Module 2: Time Value of Money in Finance

The time value of money

represents the trade-off between cash flows received today versus those received on a future date, allowing the comparison of the current or present value of one or more cash flows to those received at different times in the future.

Fixed-income instruments are debt instruments

Bond or a loan, that represent contracts under which an issuer borrows money from an investor in exchange for a promise of future repayment. The discount rate for fixed-income instruments is an interest rate, and the rate of return on a bond or loan is often referred to as its yield-to-maturity (YTM).

Discount

An investor pays an initial price (PV) for a bond or loan and receives a single principal cash flow (FV) at maturity. The difference (FV − PV) represents the interest earned over the life of the instrument.

Periodic Interest

An investor pays an initial price (PV) for a bond or loan and receives interest cash flows (PMT) at pre-determined intervals over the life of the instrument, with the final interest payment and the principal (FV) paid at maturity.

Level Payments

An investor pays an initial price (PV) and receives uniform cash flows at pre-determined intervals (A) through maturity which represent both interest and principal repayment.

Zero-coupon Bond

Bonds that do not pay interest during their life. They are issued at a discount to par value and redeemed at par. Also called pure discount bond.

The investor's sole source of return is the difference between the price paid (PV) and full principal (FV) received at maturity.

Perpetual Bond

is a less common type of coupon bond with no stated maturity date. Most perpetual bonds are issued by companies to obtain equity-like financing and often include redemption features. As N→∞

PV(Perpetual Bond) = PMT / r.

we can simplify this to solve for the present value of a perpetuity (or perpetual fixed periodic cash flow without early redemption)

PMT

= Coupon Payment Per Period

(Coupon rate × Face value) ÷ Number of payments per year

Think of PMT as just the cash you receive each time the bond pays you

PMT example

If it's a bond → PMT = coupon payment

The bond says: "3.30% coupon" "paid quarterly"

So ask yourself: 👉 "How much cash do I actually receive each time?"

3.30%×100=3.30 3.30÷4=0.825

👉 PMT = 0.825 Which means: "I receive 0.825 every 3 months"

PMT

PMT = "PayMenT each time"

Think of a bond like a salary:

Annual salary = 3.30

Paid quarterly → smaller paychecks

👉 Each paycheck = PMT

Annuity Instruments

Examples of fixed-income instruments with level payments, which combine interest and principal cash flows through maturity, include fully amortizing loans such as mortgages and a fixed-income stream of periodic cash inflows over a finite period known as an annuity.

Annuity Formula

A = r (PV) / 1 − (1 + r) −t

where: A = periodic cash flow

r = market interest rate per period

PV = present value or principal amount of loan or bond

t = number of payment periods.

Equity Instruments

such as preferred or common stock, represent ownership shares in a company which entitle investors to receive any discretionary cash flows in the form of dividends

T he price of a preferred or common share expected to pay a constant periodic dividend is an infinite series

P V t = ∑ i=1 ∞ D t / (1 + r) i

and P V t = D t / r .

D=dividend in one period

If dividends grow at a rate of g per period and are paid at the end of each period, the next dividend (at time t + 1)

P V t = D t (1 + g) / r − g =

D t+1 / r - g

r − g > 0

Terminal Value = where the value of the stock in n periods

E( S t+n ) = D t+n+1 / r-g1