regular perpetuity vs annuity?
perpetuity -> equal cash flows at constant time intervals that last forever
annuity -> stops after n payments
what is the pv of a growing perpetuity if g < r? if r > 0?
c1/r-g
C/r
the british gov has a consol bond paying 100 per year forever. assume curr interest rate is 4%. what is the value of the bond immediately after a payment is made? immediately before a payment?
after:
100/0.04
before
(1.04*100)/0.04
if the interest rate is 4% per year, what is the pv of an annuity paying 1k at the end of each of the next 10 years?
(1000/0.04) * (1 - (1/1.04^10))
if the interest rate is 4% per year, what is the pv of an annuity paying 1k for the next 10 years where the first payment is today?
(1.04*1000)/0.04 * (1 - (1/1.04^10))
c * (1+r) factor is to discount the entire annuity as though to move cash flows one period earlier
removes one period from the discounting as first cahs flow is today
if duane deposits 1k at the end of each year for the next 15 years in an account paying 2% interest per year, how much money will be in his account after 15 years?
c * (1/r)[(1+r)^n - 1]
(1000/0.02)(1.02^15 - 1)
you are 25 and decide to start saving for retirement. you plan to save 5k at the end of each year (so the first deposit will be one year from now) and will make the last deposit when you retire at age 65. suppose you earn 8% per year on your retirement savings.
a) how much will you have saved for retirement
b) how much will you have saved if you wait until age 25 to start saving (with first deposit at end of year)
a) (5k/0.08)(1.08^40 - 1)
b) (5k/0.08)(1.08^30 - 1)
a rich relative has bequeathed you a growing perpetuity. the first payment will occur in a year and will be 1k. each year after that, you will receive a payment on the anniversary of the last payment that is 8% larger than the last payment. if the interest rate is 12% per year, what is today’s value of the bequest?
c/(r-g)
1000/(0.12-0.08)
you are thinking of building a new machine that will save you 1k in the first year (at end of year). this machine will then begin to wear out so that the savings decline at a rate of 2% per year forever. what is the pv of the savings if the interest rate is 5% per year?
1000/(1.05 - (1/1.02))
1000/0.07
fred wants to save some money at the end of each of the next 10 years. he plans to save 1k at the end of the first year, an to have this increase by 4% each year. if he can earn 3% interest each year on his savings, how much money will he saved 10 years from today?
pv = c/(r-g) [1 - (1+g/1+r)^n]
= 1000/(0.03-0.04) * [ 1 - (1.04/1.03)^10] (1.03)^9
you’re saving for retirement and you decide that to live comfortably, you need to save 2 million in your rrsp by the time you’re 65. starting on your 30th birthday and continuing on every birthday up to and including your 65th, you will put the same amount into an rrsp account. if interest is 5%, how much do you need to put aside each year to have 2 million in the rrsp on your 65th?
fv = c * 1/r ((1+r)^n) - 1)
2mill/c = c * 1/0.05 [1.05^36 - 1]
2 mill/c = (1.05^35-1) / 0.05
2 * (0.05 / (1.05^36 - 1))
you’re saving for retirement and you decide that to live comfortably, you need to save 2 million in your rrsp by the time you’re 65. starting on your 30th birthday and continuing on every birthday up to and including your 65th, you will put the same amount into an rrsp account, growing by 7% per year. if interest is 5%, how much do you need to put aside each year to have 2 million in the rrsp on your 65th?
2 * 1/1.05^35 *
(1.07/1.05 - 1)/((1.07/1.05)^36 - 1)
what is internal rate of return?
interest rate that makes npv of investment = 0
if you invest 10k today in return for receiving 15k after 4 years what is the irr?
(15/10)^1/4 -1
suppose you 5k today in return for a perpetuity paying 200 after one year growing at 1.5% per year what is the irrr?
200/5000 + 1.5%
if you deposit 5k today in an account paying interest of 4% per year how many years will it take until the value of your account grows to 8k?
n = ln(c1/p)/ln(1+r)
(ln(8000/5000))/ln(1.04)
