Time Value of Money

• Interest – Market price of money

• Supply – lending rate• Demand – borrow rate• Difference – margin for

lender

– Makes values at different points in time equivalent

• Holder indifferent between payment now and payment in future

$1.00 at compound Interest

0

200

400

600

800

1000

1200

0 9 18

27

36

45

54

63

72

Years

$'s

1% (70)

2% (35)

3% (24)

4% (18)

5% (15)

6% (12)

7% (10)

8% (9)

9% (9)

10% (8)

Compound Interest Formulas

• Classification of formulas– Time direction (time is

relative)• Forward (compound)

• Backward (discount)

– Frequency of payment• Single

• Annual

• Periodic

Producer Price Index

0

50

100

150

57 63 69 75 81 87 93 99

YearIn

de

x, 1

98

2 =

10

0

Nomenclature• Vn = value at time n• Vo = value at time zero• n = number of years• i = interest rate as a decimal• a = payment made at the end of a regular interval,

e.g. an annuity• t = number of interest periods between payments

when interest period and payment period differ• nt = total number of interest periods

Nomenclature

• Single payment – compounding. – Payment made at point 0

and compounded for n years

0 n1 2 3 4 n-2 n-1

•Single payment – discounting.

–Payment made at point n and discounted back to point 0

Final (future) value of a single payment (compounding)

V20 = $60 x (1.09)20

V20 = $60 x 5.60

V20 = $336.26

Tree is worth $60 now. If it increases in value by 9% annually, what’s its estimated value in 20 years?

Vn = V0 x (1+i)n

Present value of a single payment in the future (discounting)

If a tree is expected to be worth $120 in 10 years, and you want to earn 5% interest, what’s it worth to you now?

V0 = $120 x (1/(1.05)10)

V0 = $120 x 0.6139

V0 = $73.67

V0 = Vn x (1/(1+i)n)

Rate earned

A high quality tree is worth $320 today and is expected to be worth $600 in 10 years. If I want to earn 6% on my investment should I cut it now or in 10 years?

i = (600/320)1/10-1

i = 1.88 0.1 –1

i = 1.065 –1

i = 0.065 or 6.5%

6.5% > 6% so cut in 10 years, not today

(1+i)n = Vn/V0

i = (Vn/V0)1/n -1

Assumptions for multiple payment formulas

0 1 2 3 4 n-2 n-1

Annuity payments – annual payments of an equal amount

1st payment

2nd payment

. . . . . .

nth payment

n

Assumptions for multiple payment formulas

• Compounding– First annuity payment compounded for n periods. – Last annuity payment not compounded

• Discounting– First annuity payment discounted for 1 year– Last annuity payment discounted for n years

• Year zero payments must be handled separately

0 1 2 3 4 n-2 n-1 n

Example 1

You are deciding how heavy a cut to make in a mature 30 acre stand of hardwood timber. The owner needs current income, but also wants to leave sufficient growing stock to let the stand build up in value in case his heirs need income to pay death taxes. His life expectancy is 30 years. One option is to leave 1.8 MBF of good growing stock per acre and harvest the rest. The value of this growing stock is currently $350 per MBF. You expect it to increase in value at 5% rate for the next 15 years and at 8% thereafter. Your client wants an estimate of what the timber might be worth in 30 years.

Example 1V0 = 1.8 MBF x $350/MBF = $630

V15 = V0 * 1.0515

= $630 * 2.078928

= $1,309.73

V30 = V15 * 1.0815

= $1,309.73 * 3.172169

= $4,154.67

Example 2

Continuing with example 1, your client explains that his attorney indicate that at least $250,000 should be allowed for death taxes and $150,000 should come from timber. Your client asks approximately how much growing stock would have to be left now to accumulate $150,000 in timber value.

Example 2

From example 1 1.8 MBF would provide $4,154.67 per acre at end of 30 years.

30 A * $4,155/A = $124,640

Want $150,000, so how much more is needed at year 30?

$150,000 - $124,640 = $25,360

Example 2

On a per acre basis this is,$25,360/30 = $845.33/A

Now we need to go back to year “0”

V15 = V30/(1.08)15 = $845.33/3.172169 = $266.483

V0 = V15/(1.05)15

= $266.483/2.078928 = $128

Example 2

At $350 per MBF the additional growing stock that would need to be left is,

$128/$350 = 0.366

Thus, the total growing stock needed at year 0 is,

1.8 + 0.366 = 2.2 MBF

Example 3

Your client understands why you used two compounding periods and two different interest rates, but he wants an average rate of value increase for the timber to give to his financial advisor.

Example 3

For the established goal of $150,000 at year 30 and starting with 2.2 MBF per acre wroth $350 per MBF, or $23,100 for the 30 acres, the compound rate of increase is,

i = (V30/V0)1/30 -1.0

= ($150,000/$23,100)0.0333 – 1.0

= 1.06435 – 1.0

= 0.06435 or 6.435%

Compound Annual Payment Multiplier

Vn = a * [(1 + i)n -1]/i

Assumes “a” is paid at end of each payment period, and lengths of compounding and payment periods are the same.

Example 4

Your client tells you that on average it costs him $0.60 per acre to cover the property tax, insurance, and other annual carrying costs for the woodlot. He wants to know by how much these costs reduce the value of the woodland over the next 30 years assuming the average interest rate on long-term bonds has been 8% over the last 5 years.

Example 4

You solve this problem by first determining the final value of this annuity,

V30 = $0.60 * [(1.08)30 -1]/0.08= $0.60 * 113.28= $67.97 per acre,

and then discounting this back to year 0

V0 = $67.97/(1.08)30

= $67.97/10.06= $6.75 per acre

Sinking Fund Multiplier

a = Vn * [i/((1+i)n -1)]

This is compound annual payment multiplier solved for “a.” Payments are made at the end of the payment periods and payment and compounding periods are the same.

Example 5

Your client has other alternatives for meeting his goal of having $150,000 in disposable assets at the end of 30 years. One is to make annual payments into a money market fund. If he is in the 40% tax bracket, how much would he have to pay into a fund paying a taxable return of 8% on average?

Example 5

The after-tax return is 60% of 8%, or 4.8%, and the annual payment required to net $150,000 after taxes in 30 years would be,

a = $150,000 * [0.048/((1.048)30-1)]

= $150,000 * 0.048/(4.0817 – 1)

= $150,000 * 0.156

= $2,336.37

Discounted Annual Payment Multiplier

V0 = a * [(1+i)n – 1]/[i * (1+i)n]

Assumes “a” is paid at end of each payment period, and lengths of compounding and payment periods are the same.

Example 6

By how much do annual carrying costs of $0.60 per acre reduce the value of a woodlot, i.e. what is the present value of annual payments of $0.60 per acre for 30 years?

V0 = $0.60 * [(1+i)30 – 1] / [I * (1+i)30]

= $0.60 * 9.0627 / (0.08 * 10.0627)

= $0.60 * 11.2578

= $6.75 per acre

Capital Recovery Multiplier- Installment Payment

a = V0 * [i * (1+i)n] / [(1+i)n -1]

Example 7

Your client is considering amortizing the original purchase price of the forest land over a 10-year period for purposes of his financial records. Therefore, he wants to know the annual payment required to payoff a purchase price of $1,340 per acre.

Example 7

a = $1,340 * [0.08 * (1.08)10]/[(1.08)10 – 1]

= $1,340 * (0.08 * 2.1589)/(2.1589 – 1)

= $1,340 * 0.1490

= $199.70 per acre

Example 7If the amortization charges were to be made monthly, the monthly charge would be,

a = $1,340 * [0.0067 * 1.0067120]/[1.0067120 -1]

= $1,340 * 0.0149/1.2285

= $1,340 * 0.0122

= $16.35

Example 7

It was possible to determine the monthly payments using the same formulas as for annual payments because the payment period and compounding period were the same. The annual interest rate was converted to a monthly rate by dividing by 12, and the number of payment periods was determined by multiplying the number of years by 12.

Compounded Periodic Payment Multiplier

Vn = a * [(1+i)nt – 1]/[(1+i)t – 1]

where,

n = number of payments to be made

t = number of interest periods

between payments

Example 8

Your credit union account compounds interest daily and pays interest of 9.5%. If you have $100 withheld from your paycheck each month, what would be the approximate balance in your account after 10 years?

i = 0.095/365 = 0.000260

n = 10 years * 12 months = 120

t = 365/12 = 30.4167

Example 8

Vn = $100 * [1.00026120*30.4167 – 1]/

[1.0002630.4167 – 1]

= $100 * [1.000263650-1]/[1.0002630.4167 -1]

= $100 * [2.5828 – 1] / [1.007939 – 1]

= $100 * 199.3799 = $19,9379

Example 9

Your client has acquired two other woodlots from which he will receive timber income of $1,000 every five years. He plans to place the funds in a money market account which compounds monthly and pays 9.5% annually. What will the value of the account be after 30 years?

Example 9

i = 0.095/12 = 0.007917

n = 30/5 = 6

t = 5 * 12 = 60

V30 = $1,000 *[1.0079176*60 –1]/[1.00791760 -1]

= $1,000 * 26.504685

= $26,504.68

Discounted Periodic Payment Multiplier

Vn = a * [(1+i)nt – 1]/[(1+i)t – 1](1+i)nt

where,

n = number of payments to be made

t = number of interest periods

between payments

Example 10

As part of your financial plan you need to estimate how much life insurance is necessary. As the sole provider of your family your coverage would ideally be sufficient to replace your earnings for the expected life of your spouse. Assuming your life expectancy is 35 years and your monthly salary is $2,800, how much life insurance would be needed to yield 10 percent interest compounded annally and there is no reason for your spouse not to use up the fund during his or her life.

Example 10

1.1420*0.0833 -1

V0 = $2,800 (1.10.0833-1) * 1.1420*0.0833

1.134.99 - 1 = $2,800

(1.10.0833 – 1) * 1.134.99

= $337,542

Example 10

Example 10

If insured died their spouse would receive $337,542 in insurance proceeds and buy an annuity contract paying 10% interest. The annuity would pay out $2,800 per month for 35 years. At the end of 35 years the balance in the account would be zero.

Capital Value of an Annuity

V0 = a/i

This is the present value of an annuity with n approaching infinity.

V0 = a * [(1+i)n – 1]/[i * (1+i)n]

It’s the amount needed to pay out the annuity “a” forever.

Example 10

Suppose you were not willing to risk being wrong about how long your spouse will live. How much life insurance would be needed to guarantee your spouse an annual income of $33,600 (12 x $2,800) forever if the proceeds from the insurance can be expected to earn 10%?

Example 10

V0 = $33,600 / 0.10 = $336,000

If payments were to be made and interest earned monthly the amount needed would be,

V0 = $2,800 / (0.1/12) = $336,000

Discounted Periodic Payment Multiplier

V0 = a * 1 / [(1+i)t – 1]

This is same as capital value of an annuity except the payment occurs every t years forever, instead of every year for ever. Compounding period is one year.