Time Value of Money Lecture Notes

July 9, 2014

Opportunity Cost

Money paid / received at some point in the future doesn’t have the same value as an equivalent amount paid / received today;

Why? Opportunity cost Opportunity Cost = The cost of an alternative that must be

forgone in order to pursue a certain action. In other words, it is the benefits you could have received by taking an alternative action;

In PFP, Opportunity Cost is reflected through a “Rate of Return” or “Discount Rate”

So, Opportunity Cost = the benefits you could have received (i.e. rate of return you could have earned) had you taken an alternative action;

Time Value of Money

Time Value of Money – involves restating monetary amounts (using the “rate of return” or “discount rate”) to “present” or “future” values in order to provide a common basis for comparison / measurement;

Time Value of Money is the underlying basis of measurement in Personal Financial Planning;

Time Value of Money

Opportunity cost is reflected in time value calculations as the Rate of Return or Discount Rate;

Discount rate = rate of return used to compare / equate amounts paid or received at different points in time.

Discount rate is the rate that leaves an individual indifferent between present and future amounts;

Choice of a discount rate is personal, based on best available alternative;

In order to select a discount rate, must have an understanding of what is meant by “rate of return.” How is it determined?

Understanding Rates of Return

Rates of Return can be presented in various ways, including: Holding Period Returns; Average Rate of Returns; Compound Rates of Return; Annual Percentage Rate (APR); Effective Annual Rate (EAR).

In PFP, need to understand the underlying rates of return being used for comparison / measurement purposes;

Holding Period Returns

• Holding Period Returns are the simplest measure of rate of return - reflect the change in value of an asset over the period held and any income received while holding;

• Calculate Holding Period Returns for a Single Period with cash flows at the beginning and end of the period :

r = [(P1 - P0) + D] / P0 where P = Purchase / Selling Price

D = Dividend or Interest 0 = Beginning of the Period (Purchase) 1 = End of the Period (Sale)

e.g. [($110 - $ 100) + $4 div.] / $100 = 14%

Holding Period Returns

• Difficulty with Holding Period Returns – does not account for the length of the holding period, multiple periods, or allow valid comparisons when there are different holding periods;

Annual Rates of Return Average / Arithmetic Rates of Return

Common practice involves expressing rates of return as annual measures.

Average (Arithmetic) Rates Returns are often used when considering Multiple Periods;

Difficulty with average (arithmetic) rates of returns: Calculation of averages is subject to “extreme” values in

the data; Average returns do not account for compounding effect

over multiple years;

Annual Rates of Return Compound/Geometric Rates of Return

An alternative to using Average Rates of Returns for Multiple Periods is to use Compound (Geometric) Rates of Return;

Geometric Mean = ((1+k)*(1+k)*(1+k)*….) y 1/n -1 where ‘n’ = the number of periods and ‘k’ = the periodic rate of return

Compound / Geometric rate – accounts for the compounding effect of “interest on interest” when comparing returns over multiple years;

Rates of Return

Ch. 2 - Problem # 15

Rates of Return Problem 2-15

(a) Calculate rates of return for single periods (Holding Period Returns = [(P1 - P0) + D] / P0

Stock A Time 0 1 2 3 4 5 Dividend 0 0 0 0 0 0

12-11 10-12 13-10 15-13 16-15 ` 11 12 10 13 15

Rate of Return .091 -.167 .30 .154 .067

Rates of Return Problem 2-15

(a) Calculate rates of return for single periods (Holding Period Returns = [(P1 - P0) + D] / P0

Stock B Time 0 1 2 3 4 5 Dividend 0 1 1 1 1.15 1.15

15+1-18 16+1-15 14+1-16 13+1.15-14 18+1.15-13 ` 18 15 16 14 13

Rate of Return -.111 .133 -.063 .011 .473

Rates of Return Problem 2-15

(b) Arithmetic Mean / Average Return for Stock A = (.091 - .167 + .30 + .154 + .067) / 5 = .089 or 8.9%

Note: Negative values are simply “subtracted” in the calculation. Arithmetic Mean / Average Return for Stock B = (-.111+ .133 - .063 + .011 + .473) / 5 = .0887 or 8.87%

Rates of Return Problem 2-15

(c) Geometric Mean (Compound Return) for Stock A = [(1 + .091)(1 - .167)(1 + .30)(1 + .154)(1 + .067)]1/5 - 1 = .0778 or 7.78%

Note: Negative values are “subtracted” from 1.0 to determine the compounding factor

Geometric Mean (Compound Return) for Stock B = [(1 - .111)(1 + .133)(1 -.063)(1 + .011)(1 + .473)]1/5 -1 = .0705 or 7.05%

Rates of Return Within a Period

Annual Percentage Rate (APR) - conventional method of reporting rates which ignores compounding;

Effective Annual Rate (EAR) accounts for the compounding

effect within a given year: Calc. EAR = (1+ kj /m )m – 1 where ‘m’ = the number of periods ‘kj’ = the annual rate of return

Rates of Return Within a Period Problem - Sheila Gilbert

Sheila Gilbert wants to invest $50,000 in a bank account. The following accounts are available at the local bank:

Stated interest rate Frequency of (APR) Compounding

1 10.0 % Annual 2 9.75 % Semi-Annual 3 9.50 % Quarterly 4 9.25 % Monthly 5 9.00 % Weekly

Calculate the effective annual rate (EAR) for each type of account.

Rates of Return Within a Period Problem - Sheila Gilbert

(a) EAR = (1+ kj /m )m – 1 where ‘m’ = the number of periods ‘kj’ = the annual rate of return

Rates of Return Within a Period Problem - Sheila Gilbert

(a) EAR = (1+ kj /m )m – 1 where ‘m’ = the number of periods ‘kj’ = the annual rate of return

Stated rate (APR) EAR 1 10.00 10.000 2 9.75 9.988 3 9.50 9.843 4 9.25 9.652 5 9.00 9.409

Discount Rates

Choice of a discount rate is personal, based on best available alternative;

Various factors that influence discount rates: time premium, risk, income tax and inflation;

In PFP, must often account for the effects of inflation in order to plan for a level of consumption rather thana fixed amount of $;

Therefore, in many situations we must discount future amounts back to today’s (constant) dollar;

For present / future value calculations involving real / constant / inflation-adjusted cash flows, use Real Rates of Return.

Real Rate of Return

Use real (inflation adjusted) discount rates for real cash-flows;

Real rate of return: Kr = [(1+Knom) / (1+i)] –1

where ‘i’ = inflation rate

Present and Future Values

Time Value of Money = involves restating monetary amounts (using the “rate of return” or “discount rate”) to present or future values in order to provide a common basis for comparison / measurement;

Restate amounts to present or future values to provide a common basis for comparison / measurement;

Present Values and Future Values can be calculated on a lump sum amount or periodic payments (annuities);

Present / future value calculations involving multiple periods always incorporate the effect of compound interest.

Present and Future Values

Future Value Calculations:

Lump Sum FV = PV*(1+k)t where ‘t’ = the number of periods compounded

‘k’ = the discount rate

e.g. FV $100 invested @ 5% for three years = $100 *1.05 *1.05*1.05 = $115.76 or

$100 *(1.05)3 = $115.76 or

PV = 100, N=3, I/Y=5 Calc FV = $115.76

Annuity (Reg.) FVA = ((1+k)n –1)/k * Pymt where ‘n’ = the number of periods ‘k’ = the discount rate

Present and Future Values

e.g. FV $100/yr invested @ 5% for three years = = $100 *1.05 *1.05* = 110.25 + $100 *1.05 = 105.00 + $100 = (110.25+105.00+100.00) = $315.25

or ((1+0.05)3-1)/0.05 *$100 = 3.1525 * 100 = $315.25

or PYMT = 100, N=3, I/Y=5 Calc FV = $315.25

Annuity Due FVA = ((1+k)n –1)/k) * (1+k) * Pymt where ‘n’ = the number of periods

‘k’ = the discount rate

Present and Future Values

Present Value Calculations:

Lump Sum PV = FV/(1+k)t where ‘t’ = the number of periods compounded and ‘k’ = the discount rate

Annuity PVA = [1-(1/(1+k)n]/k * Pymt where ‘n’ = the number of periods and ‘k’ = the discount rate

Financial calculators or spreadsheet applications will perform all of these computations.

Present and Future Values

Observation:

As discount rate increases, the present value of future cash flow decreases.

Because of the higher rate of return, an individual requires less $$ today to be indifferent between the present and future amounts;

Constant Growth Annuity

Where periodic payments are not equal, but expected to grow at a constant rate (pgs 31-33);

Future Value : Annuity FVCGA = {[(1+k)n - (1+g)n ])/(k-g)} * Pymt

where ‘n’ = the number of periods, ‘k’ = the discount rate, and ‘g’ = the growth rate

Present Value : Annuity PVCGA = { [1- ((1+g) / (1+k))n ] /(k-g)} * Pymt

where ‘n’ = the number of periods, ‘k’ = the discount rate, and ‘g’ = the growth rate

CGA calculations are often relevant in retirement planning;

Time Value of Money Present and Future Values

Ch # 2 Problem - Sheila Gilbert, Part 2 Problem # 2-3 Problem # 2-4

Problem # 2-14 Problem # 2-11 Problem – Kay MacDonald Problem # 3-6

Time Value of Money Problem - Sheila Gilbert (Part # 2)

Sheila Gilbert wants to invest $50,000 in a bank account. The following accounts are available at the local bank:

Stated interest rate Frequency of (APR) Compounding

1 10.0 % Annual 2 9.75 % Semi-Annual 3 9.50 % Quarterly 4 9.25 % Monthly 5 9.00 % Weekly

Calculate the effective annual rate (EAR) for each type of account. Calculate the compound value of her investments at the end of one year, three years; and five years.

Time Value of Money Problem - Sheila Gilbert (Part # 2)

(a) EAR = (1+ kj /m )m – 1 where ‘m’ = the number of periods ‘kj’ = the annual rate of return

Stated rate (APR) EAR 1 10.00 10.000 2 9.75 9.988 3 9.50 9.843 4 9.25 9.652 5 9.00 9.409

Time Value of Money Problem - Sheila Gilbert (Part # 2)

1 yr 1 $55,000 (50,000 * 1.10) 2 $54,994 (50,000 * 1.09988) 3 $54,922 (50,000 * 1.09843) 4 $54,826 (50,000 * 1.09652) 5 $54,704 (50,000 * 1.09409)

Time Value of Money Problem - Sheila Gilbert (Part # 2)

3 yr 1 $66,550 PV 50,000 I/Y= 10.00 N=3 2 $66,528 PV 50,000 I/Y= 9.988 N=3 3 $66,265 PV 50,000 I/Y= 9.843 N=3 4 $65,920 PV 50,000 I/Y= 9.652 N=3 5 $65,483 PV 50,000 I/Y= 9.409 N=3

Time Value of Money Problem - Sheila Gilbert (Part # 2)

5 yr 1 $80,526 PV 50,000 I/Y= 10.00 N=5 2 $80,482 PV 50,000 I/Y= 9.988 N=5 3 $79,952 PV 50,000 I/Y= 9.843 N=5 4 $79,260 PV 50,000 I/Y= 9.652 N=5 5 $78,385 PV 50,000 I/Y= 9.409 N=5

Time Value of Money

Problem # 2-3 – “Income for Life”

Time Value of Money Problem # 2-3

Present Value of Lump Sum Payment = $500,000

Present Value of the “Income for Life Annuity = PMT – $20,000

I/Y 4 N 40 years CPT PV = $395,855.48

It is better to receive the $500,000 now.

Time Value of Money

Problem # 2-4

Time Value of Money Problem # 2-4

The determination of the future value of the savings account balance combines the result of two FV calculations:

Future Value following the initial 3 year period: PV $ 20,000 N 12 periods (3 years, quarterly) I/Y 1.5 / period (6% p.a. / 4) FV $23,912.36

Time Value of Money Problem # 2-4

The future value at the end of the initial investment period serves as the PV at the outset of the second period:

Future Value at the end of six years: PV $ 23,912.36 N 36 periods (3 years, monthly) I/Y 0.75 / period (9% p.a. / 12) FV $31,292.80

Time Value of Money

Problem # 2-14

Time Value of Money Problem # 2-14

Assume that the intermediate cash flows are reinvested at 10%. (a) (i) (PV = $9,500, N = 5, I/Y = 10) FV = $15,300

Time Value of Money Problem # 2-14 (a) (ii) Interest Pymts = FVA (PMT=375, N=10, I/Y=5) = 4,717 Principal Pymt = = 10,000 FV = $14,717

Time Value of Money Problem # 2-14

(a) (iii) FV = [(PV=200, I/Y=10, N=4) + (PV=400, I/Y=10, N=3) + (PV=800, I/Y=10, N=2) + (PV=1,600, I/Y=10, N=1) + (PV=2,500+9,500 = 12,000, I/Y=10, N=0)] FV = 292.82 + 532.40 + 968 + 1,760 + 12,000 = $15,553 Based on the future values shown above, choose (iii).

Time Value of Money Problem # 2-14

b) The risk of the investment should be considered. (iii) has the highest risk and (i) has the lowest.

Other factors might include differential taxes on the different investments, and marketability of the investments.

Time Value of Money

Problem # 2-11

Time Value of Money Problem # 2-11

Since the perpetuity provides an annual payment forever, the value of the prize, when invested at 8%, must be sufficient to pay-out $10,000 /year: = PMT / I/Y = $10,000 / 0.08 = $ 125,000

Time Value of Money Problem # 2-11

To account for the impact of inflation, the real discount rate should be substituted for the nominal rate :

Real rate of return: Kr = [(1+Knom) / (1+i)] –1

where ‘i’ = inflation rate

Time Value of Money Problem # 2-11

Real rate of interest = [(1+.08)/(1+.03)] – 1 = .048544

The value of the “constant dollar” perpetuity: = PMT / I/YReal

= $10,000 / 0. 048544 = $ 205,998.68

Time Value of Money Problem – Kay MacDonald

Kay MacDonald is 25. She plans to retire at age 55 and she desires a before-tax income of $50,000 “real” dollars per year in retirement. She expects to live until she is 85 at which time she wants to leave a $500,000 estate to her heirs.

She can invest money in the stock market, with a long-term rate of return of 12%, or in debt securities with an expected return of 7.5%. She expects inflation to average 5% over the complete period.

How much money will Kay need at age 55 to fund her retirement goal? .

How much money will she have to save each year between now and age 55 to have this amount?

Time Value of Money Problem – Kay MacDonald

Stock Market: By investing in the stock market, Kay would earn a real rate of

return = (1+.12)/(1+.05) – 1 = 6.67% Using this real rate of return, Kay would require the following

amount at retirement:

i) PV of her annual retirement income: PMT 50,000 I/Y 6.67 N 30 (From Age 55 to 85) PV 641,587

Time Value of Money Problem – Kay MacDonald

ii) PV of her estate: FV 500,000 I/Y 6.67 N 30 PV 72,061 Kay will need $ 713,648 at retirement ($641,587 + 72,061).

Time Value of Money Problem – Kay MacDonald

Debt Securities: By investing in the debt securities, Kay would earn a real

rate of return = (1+.075)/(1+.05) – 1 = 2.38% Using this real rate of return, Kay would require the following

amount at retirement:

i) PV of her annual retirement income: PMT 50,000 I/Y 2.38 N 30 PV 1,063,457

Time Value of Money Problem – Kay MacDonald

ii) PV of her estate: FV 500,000 I/Y 2.38 N 30 PV 246,897 Kay will need $ 1,309,914 at retirement (1,063,457+ 246,897)

Time Value of Money Problem – Kay MacDonald

How much money will she have to save each year between now and age 55 to have these amounts?

Time Value of Money Problem – Kay MacDonald

Stock Market: The present value of Kay’s savings requirements at age 55

becomes the future value for Kay’s savings plan at age 25: Annual PMT required to meet her savings goal: FV - 713,648 I/Y 6.67 N 30 (From age 25 – 55) PV 0 PMT $8,015.47

Time Value of Money Problem – Kay MacDonald

Debt Securities: The present value of Kay’s savings requirements at age 55

becomes the future value for Kay’s savings plan at age 25: Annual PMT required to meet her savings goal: FV -1,309,914 I/Y 2.38 N 30 (From age 25 – 55) PV 0 PMT $ 30,411.59

Time Value of Money

Problem # 3-6 (Jonathan Letters) – Parts (a) and (b)

Time Value of Money Problem # 3-6

(a) EAR = (1+ kj /m )m – 1 where ‘m’ = the number of periods ‘kj’ = the annual rate of return

Jonathan’s mutual fund offers a 10% APR, with monthly compounding:

EAR = (1+ .10 /12 )12 – 1 = 10.47%

Time Value of Money Problem # 3-6

(b) Although the stream of future payments varies over time according to Jonathan’s income, growth occurs at a constant rate (0.5% /mo).

Therefore the FV of monthly contributions can be calculated as a constant growth annuity:

FVCGA = {[(1+k)n - (1+g)n ])/(k-g)} * Pymt where ‘n’ = the number of periods, ‘k’ = the discount rate, and ‘g’ = the growth rate

Time Value of Money Problem # 3-6

(b) Although the stream of future payments varies over time according to Jonathan’s income, growth occurs at a constant rate (0.5% /mo).

Therefore the FV of monthly contributions can be calculated as a constant growth annuity:

FVCGA = {[(1+k)n - (1+g)n ])/(k-g)} * Pymt where ‘n’ = the number of periods, ‘k’ = the discount rate, and ‘g’ = the growth rate

= {[(1+ 0.00833)120 - (1+ 0.005)120 ])/(0.00833-0.005)} * (3600/12) = {[ (2.7070 – 1.8194) ]) / (0.0033)} * 300 = 268.97 * $ 300 = $ 80,691

Time Value of Money Problem # 3-6

(b) In addition, Jonathan will have the FV of his original inheritance invested in the mutual fund:

PV 10,000 I/Y 10.47 N 10 FV 27,067

Total accumulated after 10 years = 80,691 + 27,067 = 107,758

Time Value of Money

Questions / Discussion