Time Value of money (LECTURE 2) Flashcards
Why is £100 worth more today than in 1 year’s time?
- Risk, £100 is certain now. Things we have now are more valuable than things we hope to get.
- Inflation, under inflationary conditions purchasing power declines.
- Personal consumption and investment preferences, immediate rather than delayed consumption.
TIME VALUE OF MONEY
- Money has a price, if you own it you can “rent” it out to say a bank.
- Cash should be used to earn highest return.
- Investors return comprises:
- Reward for foregoing immediate consumption.
- Compensation for risk and loss of purchasing power.
What is the role of time value in finance?
- Most financial decisions involve costs and benefits that are spread out over time.
- Time value of money allows comparison of cash flows from different periods.
SIMPLE INTEREST
With simple interest, you don’t earn interest on interest.
So each year you earn the same amount of interest (same percentage of initial investment) not percentage of current savings.
COMPOUND INTEREST
Depositor earns interest on interest.
So each year, the same percentage is taken from the compounded amount of the previous years’ interest.
FUTURE VALUE
Compounding or growth over time.
The value of a sum after investing over one or more periods.
e.g. the future value of £10,000 at 12% is £11,200
10,000 x 1.12 = 11,200
PRESENT VALUE
Discounting today’s value.
How much must you initially invest to get a certain amount of money in the future.
e.g. if you want £11,424 next year and there is a 12% rate, how much must you put in the bank today? Algebraically: PV x 1.12 = £11,424 PV= £11,424/1.12 =£10,200
PV = C1/ 1+r
where C1 is cash flow at date 1 ND R is rate of return (or discount rate).
FUTURE VALUE formula (FVn)
FVn= PV0 (1+K)^n
OR (using table)
PV(FVIFk,n)
PRESENT VALUE (PV0)
FVn[1/(1+k)^n]
OR (using table)
FV(PVIFk,n)
FUTURE VALUE of ANNUITY (FVAn)
A(1+k)^n - 1 / k
OR (using table)
A(FVIFk.n)
PRESENT VALUE of ANNUITY (PVA0)
A 1 - [1/1+k)^n] / k
OR (using table)
A(PVIFAk,n)
TIME VALUE TERMS
PV0, k, FVn, n, A
PV0 = present value or beginning amount
k = interest rate
FVn = future value at the end of n periods
n = number of compounding periods
A = and annuity
ANNUITY
Series of equal payments or receipts.
FUTURE VALUE EXAMPLE:
You deposit £2,000 today at 6% interest. How much will you have in 5 years?
PV = 2,000 k = 6 or 0.06 n = 5
[1] FVn = PV0(1+k)^n
[2] PV(FVIFk,n)
[1] 2,000 x (1.06)^5
[2] 2,000 x FVIF6%,5 = 2,000 x 1.382
= £2,676.40
What happens when you compound more frequently than annually?
- Results in a higher effective interest rate because you are earning on interest on interest more frequently.
- As a result, the effective interest rate is greater than the nominal (annual) interest rate.
- Furthermore, the effective rate of interest will increase the more frequently interest is compounded.
EXAMPLE: What would be the difference in the future value if I deposit £100 for 5 years and earn 12% annual interest compounded a) annually b) semi-annually c) quarterly d) monthly?
ANNUALY: 100 x (1+0.12)^5 = £176.23
SEMI-ANNUALLY: 100 x (1+0.06)^10 = £179.09
QUARTERLY: 100 x (1+0.03)^20 = £180.61
MONTHLY: 100 x (1+0.01)^60 = £181.67
FUTURE VALUE formula for compounding more frequently than annually.
Divide k by number of compounding periods m and multiply n by number of compounding periods m.
FVn = PV0(1+k/m)^nxm
NOMINAL INTEREST RATE
The stated or contractual rate of interest charged by a lender or promised by a borrower.
EFFECTIVE INTEREST RATE
The rate actually paid or earned
What is the general relationship between the effective and the nominal interest rate when compounding occurs more than once per year?
Effective rate > nominal rate whenever compounding occurs more than once per year.
EAR = (1+k/m)^m - 1
EAR EXAMPLE:
What is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly?
EAR:
(1+0.018/12)^12 - 1
= 19.56%
What idea is present value based on?
That a pound today is worth more than a pound tomorrow.
Other ways of saying discount rate?
Opportunity cost
Required return
Cost of capital
PRESENT VALUE EXAMPLE:
How much must you deposit today in order to have £2,000 in 5 years if you can earn 6% interest on your deposit?
PV0 = FVn[1/1+k)^n]
OR
FV(PVIFk,n)
£2,000 x [1/1.060^5]
OR
£2,000 x PVIF6%,5
£2,000 x 0.74758 = £1.494.52
What are ANNUITIES?
- Equally spaced cash flows of equal size.
- Can be inflows or outflows.
What are the two main types of annuities?
- An ORDINARY (deferred) ANNUITY has cash flows that occur at the end of each period.
- An ANNUITY DUE has cash flows that occur at the beginning of each period.
Why will annuity due always be greater than an otherwise equivalent ordinary annuity?
Because interest will compound for an additional period.
FUTURE VALUE OF AN ORDINARY ANNUITY EXAMPLE:
How much will your deposits grow to if you deposit £100 at the end of each year at 5% interest for 3 years?
FVAn = A(1+K)^n - 1 / k
= A(FVIFAk,n)
FVA = 100(FVIFA5%,3) = £315.25
FUTURE VALUE OF ANNUITY DUE EXAMPLE:
How much will your deposits grow to if you deposit £100 at the beginning of each year at 5% interest for 3 years?
FVA= 100(FVIFA5%,3)(1+k) =
£330.96
FVA = 100(3.152)(1.05)= £330.96
PRESENT VALUE OF AN ORDINARY ANNUITY EXAMPLE:
How much could you borrow if you could afford annual payments of £2,000 (which includes principle and interest) at the end of each year for three years at 10% interest?
PVA = 2,000(PVIFA10%,3)
=£4,973.70
MIXED STREAM OF CASH FLOWS
Reflects no particular pattern.
PERPETUITY
A special kind of annuity where the periodic annuity or cash flow stream continues forever.
PRESENT VALUE OF A PERPETUITY
PV= Annuity/k
OR
A/k
PV OF A PERPETUITY EXAMPLE:
How much would I have to deposit today in order to withdraw £1,000 each year forever if I can earn 8% on my deposit?
PV= £1,000/0.08
=£12,500
LOAN AMORTIZATION EXAMPLE:
Suppose a business takes out £6,000 loan at 10% interest for 4 years. The loan agreement calls for the borrower to pay certain amount (some equal instalments) so that the loan and the interest amount is fully paid after 4 years? Calculate the instalment and prepare an amortization schedule.
PVA0= A 1-[1/(1+k)^n]/k
OR
A(PVIFAk,n)
£6,000 = A(PVIFA10%,4) £6,000 = A(3.170) A= £1,892.74
