Time Value of Money

InflationConsumption PreferenceA Real Return

Assumptions

Free Cash Frictionless money market: Lending

& Borrowing Rate Equal Risk-free No borrowing or lending limit Cash flows at fixed time intervals

Forms of cash Flow

Single Amount (Lump Sum) Annuity (Series of Equal amount) Equal amount to Perpetuity Unequal Cash Flows Constantly Growing amount to

Perpetuity

The Input Variables Seek out

Present Value (PV) (Amount at the beginning of a time line) Future Vale (FV) (Amount at the end of a time line) Payment (PMT) (Amount of Annuity) Interest/Discount/Reinvestment Rate (I) Length of time Period (N)

For basic time value problems, you will need 4 input values. If you know 3, you can find the 4th variable.

For Bond valuation, you will need 5 input variables. You need to know 4 to find the 5th value.

Suppose there is Taka 12,00,000 in an account today. No money was taken out and no money deposited into the account in the last four years. If the interest rate in the account is 10.25%, how much was in the account 4 years ago?

Frequency of Compounding Semi-annual Quarterly Monthly Daily Continuously Less frequent than annual

How Long Does it Take to Accumulate Taka 1 Crore At What Rate Money Must Grow to

Make Sure That you Will Have Taka 1 Crore If You Save Taka 50,000 per year for 30 Years?

At 12 Percent, How Much Must You Save Per Year to Have Taka 1 Crore in 30 years?

Amortization

Paying off a loan in Equal Annual Installments Interest and Balance Due declines as

time elapses Payment of Principal Portion increases

with Time.

Example: Problem 40

Roger will deposit Taka 12,500 every year at the end of the year beginning this year until he accumulates Taka 70,000. Interest rate in the account in 12%. How many years will he need to accumulate the target amount?

What will be the size of last deposit?

Example: Retirement Planning You are 30 years old and have a decent job. You will work

for 35 more years. You will start saving this year and make a deposit into a retirement account every year until you retire. The interest rate in this account is 9.8% per year.

After you retire, you will live exactly 10 years. Your current living standard requires Taka 4,80,000 per year. Inflation is 7.5% per year. When you retire, you will maintain the same living standard. For simplicity, assume that you will need the same amount every year after you retire (that is, no inflation adjustment after that).

Question 1. How much do you have to accumulate in the retirement account when you retire?

Question 2. How much do you have to save per year?

Present Value of A Single Amount

Rearrange The FVi,n Equation PV = FVi,n / (1+i)n = FVi,n * (1/(1+i)n)

Using the PVIF Table

Nature of the PVIF Table

What happens as time lengthens What happens as I increases

PVIF on A Graph

PVIF Gets Smaller for Distant Amounts and Higher Rates

Figure 2: PVIF: Present Value of Taka 1

00.20.40.60.8

11.2

Time

PVIF

Discount Rate 5%

Discount Rate10%Discount Rate15%Discount Rate20%

Present Value of An Annuity Present Value of An Annuity of Taka 1PVAi,n = 1(1/(1+i)1) + 1(1/(1+i)2)+

………… + 1(1/(1+i)n) n n= PVAi,n = 1 ∑ (1/(1+i)t) = ∑ (1/(1+i)t) t=1 t =1

Present Value of Unequal Cash Flows Find Individual Present Values Sum the Present Values What about simply using the CF

function of your calculator? Finding FV of Unequal Flows by

converting The PV of Unequal Flows to A FV.

Present Value of A Perpetuity ∞ PV = ∑ Pft / (1+i)t t=1

PV = Pf / i PV = CF/i

Present Value with constant growth to infinity

PV = CFt(1+g)/(k-g)

g must be smaller than k

Valuation of Stocks and Bonds

Chapter 5The General Procedure of valuation of any financial instrument

Valuation of Bonds/Debentures Identify Cash flows

Annuity component (PMT) Maturity Value, A Single Amount (FV)

Discount Rate: The Required Rate, (I/Y)

Time Frame (N)

V = Σ Pmt/(1+r)t + M/(1+r)n

Valuation Example

Consider a 10 year bond with an annual coupon rate of 12% paid annually and has a face value of Taka 1,000. Discount rate is 14%. What is the value of such a bond now?  

Annuity (PMT) = 120 Number in the Series (N) = 10, Maturity Value (FV) = 1000Discount Rate (I/Y) = 14%

Valuation of Zero Coupon Bonds Zero Coupon Bonds Have no Annuity.

If a firm issues a 20 year zero coupon bond when the market yield is 12%, how much will the firm raise per bond?

If the firm issues 100,000 such bonds, what is the total amount the firm will raise?

What is the total payment obligation when the bond matures?

Yield to Maturity (YTM)

The Rate of Return You Lock in if You Buy The Bond Today and Hold It till Maturity

Easiest Way to Calculate is to Use Financial Calculator

YTM = Current Yield + Capital Gain Current Yield (CY) =

Coupon/Purchase Price Capital Gain (g) = (Current Price-

Initial Price)/Initial Price

Valuation of Preferred Stock For non-convertible, non redeemable

Preferred Stock: Claim to A Perpetuity

Vpr = Dpr/rpr

Valuation of Common Shares Constant Dividend Stocks(Perpetuity) Dividends Growing at A Constant

Rate(constant growth to infinity) Dividend is not Constant, The Growth

Rate is Not Constant

Valuation of Constant Dividend Stocks Very Much Like Infinite Life Preferred

Stocks

P0 = D/k

Rearrange to Find Expected Returnk = D/Po

Valuation of Stock with Dividend Growing at A Constant Rate

Gordon Growth Model n P0 = ∑ [D1(1+g)t/(1+r)t] t = 1

P0 = D1/(k-g) = D0(1+g)/k-g

Valuation of Stocks with Dividend Growing at A Non-Constant Rate

Four Steps Forecast Dividends until Dividend

Growth Becomes Constant and Stable Find the Expected Price when Dividend

Growth Becomes Stable Project the Cash Flows (Dividends Plus

Expected Price) till Dividend Becomes Stable

Find The PV of Cash Flows

Non-Constant Growth Stock Valuation: An Example Majestic Corporation is expected to

pay a dividend of Taka 10 two years from now. Dividend will increase at 50% in the 3rd year, 25% in the fourth year and then will grow at a constant rate of 8%. If stockholders require a return of 16% on this stock, what is the fair value of the stock?

Solution

Dividend Projections D1 = 0 D2 = Taka 10 D3 = Taka 15D4 = Taka 18.75 D5 = Taka 20.25

Price ProjectionP4 = D5/(k-gn) = Taka 20.25/(.16-.08) =

Taka 253.125. Cash Flows: CF1 = 0 CF2 = Taka 10

CF3= Taka 15CF4 = Taka 18.75+253.125 Po = Taka 167.20

Dividend Yield and Expected Capital Gain Dividend Yield = Expected

Dividend/Po

Capital Gains = (Expected Price – Initial price)/Initial Price.

Compute the dividend yields and capital gains in each of the next four years in the previous example.

Stock Price Equilibrium

Required Return and Expected Return Equal

If Required Return>Expected Return: Stock Is Overvalued

If Required Return<Expected Return: Stock Is Undervalued

Estimating Dividend Growth Historical Growth in Dividend Historical Growth in Return on Equity GNP Growth National Inflation Regression Slope Asset Growth (Especially for Financial

Institutions) Retention Growth= ROE(1-Payout

Ratio)