compound functions - annuities

The most common valuation problem is the valuation of future periodic income from investment properties, usually rent. In part 10 the present value of rental income for the period 0 to perpetuity was found with the capitalization in perpetuity formula:

MV = NAI * 100/CR

Where:

MV = market value

NAI = net annual income

CR. = capitalization rate as a %.

In this part we are concerned with the future and present values of periodic level payments (that is, the same amount each period) at periods less than perpetuity. Periodic means that the payments are made at regular periods for example; per annum.

The formulae covered in this part cannot be used for expected incomes that are uneven in time or amount. For complex cash flows discounted cash flow is used.

YEARS PURCHASE (YP)

The factor; 100/CR in the above capitalization formula is known as the years purchase (YP) and is the maximum factor or multiplier by which rent can be multiplied to obtain market value.

EXAMPLE

A capitalization rate of 8% pa has a years purchase of 100/8 =12.5.

Years purchase is commonly used in business valuations and can be equated with the break even point or payback period used in feasibility studies.

For valuation purposes the capitalization rate expressed as a percentage is the most commonly used measure of return on investment. However, it cannot be compared directly with opportunity cost investments such as bank interest rates and the return on government bonds, as it does not take into account an important part of real estate investment return; capital gains.

FUTURE VALUE OF 1 PER ANNUM (FVPMT)

For example, to determine the future value of 1 pa over 20 years at 12% pa:

STEP 1:

Determine the base: b = 1+i = 1.12

STEP 2:

Determine the future value factor (FV):

FV = bn = 1.1220 = 9.646

STEP 3:

The future value of 1 per annum (FVPMT) can be found with the following formula:

FVPMT = (FV 1)/(i/l00)

Where:

FVPMT = future value of 1 pa factor

FV = future value of 1 = interest rate as a % eg 5% is 5

Therefore, the FVPMT for the expected cash flow above:

FVPMT = (9.646 1)/ 0.12 = 72.05

STEP 4:

Multiply the FVPMT factor by the periodic income for example, \$5 000 pa:

FVPMT(5000) = 72.05 * \$5 000 = \$360 250

The FVPMT factor shows that 1 per annum will accumulate to 72.052 if invested over 20 years at 12%pa. This is shown on the following time scale:

1 pa @ 12% = 72. 052

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

years

If the net rental income for the above property is \$200 000 per annum the future amount is:

FVPMT(200 000)= 72.052 * 200 000 = 14 410 500

say \$14 400 000

PRESENT VALUE OF 1 PER ANNUM (PVPMT)

The most useful compound formula in valuation practice is the present value of a future periodic level cash flow. For example, the present value of rents due under a lease agreement.

EXAMPLE

The expected net rental income from a 10 year lease is expected to be

\$10 000pa.

Determine the present value of \$10 000 pa for 10 years at 9 % pa.

STEP 1:

Determine the base: b= 1+i = 1+0.09 = 1.09

STEP 2:

Determine FV: FV = 1.0910 = 2.367

STEP 3

Determine PV: PV = 1/FV = 1/2.367 = 0.4224

STEP 4:

PVPMT is found with the following formula: PVPMT = (1 - PV)/i

Therefore, for the above example:

= (1 - 0.4224)/0.09 = 6.418

STEP 5:

Multiply PVPMT by the annual income:

PVPMT(10000) = 6.418 * 10000 = \$64180 say \$64 200

Therefore, the present value of a future rental stream of \$10 000 over 10 years at 9% pa = \$64 200.

REPLACEMENT OF CAPITAL IN REAL ESTATE INVESTMENTS

For a normal investment rate of return it is assumed that the capital invested is being replaced at the same rate of interest as that for the return on investment:

LEASEBACK EXAMPLE

A new factory premises is built for a company and then leased back to that company for the life of the building.

Life of building: 20 years

Required return on investment: 12%pa

Cost to build: \$100 000

The PVPMT factor is:

base = 1+i = 1.12

FV = 1.1220 = 9.646

PV = 1/FV = 1/9.646 = 0.1037

PVPMT = (1-PV)/i = 7.469

The required annual payment (rent under the leaseback finance agreement) is:

100 000/7.469 = \$13 389 pa

The required return is 12% and the money invested by the financier in the leaseback is \$100 000. Therefore, the required annual return on investment should be \$100 000 * 12% = \$12 000.

However, using the compound formula, the actual return on investment is \$13 389 that is, an extra \$1389 pa.

Why the extra amount?

Before we answer, which of the following investments would you prefer, assuming that the risks applicable to both are the same:

• the leaseback agreement as outlined above; or

• investing in a bank account that returns 12% per annum?

The bank account!

At the end of 20 years the investor will have his/her original capital of \$100 000 intact whereas under the leaseback agreement the building will have NIL value in 20 years time. Therefore, investors in depreciating assets such as buildings, must allow for the replacement of capital over the life of the investment.

REPLACEMENT OF CAPITAL   - THE SINKING FUND

Investors in buildings allow for the replacement of capital over the life of the building by way of a sinking fund. The sinking fund is a special fund set aside with a periodic payment out of the rents, sufficient to amount to the value of the building over its expected life. Therefore, at the end of that period, the building investor will have sufficient funds to rebuild the building and start all over again.

EXAMPLE:   BODY COPORATE OF A STRATA (UNIT) SCHEME

A typical sinking fund problem is the statutory requirement of the body corporate (owner’s association) of a strata (unit) title plan to replace for example, carpets in the common property at the end of their life. The carpets have a life expectancy of 15 years and are expected to cost

\$200 000 to replace. The relevant bank account is paying 12%pa.

The body corporate has asked you to determine how much their annual contribution to a sinking fund should be.

The sinking fund factor (SFF) is that annual payment sufficient to amount to \$1 over the life of the depreciating asset to be replaced. Therefore, it is the reciprocal of the FVPMT factor:

SINKING FUND FACTOR

SFF = 1/FVPMT

Where:

SFF = sinking fund factor

FVPMT = the future value of 1pa

To calculate the necessary contributions to the body corporate's (owners’ corporation) fund:

STEP 1

Calculate FVPMT:

base = 1.12

FV = 1.1215 = 5.474

FVPMT = (FV1)/(i/100)

= (5.474)/0.12 = 37.28

STEP 2

Determine the sinking fund factor (SFF):

SFF = 1/37.28 = `0.0268

STEP 3

Determine the sinking fund contribution:

200 000 * 0.0268 = \$5 360pa

That is, \$5 360 pa will amount to about \$200 000 (error due to rounding) over 15 years at 12%pa.

SINGLE RATE ASSUMPTION

In the leaseback example above, the question was raised about the apparent surplus when the company paid a rent of \$13 389 pa. The required rate of return was \$12 000 pa (12%) leaving a residue of \$1389pa to replace the building by way of a sinking fund:

FVPMT(1) = 72.05

FVPMT(1388.67) = 72.05 * \$1389

= \$100 077

say \$100 000

(difference due to rounding)

Therefore, a single rate factor includes a sinking fund component replacing the investment amount at the SAME rate as the required return on investment. In the above example; 12%pa.

See DUAL RATE FACTOR

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