Money-Time Relationships

Part II – Tricks and Techniques

Overview

• Important assumptions and how to break them

• Deferred annuities– Slow (but simple) use only P/F

– Faster - Combining P/F and P/A

• Linear Gradients– P/G, A/G, etc.

– Sometimes P/F is easier

• Exponential growth (or decay)

• Compounding intervals more often than cash flows

• Variable interest rates require F/P or P/F

Important Assumptions

Factor Assumption Violations

All (P/F,F/P,P/A,etc.) Constant interest rate for N periods

Variable interest rates

(5% for 3 years, then 6% for 2 years, then 4% for 6 years)

(P/A,i%,N)

(F/A,i%,N)

Constant cash flows Cash Flows that grow or shrink

(P/A,i%,N) A/P

(F/A,i%,N) A/F

Cash flow interval matches interest compounding interval

Interest compounded more often (monthly) than cash flows (quarterly/yearly)

(P/A,i%,N) A/P

(F/A,i%,N) A/F

Cash flows start at end of year 1 and finish at end of year N

Cash flow starts at a future year (year 10)

Dealing with Violations IProblem Solution TechniquesChanges in i% Find breakdown of problem where i

% is piecewise-constant. Use P/F or F/P only. Avoid using P/A, etc.

Variable Cash Flows

($A grows or shrinks)

If $A is piecewise-constant, with changes in the level, use deferred annuity technique.

If $A changes every period, shortcuts exist only for special cases (linear, exponential).

For the general case, you can always use P/F or F/P and sum up the results.

Dealing with Violations II

Problem Solution Techniques

Interest compounded more often (monthly) than cash flows (quarterly/yearly)

Use time periods that match the cash flows.

Adjust the monthly interest rate I up to the new period.

I* = (1+I)4-1 I*=(1+I)12-1

First Payment of Annual Cash flow starts at a future year, not year 1 (example: need PV of cash flow that starts at year 10 with last payment in year 20)

Use deferred annuity formula.

Apply appropriate combinations of P/F and P/A.

example: (P/F,i%,9)*(P/A,i%,10)

Deferred annuitiesCash flow starts at beginning of year J+1Last payment at end of year N

0 J J+1 N

$A/year

Present Value $P = (P/F,i%,J)*(P/A,i%,N-J)*A

$P ?

Why not just P/A?

0 J J+1 N

$P ?$A/year

(P/A,i%,N-J)*$A gives the value of the cash flow in units of “year J dollars”. This is probably not what you wanted as your final result.

Remember: P/A gives a dollar value that is timed one year before the start of the cash flows. You need to use P/F or F/P to move this value to other years!

That’s why $P = (P/F,i%,J)(P/A,i%,N-J)*$A

Variable cash flows

• For general cases, use P/F or F/P• For linear cases (e.g. –3000,-1000,1000,3000,5000), there is a gradient

method involving gradient factors P/G, F/G, etc.• For exponential cases, (e.g. 1000,1100,1210,…)

there is a convenience interest rate method• Use of excel together with P/F or F/P is often the

best solution technique. If you are confused, then use P/F or F/P together with a table. This keeps the analysis simple and easy to follow.

Exponential Decay ExampleA watch manufacturer expects a revenue of

$100,000 for the first month. The revenue declines by 10% each month and ends after the 12th month.

Calculate the Present Value given i=1%/month.

0 1 2 3 4 5 6 7 8 9 10 11 12

$P?

$100000$59049

$31,381

Exponential Growth/Decay Slow but simple (Excel + P/F)

Time (months) Cash Flow P/F factor Contribution to PV1 $100,000 0.9901 $99,0102 $90,000 0.9803 $88,2273 $81,000 0.97059 $78,6184 $72,900 0.96098 $70,0555 $65,610 0.95147 $62,4266 $59,049 0.94205 $55,6277 $53,144 0.93272 $49,5688 $47,830 0.92348 $44,1709 $43,047 0.91434 $39,359

10 $38,742 0.90529 $35,07311 $34,868 0.89632 $31,25312 $31,381 0.88745 $27,849

PV= $681,235(sum)

Exponential Growth/Decay Convenience Interest Rate Method

Another way to calculate annuities with a growth (or decay) factor is to adjust the interest rate factor and use a special formula.

Common ratio =f = (Ak- Ak-1)/ Ak-1 = -0.10

“Convenience rate” icr=[(1+i)/(1+f)]-1

=[1.01/0.90]-1=0.1222

Special formula PV = A1(P/A, icr%,N)/(1+f) =

= ($100000)(6.132)/(0.90)=$681,333

Exponential Growth/Decay Convenience Interest Rate Method

Common ratio =f = (Ak- Ak-1)/ Ak-1 = -0.10

“Convenience rate” icr=[(1+i)/(1+f)]-1=[1.01/0.90]-1=0.1222

PV = A1(P/A, icr%,N)/(1+f) == ($100000)(6.132)/(0.90)=$681,333 Compare with more careful excel+P/F method:

$681,235 (difference is due to rounding 6.132)

Linear Gradients

Cash flow increases or decreases _linearly_

(by the same _amount_ each period)

1 2

3 4 5

-3000-1000

10003000 5000 7000

6

Investment loses 3000 in year 1, 1000 in year 2, but earns 1000,3000,5000,7000 in years 3-6. What is the PV at i=8%?

Gradient Factors

0 1 2 3 4 5 …………. N

$0 $0

$1$2

$3$4

$(N-1)

(P/G,i%,N) is the present value (units: year 0$) of this cash flow(F/G,i%,N) is the future value (units: year N$) of this cash flow(A/G,i%,N) is the annuity value (units: $/year) of this cash flow

Year

Linear Gradients

1 2

3 4 5

-3000-1000

10003000 5000 7000

6

=

+

-3000

2000 4000 6000 8000 10000

annuity

Simple gradient

Linear Gradients

1 2

3 4 5

-3000-1000

10003000 5000 7000

6

Cash flow = annuity (-3000) + gradient (2000/year)

Gradient always starts at year 2.

PV = -3000 (P/A,8%,6) + 2000(P/G,8%,6)

(p.632) = (-3000*4.6229) + 2000 (10.523)

=-$13869 + $21046 = $7177

60 Seconds Investment ChallengeLet i=2%/month. There are two investments, A and B.

An investment costs $300 in terms of today’s dollars.

0 1 2 3 4….N….. 24 months

$0 $1 $2 $3 ….$N… $36 (end)

INVESTMENT A

INVESTMENT B

0 1 2 3 ….N….. 36 months

$21 $21 $21 $21 $21$0

Do you want to swap $300 for the PV of A, or the PV of B?

$21 (end)

Investment Challenge: Analysis of A

Let i=2%/month. The investment costs $300 in terms of today’s dollars.

$0 $1 $2 $3 ….$N… $36 (end)

INVESTMENT A

0 1 2 3 ….N….. 36 months

PV = $1 * (P/G,2%,36) + $1 * (P/A,2%,36)

= $392.04 + $25.49 = $417.53

Did you forget that cash flow for P/G begins in year 2? In this example, this mistake cost you money!

Investment Challenge:Analysis of B

Let i=2%/month. The investment costs $300 in terms of today’s dollars.

0 1 2 3 4….N….. 24 months

INVESTMENT B

$21 $21 $21 $21 $21$0

Evaluating B is simple, because it has a constant cash flow of $21. It starts in year 1, so we can use the P/A formula.

$PV = $21 * (P/A,24,2%) = $21 * 18.9139 = $397.20

Compounding intervals more often than cash flows

Solution: Change interest rate period to match cash flow period

Example: Interest compounded every month at 1%/month, but cash flows are every six months. Use i*=[(1+i)^N]-1=1.01^6-1

=1.06152-1

=6.152%/6-month period

Savings account example: Problem

Every month your bank pays 0.25% interest on your savings account balance.

Every 3 months you deposit $5000. How much do you have after 3 years?

Note: this is a F/A problem, except that the compounding interval of 1 month does not match the deposit (cash flow) interval of 3 months.

Savings account example: Analysis

Step 1: Change interest rate to 3-month rate:i*=[(1+.0025)3-1]=.0075187Step 2: Determine N. If a period is 3-months, then we

have N=12 periods in 3 years.Step 3: Notice that the cash flow every period is

constant, so we can use the FV formula, $FV = $5000 * (F/A,i*,12)Final Step: Calculate the F/A formula.$FV =$5000 * [(1.0075187)12-1]/(0.0075187)=$5000*12.50888=$62544.42

Variable interest rates and F/P (or P/F)

Rule: Use a separate F/P (or P/F) for each group of years or periods where the interest rate is constant.

Example: You have $5000 today. For 3 years you invest it at 4% per year, then for 5 more years at 5% per year, then 2 more for 3% per year. The future value is

$FV = $5000 * (F/P,4%,3) * (F/P,5%,5) * (F/P,3%,2) = $5000 * 1.1249 * 1.2763 * 1.0609

Summary• We learned some tricks for finding present and

future values in special situations.• The “trick that always works” is to make a table of

all cash flows in excel, apply appropriate P/F or F/P factors, and add up the result.

• To apply shortcuts for linear or geometric cases, one must pay careful attention to detail.

• Next week – Chapter 4– more applications– Return on investment (finding the i%)– comparing machines, investment plans, etc.