1: Time Value of Money & Power of Compounding

Mini Chapter One

Time Value of Money &
Power of Compounding

Time value of money

Understood literally this means the value of a dollar erodes because of inflation and with passage of time. Value of a dollar in the future is less than the value of a dollar today if we are in a positive interest rate (motivated by inflation) environment. To clarify further – nominal interest rates above inflation i.e. positive real yields would add to the purchasing power of money (assuming it’s invested) but negative real yields would erode it which is when consuming the dollar today is better than investing it.

Future value (FV) of a dollar is a function of the time period and the expected return over that period. It’s this expected return that we refer to as the discount rate to calculate the equivalent present value (PV) of the future cash flows.

Expected return can take the form of simple or a compounded interest rate. I’ll skip any discussion on simple interest and focus instead on the power of compounding. Intuitively interest on interest is a reflection of a compounded return, which can be mathematically expressed as:

\[ FV = PV \times {(1 + R_t )}^t \]

Where R_t = Annual Rate of Interest, t = time period in years

The above formula when used to calculate PV of a future cash flow,

\[ PV = \dfrac{ FV}{ {(1 + R_t )}^t } \]

And the term \(\dfrac{ 1}{ {(1 + R_t)}^t }\) is referred to as the discount factor which when multiplied by the respective future cash flow will give the present value of the same.

For higher frequency of compounding the formula above will be amended to:

\[ FV = PV \times {\{ 1 + \dfrac{ R_t }{f} \}}^{( t \times f)} \]

Where f is the frequency of compounding i.e. f = 2 for semi-annual compounding, f= 4 for quarterly compounding, f = 12 for monthly compounding, f = 365 for daily compounding.

Continuous compounding

In today’s markets we have seen increasing use of daily compounded rates (OIS). which would be depicted by:

\[ FV = PV \times {\{ 1 + \dfrac{ R_{ois} }{ 365} \}}^{365} \]

Where, R_ois is the annual interest rate compounded daily.

Trivia. Overnight Indexed swap (OIS) markets mimic daily funding rates hence daily compounding of the floating rate is done only for business days and not calendar days. In other words simple interest accrual is followed for Friday over Monday and or over public holidays.

In the extreme case where we want to compound every instant (compounding frequency x) we need to solve the above equation as:

\[ FV = \underset{ x \rightarrow \infty }{\lim} \{ PV \times {[ 1+ \dfrac{R_t }{x} ]}^{xt} \} \]

\[ FV = \{PV \times e^{R_t \times t} \} \]

Concept of YTM or IRR

• The popular “internal rate of return” (IRR) term is used to refer to the singular yield (same as the reinvestment yield) for calculating returns on multiple cash flows. Note that IRR and Yield to Maturity are interchangeable for defining the expected return on investments.
• Present Value of cash flows (assuming annual compounding) is denoted as:

\[ PV = \dfrac{C}{ {( 1 + IRR)}^{t_1} } + \dfrac{C}{ {( 1 + IRR)}^{t_2} } + .... + \dfrac{ ( TV + C )}{ {( 1 + IRR)}^{t_n} } \]

Where,
PV: Present Value, TV: Terminal Value, C: Coupon
t₁,t₂…: intermediate cash flow periods, tₙ: Tenor (Final Cashflow time period),
IRR: Internal Rate of Return

A key assumption for the concept of IRR is that all intermediate cash flows are reinvested at an average rate equal to the IRR itself to generate the same compounded return. As an example a 5% annual coupon bearing 5y bond with an IRR or YTM of 4% (annually compounded) has a present value of:

\[ PV = \dfrac{5}{ {1 + 4\%} } + \dfrac{5}{ {( 1 + 4\% )}^2 } + \dfrac{5}{ {( 1 + 4\% )}^3 } + \dfrac{5}{ {( 1 + 4\% )}^4 } + \dfrac{ ( 100+ 5)}{ {( 1 + 4\% )}^5 } = 104.452 \]