## Formulas for Investment Yield & Return

Internal Rate of Return

The Internal Rate of Return (IRR) is probably one of the most common ROI metrics used in corporate finance. The IRR is quite closely related to the net present value (NPV) method, in that, like NPV, it also involves discounting future cash flows. The internal rate of return (IRR) of a particular investment is the discount rate that, when applied to its future cash flows, will produce an NPV of precisely zero.

The formula for calculating IRR is

0 = CF0 + CF1/1+IRR + CF2/(1+IRR)2……CFn/(1+IRR)N

Holding Period

Holding period return is defined as the return earned by an investor over a specific investment period. It is a simple calculation used to compare the rate of return against the target rate of return.

HPR= Income + (End of period value – initial value) / Initial Value

Consider a bill purchased for \$30 and sold for \$33 six months later, in which \$0.50 dividend is received. What is the holding period?

In case we wish to calculate the annualized holding period return, we use the formula

HPR= {Income + (End of period value – initial value)/Initial Value}1/n – 1

Where n = no. of years

Usually, we use the annualized holding period return for calculations. The calculation above does not take the time factor rather takes the money per unit factor into consideration. It is a simple and basic measure to calculate the return on a particular investment.

Money Weighted Returns

The Internal Rate of Return calculated after incorporating all cash inflows and outflows is called money weighted rate of return. As seen from the definition, it takes into account the internal rate of return. The steps to calculate the money weighted rate of return are:

1. Find out all inflows and outflows
2. Net the cash flows and set PV of inflows – PV of outflows
3. Solve R to find the rate of return

An investor buys a share for \$100.(t=0). At t=1, he buys one more share at \$120. At t=2, he sells both shares for \$130 each. At the end of each year in the holding period, he is paid \$2 per share dividend. What is the money weighted rate of return?

T= 0 +100 = +100

T=1 -2+120 = +118

T=2 -4-130*2 = -264

PV inflows = PV outflows

100+118/(1+r) = 264/(1+r)2

Solving further, we get the rate of return is 13.86%

Time Weighted Return

It measures compounded growth. It is a process of averaging values over time. The steps to calculate the time weighted return for an investment are:

1. Value the portfolio foregoing the addition or withdrawals
2. Calculate the HPR for all sub periods
3. Calculate the product of (1+HPR) for each sub period. Use geometric mean if necessary

An investor buys a share for Rs 100.(t=0). At t=1, he buys one more share at Rs 120. At t=2, he sells both shares for Rs 130 each. At the end of each year in the holding period, he is paid Rs 2 per share dividend. What is the time weighted rate of return?

HPR1 = {(120+2)/100}-1= 22%

HPR 2 = {(260+4)/240}-1 = 10%

Thus calculating, we have the time weighted return as 15.84%

The time weighted return is preferred and widely used in the industry because it is not affected by the timing of cash inflows and outflows. In case of this illustration, the money weighted return gave more importance to HPR2.

The money weighted return calculation depends on whether funds contributed to a portfolio are at a favourable time or during a poor portfolio performance. The time weighted return removes this warp. The money weighted return is more suitable only in cases where there is a complete control exercised over the inflows and outflows.

Bank Discount Yield

U.S Treasury Bills (T-Bills) are quoted on a bank discount basis, which is not based on purchase price, but takes into account the face value of the instrument. The bank discount basis annualizes, based on a year (360 days), the discount as a percentage of face value. This method assumes no compounding. The formula for calculating the annualized yield on a bank discount basis is

R = D/F * 360/t

Where:
D = Dollar Discount
F= Face Value
T= remaining no. of days till maturity

Calculate the bank discount yield for a T Bill priced at \$99,000 with a FV of \$100000 and 100 days till maturity.

R = 1000/100000* 360/100 = 3.6%

Please note that the yield quoted here does not signify return earned by an investor.

Holding Period Yield

The total return earned by an investor between the purchase date and sale date is called the holding period yield. The formula to calculate HPY is

HPY = {Income + End Value/Initial Value}- 1

Calculate the holding period yield for a T Bill priced at \$99,000 with a FV of \$100000 and 100 days till maturity.

HPY = 1000/99000 = 1.01%

Since there is no income, the value of Income = 0.

Effective Annual Yield

This yield takes into account the compound interest, is an annualized value and is based on a 365 day year.

The formula to calculate EAY is:

Effective Annual Yield = (1+HPY) 365/t – 1

Calculate the EAY for a T Bill priced at \$99,000 with a FV of \$100000 and 100 days till maturity.

HPY = 1.01%

EAY = (1.0101)365/100 – 1 = 4%

Money Market Yield

The money market yield is equal to the the annualized HPY, assuming a 360 day year. The money market yield is computed on basis of the purchase price. Formula is:

R = HPY*360/t

What is the money market yield for a 100 day tbill that has a bank discount of 3.6%?

R = 360*0.036/(360-(100*0.036)) = 12.96/356.4 = 3.3636%

Converting Among Holding Period Yields, Money Market Yields, and Effective Annual Yields

Once we have established HPY,EAY and R, we can use one for the basis of calculating the other two.

Money Market to HPY = Money Market * No. of days till maturity/360

HPY to EAY = (1+HPY)365/days till maturity – 1

EAY to HPY = (1+EAY) days till maturity/365 -1

Assume you purchase a Tbill that matures in 150 days for a price of \$98,000. The money market yield is 4.89%. Calculate HPY & EAY:

Money Market to HPY = R*150/360 = 2.041%

HPY to EAY = (1+0.0204)365/150 – 1 = 5.039%

Bond Equivalent Yield

2* Semi annual discount Rate = Bond Equivalent Yield.

The formula is such because on US Bonds , the annual coupon interest is paid in 2 semi annual payments. The steps to calculate are:

1. Convert to Semi Annual
2. Double it

A 3 month loan has a HPY of 2%. What is the yield on a bond equivalent basis?

Semi Annual = 1.022 – 1 =4.04%

Bond Equivalent Yield = (2*4.04 = 8.08%)