Yield measures for T-bills

Yields on Treasury bills are expressed as a discount from face value and their maturity is often less than a year.

Bank discount yield (BDY)
Annualized discount from face value. It is not a meaningful measure of the return earned by investors.

BDY = [(F-P)/F ] x (360 / t).

Where:
F- Face value
P- Purchase price
t - Days to maturity

Problem:
· Use face value as denominator but not purchase price
· Based on 360-day but not 365 day
· On simple interest basis, but not compounding

Holding period yield (HPY)
The total return received from holding an asset or portfolio of assets. Holding period yield is calculated as the sum of all income and capital growth divided by the value at the beginning of the period being measured.

HPY = (P - F) / P

Problem:
· Difficult to compare with other instruments with different time frame.

Effective annual yield (EAY)
Take account of compound interest and is on 365 day basis.

EAY= (F / P)^{(365 / t)} – 1

Problem:
· Implicit reinvestment assumption of all yield to maturity type measures.

Money market yield (MMY)
Its assumption of 360 day year makes T-bills comparable to yield bearing money market instruments that are applied on 360 day basis. It equals to the annualized HPY.

MMY= [(P - F) / P] x (360 /t)

Problem:
· On simple interest basis, but not compounding

Bond Equivalent Yield (BEY)
So-called Annual Percentage Rate (APR ), converting the yield of a money market instrument, such as a Treasury bill, into the equivalent yield of a Treasury bond in order to compare efficiency ( equivalent annual yield of a bond).

The BEY is the yield that is quoted in newspapers.

BEY = [(F-P)/P] x (360/t)

Problems:
· Use simple interest to annualize


The yields are inter-convertible:

EAY= (1+HPY) ^{(365 /t)} –1
EAY=[1+BEY(t/360)] ^{(365 /t)}
MMY= (HPY)( 360/t)
MMY= 360x(BDY)/ {(360-t)xBDY}

Yield of Zero-coupon bond:
Price zero-coupon bond with N years remaining
P= F / (1 + Y/2)^{(2 x N)}
Yield of zero-coupon bond
Y= {(F / P)^{[1 / (2 x N)]} - 1}x2

Dollar-weighted rate of return vs. Time-weighted rate of return

Dollar-weighted (or money-weighted) rate of return (DWR)
Dollar-weighted rate of return is the internal rate of return (IRR) of all the cash flows into and out of a portfolio, or the discount rate that makes the present value of inflows equal to that of outflows.

For asset portfolio, the cash flows include the purchase and sale of stock as well as dividends received.

DWR = Sum[CF₁/(1+r)^t] = C₀

Notes:

• If new fund adds to an investment portfolio when it is performing poorly, dollar-weighted method will end to be depressed.
• If new fund distributes to portfolio in favorable time, dollar-weighted method will tend to elevate the performance.
  

Time-weighted rate of return of portfolio (TWR)
Measure the compound growth rate of $1 over a specified measurement period. It is also called the "geometric mean return," as the reinvestment is captured by using the geometric total and mean.

TWR = [(P₁-P₀)/P₀] [(P₂-P₁)/P₁]…. [(P_n-P_n-1)/P_n]

Steps to calculate time-weighted rate of return

1. Value the portfolio immediately prior to any significant addition or withdrawal of funds
2. Break the overall evaluation period into equal subperiods based on the dates of inflows & outflows
3. Calculate Holding period return (HPR) for each subperiods
4. Link HPRs by multiplying, to derive the geometric mean
   

Notes:

• TWR is a preferred measurement as it is not be affected by the timing of cash flows and can compare the returns of investment managers.

Holding Period Return (HPR)

Measure the total return received from holding an asset or portfolio of assets, i.e. the sum of all income and capital growth divided by original investment, on a per-dollar-invested basis

HPR = (P1-P₀+D₁)/ P₀

Notes:

• Difficult to compare returns on different investments with different time frames.
• To make comparisons for them, the annualized calculation is used.

Annualized HPR : [(P₁-P₀+D₁)/ P₀]^1/Y - 1

Present Value (PV) vs. Future Value (FV)

Single sum of money
A sum of money in a specific point of time

PV = FV / (1 + r)ⁿ
FV = PV x (1 + r)ⁿ

Where:
r – periodic interest rate
n – no. of compounding periods

Ordinary annuity
A finite set of sequential equal sum of payments at the end of each compounding period.

PV_ordinary = PMT / (1 + r)¹ + PMT / (1 + r)² + PMT / (1 + r)³ + …..+ PMT / (1 + r)ⁿ
= PMT{[1-1/(1+r)ⁿ]/r}

FV_ordinary = PMT(1 + r)^n-1+ PMT (1 + r)^n-2 + PMT (1 + r)^n-3 + …..+ PMT(1 + r)⁰
= PMT{[(1+r)ⁿ-1]/r}

Annuity due
A finite set of sequential equal sum of payments at the beginning of each compounding period.

PV_Due = PMT + PMT / (1 + r)¹ + PMT / (1 + r)² + PMT / (1 + r)³ + …..+ PMT / (1 + r)^n-1
= PMT(1+r){[1-1/(1+r)ⁿ]/r}

FVDue = PMT(1 + r)ⁿ+ PMT (1 + r)^n-1 + PMT (1 + r)^n-2 + …..+ PMT / (1 + r)¹
= PMT(1+r){[(1+r)ⁿ-1]/r}


Ordinary annuity and annuity due conversion

PV_Due = (1+r) PV_ordinary
FV_Due = (1+r) FV_ordinary

Perpetuity
A set of never-ending sequential equal sum of payments.

PV_Perpetuity =PMT/(1+r)¹ + PMT/(1+r)²+ ….. = PMT / r

Continuous compounding
Earning interest on top of interest constantly and continuously.

PV=FVe^-rt
FV=PVe^rt

Where,
r – continuously compounded rate
r = ln(FV/PV)/t.

A series of unequal cash flows
PV = CF1 / (1 + r)¹ + CF2 / (1 + r)² + CF3 / (1 + r)³ + …..+ CFn / (1 + r)ⁿ
FV = CF1 (1 + r)^n-1 + CF2 (1 + r)^{n- 2} + CF3 / (1 + r)^n-3 + …..+ CFn

My tips:

• The cash outflows for mortgage repayment are always at the end of each period and that for education fees for students are at the beginning of each school year unless specified otherwise.
• Be aware that the compounding periods may not necessarily be annual.

PV = FV / (1 + r/m)^mxn

where: m-periods per year; n- years in the future

• Use financial calculator to solve the unknown variable. Be remembered to change the BEG/END mode.

Net present value vs Internal rate of return

Net present value (NPV) and Internal rate of return (IRR) are used to determine whether to accept a project or not.

Net Present Value (NPV)
Net present value is the difference between the present value of cash inflows and the present value of cash outflows. It is used in capital budgeting to analyze the profitability of an investment or project.

NPV= sum[CF_t/(1+r)^t]-C₀
Where:
CF_t– cash flow in the time t
C₀ – initial investment
r – periodic interest rate

NPV rule:

• Accept all independent projects with NPV greater than 0 as they add value to shareholder. In case of mutually exclusive projects, the project with the highest NPV should be chosen.

Advantages:

• Direct measure of the dollar contribution to the stockholders.
• NPV method is preferable for non-normal cash flows (e.g. negative cash flows)

Disadvantage:

• Does NOT measure the project size.

Internal Rate of Return (IRR)
The discount rate makes the net present value of all cash flows from a project equal to zero. The higher a project's internal rate of return, the more desirable it is to undertake the project. IRR can be used to rank several prospective projects a firm is considering.

NPV= Sum[CF_t/(1+r)^t]–C₀

r = internal rate of return (IRR)

IRR rule:

• Accept all independent projects with IRR greater than cost of capital. In case of mutually exclusive projects, the project with the highest IRR should be chosen.

Advantages:

• Show you the return rate on the original money invested.

Disadvantages:

• Give you conflicting answers when compared to NPV for mutually exclusive projects.
• Multiple IRR for non-normal cash flows (when the project operates at a loss or the company needs to contribute more capital)

Conflicts between NPV and IRR Methods
For an independent project that the decision to invest in a project is independent of any other projects, both the NPV and IRR will always give the same result.

For mutually exclusive projects that the decision must be one project or another, NPV and IRR can lead to a conflicting result due:

• Different timing of cash flows
• Different project sizes.
  
  
  
  If the cost of capital greater than the crossover rate, NPV and IRR provide the same result;
  If the cost of capital smaller than crossover rate, conflict exists and use the NPV decision;
  
  Notes:
• NPV assures reinvestment at cost of capital; IRR assures reinvestment at IRR.
  

1.1.1 Interest Rate, Stated annual interest rate (SAR), Effective annual Rate (EAR)

The interest rate is interpreted as the internal rate of return, required rate of return, discount rate or opportunity cost .

Interest Rate = Real risk-free rate + Expected inflation + Risk premium (compensate distinct types of risk.

Stated annual interest rate (SAR)
It does NOT account for the effect of compounding within the year. \

SAR= no. of periods in a year (m) x periodic interest rate (r_p)

Effective annual rate (EAR)
It does take the compounding effect in a year into account
EAR = (1 + r_p)^m – 1

SAR and EAR conversion
SAR = [(1 + EAR)^1/m - 1] x m

Notes:
- Assuming continuous compounding at the stated interest rate, EAR = e^SAR-1

Time Value of Money (TVM)

Time value of money is one of the main principles of financial valuation. This principle is used to support corporate finance decisions and estimate the fair value of fixed income, equity, or any other type of security or investment.

Quantitative Analysis

The fundamentals of the quantitative are essential in almost any type of financial analysis. This session introduces two quantitative analytical tools:

• Time value of money
• statistics and probability theory