10_FM_3_tn_1pp

Financial Mathematics


Jonathan Ziveyi1

1University of New South Wales

Risk and Actuarial Studies, Australian School of Business

[email protected]

Module 3 Topic Notes

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Plan

Module 3: Loan Valuation and Project Appraisal TechniquesIntroductionAllowing for TaxAnalysis of Loan Schedules and RepaymentsSinking FundsLoans at a Flat Rate of InterestLoan Valuation ExampleFixed Income Securities and BondsPricing BondsBond Valuation ExampleDefinitions of Yield, IRR and MIRR RatesInvestment Decision CriteriaSensitivity of Results and Duty of DisclosureProject Appraisal Example

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Module 3: Loan Valuation and Project Appraisal Techniques

Introduction

Evaluation of a project Objective of a project appraisal:

value a given project: how much is it worth?

compare different projects based on certain criteria:which project is the best?

make a recommendation based on certain criteria:should we invest in that project?

This involves determining net cash flows:

gains: sales salvage value of assets

minus costs: expenses transaction costs taxes depreciation of assets cost of debt

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Introduction

Financing a project There are several ways of financing a project:

for an individual personal wealth personal loan

for a company equity (shares) debt (loans, bonds)

for a government taxes debt (treasury bonds)

The analysis of loans and bonds is necessary in order to be able tobuild the cash flow model. Note that bonds are nothing else thanlarger scale, tradable loans.

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Introduction

Plan of this module

1. Introduction

2. Allowing for Tax

3. Analysis of Loan Schedules and Repayments

4. Sinking Funds

5. Loans at a Flat Rate of Interest

6. Loan Valuation Example

7. Fixed Income Securities and Bonds

8. Pricing Bonds

9. Bond Valuation Example

10. Definitions of Yield, IRR and MIRR Rates

11. Investment Decision Criteria

12. Sensitivity of Results and Duty of Disclosure

13. Project Appraisal Example5/66



Allowing for Tax

Allowing for tax Tax is a very important consideration whenanalysing a cash flow:

when and how much tax is paid influences the profitability of asecurity or project

the tax rate depends on the type of cash flows: income (e.g. interest, dividends, rents, . . . ), or capital gains (e.g. increase of the value of a share or property,

above par redemption payments, . . . )

the tax rate depends on the individual considered (person orcompany)

tax is usually paid with a lag that also depends on theindividual considered

income and capital losses are usually allowed to be offsetagainst gains to derive tax benefits

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Allowing for Tax

Allowance for taxation in price/yield calculations

tax payments are nothing else than additional (negative) cashflows

in case of losses that can offset gains, tax benefits can beadded as positive cash flows(the government wont pay any money, but a loss means taxthat otherwise would have been paid will not be paid)

in many cases price and yield calculations allowing for tax canbe done analytically (using financial mathematics formulae),"by hand" and using a calculator

larger/more complicated models can be easily done using aspreadsheet model or other relevant software.

transaction costs are similar costs that need to be allowed for

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Allowing for Tax

Depreciation Schedules and Tax Many projects involve aninvestment in capital equipment. For taxation purposes this isdepreciated usually on two (alternative) bases:

Prime Cost (level over life of equipment), or

Diminishing Value (constant percentage of written down valueWDV)

Taxable income is income minus expenses:

expenses include interest costs and depreciation

net cash flow is the cash payments less taxation expense

in case of deferral (or lag) for taxation payments, treat as twodifferent cash flows

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Analysis of Loan Schedules and Repayments

Loans Definitions:

Consider a loan of amount L made at time 0 with repaymentsof K1,K2,. . . ,Kn at times 1, 2, . . . , n

Equation of value

L = K1v + K2v2 + . . . + Knv

n at effective rate i .

Each loan repayment Kt can be decomposed into a principal component (which amortises the loan) an interest component (which pays the interest due since the

last repayment)

The amount that still need to be reimbursed after a paymentis called the outstanding balance

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Denote:

It the interest component of the tth payment

PRt the principal repaid in the tth payment

OBt the outstanding balance immediately after the tth

payment

Interest in tth payment is simply the previous outstanding balancemultiplied by the rate of interest

i OBt1

Principal repaid should just be the difference between the actualpayment and the interest component

Kt It

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If we work recursively we have

at time 0OB0 = L

at time 1

I1 = iOB0 = iL

PR1 = K1 I1 = K1 iOB0

OB1 = OB0 (1 + i) K1 = OB0 (K1 iOB0)

= OB0 (K1 I1) = OB0 PR1

and then we move forward to the next time period

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In general we have

It+1 = iOBt

PRt+1 = Kt+1 It+1

OBt+1 = OBt (1 + i) Kt+1 = OBt (Kt+1 It+1)

= OBt PRt+1

Total repayments

KT =n

1Kt

total interestIT =

n1It

and

L = KT IT =n

t=1

PRt

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Numerical example Example: Consider a loan of $1000 repaid by 5equal installments of principal and interest at the end of each yearfor 5 years with an interest rate of 5%. Determine the repayments.

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Loan Schedule In practice it is often much easier to set out all theinformation in a "loan schedule" providing information (for eachperiod) on:

Payments

Interest Due

Principal Repayments

Principal Outstanding

(and any other important items)

This is usually presented in a table computed with the help of R ora spreadsheet.

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Example For the $1000 5 year loan with level repayments, what arethe interest and principal components in each year? Give arepayment schedule.

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In order to determine a given line of the loan schedule, one needsonly the principal outstanding at the beginning (or the end) of theperiod. This can be determined directly via:

the prospective method,

OBt =

ns=t+1

Ksvst {= Kant i if repayments are equal}

or the retrospective method

OBt = L (1 + i)t

ts=0

Ks(1 + i)ts {= L (1 + i)t Kst i}

Both methods yield the same result.

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Numerical example (retrospective method) Consider a loan of$1000. For the first year the repayment was $200, and the interestcharged was 5%For the second and third years the repayment was $150 p.a., andinterest charged was 4% p.a. What is the loan outstanding at theend of the third year?

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Sinking Funds

Sinking Funds Consider the following situation:

Company A has borrowed an amount L (from a bank, byissuing a bond, etc. . . ) and will need to reimburse the loanafter n years

in the mean time, it needs to pay interest at a rate i each yearto the lender(s)

Company A wants to set up payments to a fund that willaccumulate to the amount of the loan at time n in order toensure the reimbursement

this fund earns interest at a rate j not necessarily equal to i .Usually, j < i .

Such a fund is called a sinking fund.

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Sinking Funds

In order to accumulate to L, level payments to the sinking fundneed to be equal to

L

sn j,

which means that the total payment for each time unit is

iL +L

sn j.

The first component is the interest component, paid to the lender,and the second is the principal component, paid to the sinkingfund.

Does a sinking fund lead to higher repayments than when theloan is reimbursed gradually using the amortisation method?

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Sinking Funds

Sinking fund example A loan of $1000 is to be repaid by 5 annualpayments, beginning one year after the loan is made. The lenderwants annual payments of interest only at a rate of 7% andrepayments of the principal in a single lump sum at the end of 5years.The borrower can accumulate principal in a sinking fund earning anannual interest rate of 6%, and decides to do this with 5 leveldeposits starting one year after the loan is made. Determine therepayment and model the cash flows of this transaction in aspreadsheet.

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Sinking Funds

Example

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Loans at a Flat Rate of Interest

Loans at a Flat Rate of Interest The interest charge I is given by

I = L f n

where:

L is the loan amount

f is the flat rate of interest

n is the duration of loan (in time units of the flat rate ofinterest)

Loan Repayments R are given by

R =L + I

n

where n is the number of level instalments.

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Numerical example A lawnmower worth $400 is offered for sale onthe following terms:10% deposit, flat interest of 10% p.a. with monthly repaymentsover 30 months.Determine the repayment and the effective annual rate of interest.

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Usage Easier to understand, but presents serious problems:

the "real" rate of interest is usually much higher than whatthe flat rate suggests

flat rate loans do not encourage earlier payments (the amountof interest that has to be paid is fixed)

Flat rates of interest are not used everywhere:

because of the problems described above, it is forbidden issome countries (mainly developed, such as in Australia)

however, it is widely used in developing countries (mainly bymicrocredit institutions)

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Loan Valuation Example

Example - Loan Valuation - Spreadsheet A loan of nominal amount$500,000 was issued bearing interest of 8% per annum payablequarterly in arrears. The loan will be repaid at $105% by 20 annualinstallments, each of nominal amount $25,000, the first repaymentbeing ten years after the issue date. An investor, liable to bothincome tax and capital gains tax, purchased the entire loan on theissue date at a price to obtain a net effective annual yield of 6%.Find the price paid, given that his rates of taxation for income andcapital gains are 40% and 30% respectively.What is the price paid allowing for taxation? Develop a spreadsheetmodel for the loan allowing for both income and capital gainstaxation.

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Model with Tax on Interest and CGT This is much morecomplicated.

price depends on CGT and CGT depends on price. . .

capital repayments occur over time

We can solve this with a spreadsheet.

Set a dummy figure as the price

Given a price we can determine if a CGT is due for eachrepayment:

CGTt = 30%

(actual paymentt

face value reimbursedt500000

P

)+

we have then (net receipts):

CFt = APRt + It TIt CGTt

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We solve then for the correct price:

Calculate the PV of the net receipts

Calculate in a cell the difference between the PV and the Price(which should be equal)

Using the solver, target a difference of 0

You may need to constraint the interest rate and the price tobe positive.

Note:(x y)+ = max(x y , 0).

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Fixed Income Securities and Bonds

Fixed Income Securities Broad range of securities with fixedincome:

bonds (or notes, or debentures), issued by the government private companies

types of bonds short term (e.g. Australian Treasury note, or promissory note)

vs long term (e.g. Australian Treasury bond) virtually risk free to very risky (junk bonds) coupon bonds or zero-coupon bonds (ZCB) indexed bonds, or real return bonds

but also certificates of deposit (tradable or not) . . . (see readings)

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Government BondsGovernment Bonds:

borrow money from investors to fund spending plans

also provide low risk securities (liquidity on the market, anddetermination of the structure of interest)

both short and long term(in Australia: Treasury Notes and Treasury Bonds)

consist of both coupon and capital payments(in Australia: usually interest only until maturity)

for (Commonwealth) Government Bonds in Australia usually semiannual coupons coupons are paid on the 15th of each relevant month. yields are quoted as nominal p.a. with the same frequency as

the coupon payments

Other conventions: see Broverman and Sherris30/66




Bond basics Pays coupon (interest) to purchaser a certain numberof times a year, of amount

Fc

ppaid p times per year

where

F is the face value (par value)

c is the annual coupon rate

p is the frequency of payments

Payments will continue during the term to maturity of the bond(denoted by n).

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Maturity, redemption and principal amortisation

Maturity is the date when the bond is redeemed(reimbursed)

The redemption amount FR at maturity is not always equal tothe face value. We have R = 1: the bond is redeemed at par R < 1: the bond is redeemed below par R > 1: the bond is redeemed above par

Sometimes, principal is reimbursed before maturity. Again, theamount of face value can be reimbursed at par, orbelow/above par.

A bond is essentially a loan that is amortised in a single lumpsum payment (at maturity) and/or by earlier payments.

Being able to differentiate between face value redemption andcapital gain/loss (above/below par reimbursements)is important for accounting, yield and tax purposes.

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Numerical example A 3 year bond with a face value of $100,000pays annual coupons at a rate of 10% p.a. The bond is

1. entirely redeemed at maturity with a payment of $120,000;

2. redeemed by 2 payments of $65,000 each at the end of thesecond and third year, each for half of the bonds face value.

For both cases, establish a loan schedule showing interestpayments, principal repayments and capital gains.

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Pricing Bonds

The price of a bond

as usual for securities, the price of a bond is essentially thepresent value of its future cash flows

the rate, called yield, at which cash flows are discounted is acritical assumption

it is usually quoted along with the price of the bond (bothvalues are equivalent ways of quoting the price of a bond)

it is usually of the same type as the coupon rate (semiannualnominal for US/CA/AU, sometimes also annual in EU)

the yield usually depends on the current structure of interest,as well as the risk associated to the bond as perceived by themarket (note also some rating agencies rate bonds AAA to C)

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Pricing Bonds

On coupon dates For a bond at yield i (p) = pi whose redemptionand face values are the same (R = 1) we have:

P = Fcanp i + Fvnpi

= Fcanp i + F (1 ianp i)

= F + F (c i) anp i

A par bond must have c = i

A bond with c > i trades at a premium

A bond with c < i trades at a discount

If R 6= 1, the price is the PV of the future CF (simple application ofcompound interest techniques)

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Pricing Bonds

Between Coupon dates

in practice bonds are traded between coupon payment dates

the seller will require the interest accumulated since the lastcoupon date to be paid by the buyer

if the sale is too close to the next coupon payment (inAustralia, 7 days or less), the bond becomes ex-interest,which means that the next coupon payment will still be paidto the seller, even if the bond is not his property any more

the general pricing approach is to discount the bond cash flowsto the next coupon payment date (including the couponpayment at that date), and then further discount this presentvalue this to the (prior) sale date

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Pricing Bonds

The RBA formulaThe RBA uses the following formula to value Treasury Bonds whenmaturity is between n and n + 1 semesters:

P = vf

d

i [C + Gan i + 100vn]

where:

C is the next coupon payment (zero if ex-interest)

G is the regular semi annual coupon payment

f is the number of days until the next payment

d is the number of days in the current half year

(this is identical to the general formula given in Broverman)

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Pricing Bonds

Market price Two bonds with the same cash flows and the sameyield will have a different purchase price if coupons payment datesare different, which may be confusing. Hence, bonds are usuallyquoted at a market price. We distinguish:

Price-plus-accrued: the price with accrued interest (coupon) - see previous slide the purchase price also: "dirty price", "full price", or "flat price"

Market price price as quoted ( smoothed price) accrued interest is removed

market price = dirty price accrued interest = P tFc

also: "clean price"

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Pricing Bonds

Numerical example Consider a Government bond paying semiannual interest of 10% p.a. on 15-April and 15-October each year.It is redeemable at par on 15 Oct in 6 years time. Find the purchaseand market prices to yield 8.5%p.a. (semi-annual) on 30 June.

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Pricing Bonds

Optional redemption dates In general the redemption date may:

be fixed (most of the bonds)

vary at borrowers option on or after certain date no final date (undated) between two dates

at lenders option

An uncertain redemption dates means that lenders (buyers) canteasily determine yields at the purchase date. In such a case, theycan still determine:

a maximum price, for given yield, or

a minimum yield, for given price

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Bond Valuation Example

Example - Loan Valuation - "by hand" [UNSW Final Exam 2006] Abond with a nominal face value of $100, 000 is redeemable by twopayments, one in 5 years time and the other in 10 years time. Thepayment in 5 years time is for a nominal amount of $40, 000 and in10 years time for a nominal amount of $60, 000. Redemptionpayments are payable at $105 per $100 nominal face value.Coupons are paid on the bond at 6% p.a semi-annually based onthe nominal amount outstanding. Tax is paid on the coupons at arate of 30% and tax is paid on capital gains at a rate of 15%.Capital losses are assumed to be offset against other capital gainsof the investor.

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1. Determine the price to be paid by an investor to earn a grossyield of 6.5% p.a. (semi-annual).

2. Determine the price to be paid by an investor to earn a net oftax (after tax) yield of 5% p.a. (semi-annual) allowing only fortax on the coupons.

3. Determine the price to be paid for the bond to yield an net oftax return of 4% pa. (semi-annual) allowing for tax oncoupons and capital gains.

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1. Price to earn a gross yield of 6.5% p.a. (semi-annual) (note theyield and coupons are semi-annual so work in half years)

Price =0.06

2(40,000) a10 + 40,000 (1.05) v

10

+0.06

2(60,000) a20 + 60,000 (1.05) v

20 at6.5

2%

= 1,200 8.422395 + 42,000 0.726272

+1,800 14.539346 + 63,000 0.527471

= 10,106.874 + 30,503.431 + 26,170.823 + 33,230.689

= 100,011.82

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2. Gross yield of 5% p.a. (semi-annual), tax on the coupons

Price = (1 0.3)0.06

2(40,000) a10 + 40,000 (1.05) v

10

+ (1 0.3)0.06

2(60,000) a20 + 60,000 (1.05) v

20 at5.0

2%

= 840 8.752064 + 42,000 0.781198

+1260 15.589162 + 63,000 0.610271

= 7,351.7337 + 32,810.333 + 19,642.3445 + 38,447.0694

= 98,251.48

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After tax return of 4% p.a. (semi-annual), tax on coupons andcapital gains

Price = (1 0.3)0.06

2(40,000) a10

+40,000 (1.05) v10 0.15

(40,000 (1.05)

40000

100000P

)v10

+ (1 0.3)0.06

2(60,000) a20

+60,000 (1.05) v20 0.15

(60,000 (1.05)

60000

100000P

)v20

at4.0

2%

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We have

P = 840 8.982585 + [42,000 0.15 (42,000 0.4P)] 0.820348

+1260 16.351433 + [63,000 0.15 (63,000 0.6P)] 0.672971

and thus

(1 0.049221 0.060567)P = 7,545.371 + 29,286.4236

+20,602.8056 + 36,037.597

P =93,472.197

0.890212= 105,000.

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Definitions of Yield, IRR and MIRR Rates

Definitions Yield rate

effective "average" rate of interest over the whole (time)length of an investment:

yield rate =

(accumulated value

investment cost

)1/length of investment 1

Net Present Value (NPV)

present value of inflows (gains) minus outflows (expenses andinvestment costs), or net cash flows

must be calculated using a relevant rate of interest(reflecting risk and cost of capital)

Internal Rate of Return (IRR)

rate of interest such that the NPV is 0

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Numerical example Consider the following two cash flows:

Option 1 Option 2

0 -1000.00 -1000.001 100.00 533.202 200.00 350.003 300.00 250.004 400.00 150.005 500.00 50.00

For option 1, the IRR is 12.01%. Consider the yield:

(Inflows accumulated @ IRR

1000

)1/51 =

(1762.90

1000

)1/51 = 12.01%

Yield = IRR!

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What if it is not possible to reinvest inflows at a rate equal to IRR?For option 1

(Inflows accumulated @ 3%

1000

)1/51 =

(1561.37

1000

)1/51 = 9.32%

the yield is much lower

but the IRR does not change

if the reinvestment rate is different from the IRR, the yield isnot equal to the IRR!

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Numerical example

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Reinvestment rates NPV and IRR

assume a homogeneous rate of interest for all cash flows:

(Accum. value @ IRR) = (Invmt cost)(1 + IRR)length of invmt

do not allow for a different reinvestment rate

Solution

MIRR:

(Accum. value @ reinv. rate) = (Invmt cost)(1+MIRR)length of invmt

MIRR is a modified yield that takes into account thereinvestment rates

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Multiple IRR If cash flows are non conventional (change sign morethan once), there may be several IRR...

Example:

t CFt0 -591 1542 -99

= NPV(i):

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Investment Decision Criteria

Decision criteria

1. Payback period the amount of time until repayments accumulate (without

interest) to the initial investment

2. Discounted payback period the amount of time until discounted repayments have a higher

PV than the initial investment same idea as payback period, but taking the time value of

money into account

3. NPV (net present value) the present value of net cash flows

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4. IRR (internal rate of return) the rate of interest such that the NPV is 0

5. MIRR a modified IRR that takes into account reinvestment rates

6. Profitability index the ratio

(PV of repayments) / (initial investment) remember the NPV is the difference:

(PV of repayments) - (initial investment)

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7. Dollar-weighted rate of return The simple rate of interest such that the NPV is 0

8. Time-weighted rate of return returns over subsequent periods are compounded to yield an

average return particularly used by investment funds to transform monthly

returns into longer term returns (semesterly, annual, . . . )

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Numerical exampleIn this example, what is the decision that the various decisioncriteria that were introduced would yield?

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Sensitivity of Results and Duty of Disclosure

Sensitivity of Results: Example Suppose your company isconsidering the purchase a small insurer. You forecast the followingcashflows for this insurer over the next 5 years:

Premiums: 100m p.a.

Claims: 80m p.a.

Expenses: 5m p.a, increasing at 3% p.a.

Assume that these are all incurred at the middle of the year onaverage.

At the end of the 5th year the business will be sold for a totalof 10m.

Find the NPV of this project at 6% p.a.

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Sensitivity of our results to

discount rate assumption?

expense increase rate?

premium and claim changes?58/66




7. ReportingA Member must ensure that his or her reporting (whether oral orwritten) in respect of Professional Services provided:

(a) is appropriate, having regard to:

1. the intended audience;2. its fitness for the purposes for which such

reporting may be required or relevant;3. the likely significance of the reporting to its

intended audience;4. the capacity in which the Member is acting; and5. any inherent uncertainty and risks in relation to

the subject of the report;

(b) complies with any relevant Professional Standards.

Institute of Actuaries of Australia Code of Professional Conduct(November 2009, Section 7)

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Assess Extract of professional standards on economic valuations:

4.2 Scope of economic valuation[. . . ]The Member should ascertain the materiality limits that apply tothe economic valuation bearing in mind:

the quality of the data;

the intended use(s) of the economic valuation;

the degree of uncertainty; and

the sensitivity of the overall result to different assumptions.

Institute of Actuaries of Australia Guidance Note 552 on EconomicValuations (July 2004)

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And then communicate

3.3 TransparencyThe models, methods and assumptions used for the economicvaluation should, as far as practical, be transparent, enablingvaluation results and sensitivities in the results to changes inparticular assumptions to be understood by the intended users ofthe economic valuation.

Institute of Actuaries of Australia Guidance Note 552 on EconomicValuations (July 2004)

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Project Appraisal Example

Project Appraisal Example A company considers buying equipmentand then leasing it out to third parties. This project has thefollowing variables:

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Loan schedule

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Taxable income

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Net cash flows

NPV @ 18%: $161,745IRR: 22.06%

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NPV check

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10_FM_3_tn_1pp

Documents

tax tax

project objective

given project

loan valuation example7

tax rate

taxanalysis of loan

bond valuation example10

person orcompany tax