Chapter 17

Risk, Return, and the Time Value of Money

Relationship between risk and return (Fig. 14.1):RiskRequired Rate of ReturnRisk-free RateTypes of Risk

Business RiskFinancial RiskPurchasing Power of RiskLiquidity Risk

Risk

Time Value of Money:A dollar in hand is worth more than a dollar to

be received in the future because it can either be consumed immediately or or put to work to earn a return.

DiscountingCompounding

Time Value of Money Tables

Col. 1: FV=PV (FVIF n= , i=) Future value of $Col. 2: FVA=A (FVAIF n= , i=) Future Value of an AnnuityCol. 3: SFP=FV (SFIF n= , i=) Sinking Fund PaymentCol. 4: PV=FV (PVIF n= , i=) Present Value of $Col. 5: PVA=A (PVAIF n= , i=) Present Value of an AnnuityCol. 6: PMT=PVA (PMTIF n= , i=) Amortization Payment

OLB = Outstanding Loan Balance = PV of remaining payments discounted at the contract interest

rate

OLB = PVA = Pmt (PVAIF n= , i=)

NPV = Net Present Value = PV of CF’s discounted at investor’s required rate - cost; if NPV

> 0 project equals or exceeds investor’s required rate of return.

Time Value of Money Formulas:Future value of a lump sum

Compound interest (Table 14.1)

FV = PV (1+i)n = PV (FVIF n,i)

PV FV = ? n = Known i = known

Future value of $ : Col. 1

If you invest $10,000 today which will earn 10% interest for 40 yrs., what sum will you accumulate?

FV=PV(FVIFn, i) = 10000(45.259) =$452,592.56

Present value of a lump sumPV = FV/(1+i)n = FV (PVIFn, i)

PV = ? FV n = Known i = known

Present Value of $: Col. 4You will inherit $1,000,000 forty years from now. If you “discount” money at 10%, what is the million dollars worth today?PV=FV(PVIFn, i) = 1,000,000(.022095)

= $22,095

Present Value of an Annuity:1 PVA = A 1 - _ 1__ = A(PVAIFn, i)

__(1+i)n

i

2 Example: PVA = ?

A =

K

now

n

A =

K

now

n

n = Known i = known

Present Value of an Annuity: Col. 5Your wealthy grandparents have setup a trust fund for you that pays out $10,000 at the end of each year for the next forty years. You want to borrow against this fund. If money is discounted at 10%, what is the present value of your trust fund?PVA=A(PVAIFn, i) = 10,000(9.779051)

= $97,790.51

Future Value of an annuity:1FVA= A _(1+i)n - 1__ = A(FVAIFn,

i)

i

2 Example:

A =

K

now

n

A =

K

now

nFVA = ?

n = Known i = known

Future value of an Annuity: Col. 2

If you invest $1,000 at the end of each year that earns 10% interest, how much will you accumulate after 40 deposits?

FVA=A(FVAIFn, i) =1000(442.592)

= $442, 592

Sinking Funds:1SFP= FVA (1+i)n = FVA(SFIFn, i)

i

2 Example:SFP

= ?

SFP

FVA = Known

n = Known i = known

Sinking Fund Payment: Col. 3Forty years from now you wish to have accumulated $1,000,000. If you can earn 10% annually, how much will you have to deposit annually?SFP=FV(SFIFn, i) = 1,000,000(.002259)

= $2,259

Amortization Payment: Col. 6You just borrowed $100,000 to buy a house. The loan will be repaid annually for forty years at 10% interest. What will your annual payment be?PMT=PVA(PMTIFn, i) = 100,000(.102259)

= $10,225.90Loan Amt. x 40 yrs.

409,036-100,000 prin. 309,036 int.

exp.

Mortgage Payments:1 Pmt= PVA __ i___ = PVA(PMTIFn, i)

1 - _ 1__ (1+i)n

2 Example3 Monthly Compounding

PM

T

= ? PM

T

= ?PM

T

= ? PM

T

= ? PM

T

= ?PV

A =

Know

n =

Orig.

Loan

n = Known i = known

Monthly CompoundingAlgebraic formulas that adjust the six basic

calculations (FV, PV, PVA, FVA, SFP, PMT) are found on p. 302 in your text.

A handout of the compound interest table (10%) with monthly interest factors will be provided.

Previous Example:

$100,000 mortgage, 40 yrs @ 10%. What would monthly payments be? Use same formula, but interest factors for monthly compounding.PMT=PVA(PMTIFn=480, i=10%) = 100,000(.008491) =$849.10/mo.

Calculating the outstanding loan balance (OLB) on an amortized loan:OLB = PV of remaining payments discounted at

contract rate.Ex.: you borrow $100,000 to buy a home at

10% interest, 30 years, monthly payments. What would the OLB be after the 60th payment?

1. PMT=PVA(PMTIFn360, i=10%) = 100,000(.008776)

= $877.602. OLB=PVA=A(PVAIFn=300, i=10%) = 877.60

(110.047230) = $96,577.45

Financial Decision Rules: NPV and IRRNet Present Value decision rule (NPV)Internal Rate of Return decision rule (IRR)Examples of NPV and IRR rules (Table 14.2)

NPV Example:

You have a chance to invest in an apartment complex which will generate annual cash flows of $48,000. The property can be purchased for $500,000 today and you expect to sell it after 5 yrs for $600,000. Will this property be a wise investment?

Time Cash Flow PVIF @ 10% PV 0 -500000 1.000000 -500000 1 48000 0.909091 43636 2 48000 0.826446 39669 3 48000 0.751315 36063 4 48000 0.683013 32785 5 648000 0.620921 402357

NPV +54510OR

PV = 48000(PAVIFn=5, i=10%) + 600000(PVIFn=5, i=10%) = 48000(3.790787) + 600000(.620921) = 181,958+372,553 = 554,511 = PV - Cost = NPV 554,511 - 500000 = +54,511Accept Project