Quantitative Methods - Time Value of Money

So looking closer at Reading 4 - Ethics - GIPS, even with substantial 'optional' sections the non-optional sections look like a real pain containing among other things several page long lists of defined terms and examples of compliant reporting.  Given the time sink to go into that much detail I am going to forgo that section for now and see what Schweser says we should know about GIPS and how that compares to the CFAI.

I have to say on the whole by this point, I've actually enjoyed the CFAI materials - I feel they are very thorough (more than you need) but that's comforting for getting a little more cushion on exam day.  If the practice questions on the Code/Standards were indicative, I'm feeling pretty happy with my Ethics knowledge and just want to sharpen up a few things (see previous post).

Next I'm moving on to the first section of the quantitative methods, Time Value of Money.  This should mostly be review given my finance background but it will be nice to refresh some of these topics and give a nice chance to get some calculator practice.  This is reading 5 in the CFAI materials and is about 60 pages of material.

Start - 1:00 PM

Quantitative Methods - Time Value of Money

"The candidate should be able to:

a. interpret interest rates as required rates of return, discount rates, or opportunity costs;

b. explain an interest rate as the sum of a real risk-free rate, expected inflation, and premiums that compensate investors for distinct types of risk;

c. calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;

d. solve time value of money problems for different frequencies of compounding;

e. calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;

f. demonstrate the use of a timeline in modeling and solving time value of money problems. (Institute 255)"

Institute, CFA. Level I 2012 Volume 1 Ethical and Professional Standards and Quantitative Methods, 7th Edition. Pearson Learning Solutions. <vbk:9781256112754#page(255)>.

Intro

• Individuals value a given amount more highly the earlier it is received

Interest rates

• Interest rate r is a rate of return that reflects the relationship between different dated cash flows
• If 9,500 today is equivalent to 10,000 in one year, the discount rate is 500/9,500=5.26%
• Three interpretations
• Required rate of return - i.e. minimum you must receive to accept the investment
• Discount rate - almost interchangeable with term above - how much you discount the future cash flow each year
• Opportunity cost - what you would forgo by not doing it (a return of 5.26%)
• Rates are set in the market - investors supply, borrowers demand
• Interest rate components:
• r = real risk free interest rate + inflation premium + default risk premium + liquidity premium + maturity premium
• Real risk free interest rate - single period interest rate for a completely risk free security if no inflation is expected.  
• In economic theory, it is the time preference for current versus real future consumption.
• Inflation premium - compensates investors for expected inflation
• Average expected inflation over the maturity of the debt
• Real risk free + inflation = nominal risk free (rf) rate
• 90-day T-bill gives you the nominal rf rate for that period
• Default risk premium - issuer might not pay out
• Liquidity premium - compensate investor for lost liquidity (e.g. small issue bonds)
• Maturity premium - compensate investor for higher sensitivity to interest rate changes as maturity is extended (all else equal)
• Ex look at a long and short T-bill - difference is maturity premium (and likely some inflation premium as well) - risk free, default, and liquidity are probably same

Future Value of a Single Cash Flow

• FV1 = PV(1 + r)
• 1 represents time periods
• You gain interest on both principal and interest
• Interest on the principal is called the simple interest
• Importance of compounding increases as you increase the interest rate
• Multiple periods:
• FVN = PV(1 + r)^N
• Note: Interest rate and N must be compatible!  Must be in same time units
• For a given rate, FV increases with N
• For a given N, FV increases with r
• Be careful to use calendar years
• Ex. if you receive 10 mm 5 years from now, and want to know the FV at 15 years at 9% interest rate, it is (1.09^10)*10
• Timelines can be helpful for complex cash flow problems

Frequency of compounding

• Stated annual rate = monthly rate * 12
• Strictly a quoting convention: (1 + r) is NOT a future value factor when compounding is more than annual
• When compounding more than once per year:
• Divide interest rate number of compounds m per year
• Multiply periods by number of compounds m per year
• Thus
• FV = PV (1 + (r / m)) ^ (Nm) 

Continuous compounding - introducing e

• If compounding is continuous (i.e. infinite periods), use FVN = PVe^(rsN)
• e^(rsN) is the key component.  rs is annual rate, N is number of years
• Example
• 10,000 compounded continuously at 8% for 2 years
• 10,000*e^(2*.08) = 11,735.11

Stated vs. effective rates

• EAR = (1 + periodic rate)^m - 1
• m is periods per year
• Can solve for periodic rate using sqrt/fractional powers
• For continuous, you need to use natural logs - ln
• Ex, calculate the continuous compound rate equal to an EAR of 8.33%
• 8.33% = e^(rs) - 1
• 1.0833 = e^(rs)
• ln(1.0833) = ln(e^(rs))
• 0.08 = rs

Future value of a series of cash flows

• Annuity - finite set of level sequential cash flows
• Ordinary Annuity - first cash flow is at t=1
• Annuity Due - first cash flow at t=0
• Perpetuity - a set of level never-ending cash flows, first cash flow at t=1
• Example
• FV of a 5 year 1,000 annuity at 5%
• PV=0, n=5, i=5, pmt=-1,000
• FV=5,525.63
• FVN = A * {[(1+r)^n - 1]/r}
• Term in {} is the annuity factor

Unequal cash flows - bring each cash flow to its future value at year t and sum the future values

Present Value of a series of cash equal flows

• Suppose you want to buy an asset that pays 1,000 per year for 5 years at 12% rate.
• FV=0, PMT=1,000, n=5, i=12
• PV=-3,604.78
• Annuity due
• Can view as a lump sum today plus an ordinary annuity thereafter

Present value of an infinite series of equal cash flows

• PV = A/r
• Note that a stock paying constant dividends is similar to a perpetuity
• This assumes first payment as at t=1
• Example (tricky)
• Level 100 per year perpetuity, first pmt at t=5.  What is PV (i.e. at t=0) at a 5% discount?
• FV at t=4 (not t=5) is 100/.05=2,000
• Solve FV=2,000, i=5, n=4, PMT=0
• -1,645.41
• Remember the perpetuity is valued as of the beginning of that year - first payment is as of t=5, so the value is as of t=4
• To solve a 'delayed' annuity, subtract one perpetuity from the other 

Present value of a series of unequal cash flows

• Must first find PV of each individual cash flow and then sum
• Usually use a spreadsheet

Solving for rates, number of periods, of size of payments

• Growth
• g = (FV/PV) ^ (1/n) -1
• Remember when calculating using HP12c, rate (i) comes out stated as a percent
• ex. i - 0.027 means 0.027%
• Do not need to compute intermediate growth rates to get the CAGR, but it can be interesting information
• Rule of 72 - divide 72 by interest rate to get number of years to double (rough rule)
• Solving for number of periods
• This involves natural logs - think I will just leave this to HP12c functions to solve for n etc.
• Note: appears that the HP12c only currently solves to next full period - e.g. if something takes 10.24 years it shows as n=11 in the HP12c - need to fix this 
• Solving for size of payments
• Note - when calcing monthly mortgage pmts, house price is PV not FV

Cash flow additivity principle - cash flows in same periods can be added to each other in PV calcs, and likewise for an uneven annuity you can tease out the even portion and then take care of the rest

End of reading 5.

Time - 3:30 pm

2.5 hours