Tutorial: Financial Math & Instrument Valuation

What is an Interest Rate? Cashflow? Present Value? Porfolio Internal Rate of Return?

Interest Rates

In a certain world, the interest rate is called the risk-free rate. For investors preferring current to future consumption, the risk-free interest rate is the rate of compensation they require to postpone current consumption. ie. T-bills rate.

In an uncertain world, risks muddy interest rates. Common risks include

  • Inflation risk: when prices are expected to increase, leaders charge not only an opportunity cost of postponing consumption but also an inflation premium that takes into account the expected increase in prices.
  • Default risk: companies exhibit varying degrees of uncertainty concerning their ability to repay lenders. Lenders therefore charge interest rate to incorporate default risk.

3 ways to interpret interest rates

  • Required rate of return: this is the return required by investors or lenders to postpone their current consumption.
  • Discount rate: this is the rate used to discount the future cash flows to allow for the time value of money (that is, to bring future value equivalent to present value).
  • Opportunity cost: this is the most valuable alternative investors give up by choosing what they could do with the money.

Effective annual rate (EAR): it is the annual rate of interest that takes full account of compounding within the year. For example, a $1 investment earning 8.1% compounded semiannually actually earns an EAR of (1 + 0.081/2)^2-1 (with the annual interest rate is 8.1%).



A single cash flow: FV = PV * (1+r/m)^(m*N), FV = PV * e^(r*N) for infinite compounding.

A series of equal cash flows:

  • Annuity: a finite set of sequential cash flows, all with the same value.
  • Ordinary annuity: the payments occur at the end of each period.
    • FV = C * ( ((1+r)^N-1) / r )
    • PV = C * ( (1-1/(1+r)^N) / r)
    • Perpetuity: an ordinary annuity that extends indefinitely: PV = A/r
  • Annuity due: the payments occur at the beginning of each period. Multiply {PV, FV} vaues for ordinary annuity (see above) with (1+r)

A series of uneven cash flows

  • PV: calculate the present value of each individual cash flow and then sum the respective present values.
  • FV: PV(from above) * (1+r/m)^(m*N)


Net Present Value: PV of cash inflows - PV of cash outflows. The higher, the better.

Internal Rate of Return: discount rate which makes NPV = 0. The higher, the better. Two downsides with IRR as a measure

  • Reinvestment: calculation of the IRR assumes that all project cash flows can be reinvested to earn a rate of return exactly equal to the IRR itself. (If reinvested at a higher rate, the realized return will be greater than the IRR.)
  • May contradict NPV in certain sitiuations. NPV should rule.
    • Scale: two different projects of greatly differing scale - one that requires a relatively small investment and returns relatively small cash flows, compared to another that requires a much larger investment and returns much larger cash flows. NPV should rule.
    • Timing: two different projects whose cash flows are timed very differently - one that receives its largest cash flows early in the project versus another that receives its largest cash flows late in the project.

Porfolio IRR

  • The dollar-weighted rate of return: takes into account the timing and amount of cash flows. If funds are added to a portfolio when the portfolio is performing well (poorly), the dollar-weighted rate of return will be inflated (depressed).
  • The time-weighted rate of return: measures the compound growth rate of $1 initial investment over the measurement period. aka geometric mean return - returns are averaged over time, thus not affected by the timing of cash flows. Assume all cash distributions are reinvested.

Yield measures

Bank discount yield: for U.S. T-Bill only

  • R[bd] = (Par-Price)/Par x 360/t. Downsides as a measure of the return on investment
    • based on the face value, not on the purchase price (ie. cost)
    • annualized using a 360-day year, not a 365-day year
    • ignores compunding (ie. the effect of interest on interest)
  • Holding Period Yield: HPY = (Par-Price+Interest)/Price
  • Effective Annual Yield: EAY = (1+HPY) ^ (365/t) - 1
  • Money Market Yield (aka CD equivalent yield): annualized HPY on the basis of a 360-day year and uses simple interest
    • vs R[bd]: (360 x R[bd]) / (360 - t x R[bd])
    • vs HPY: MM = HPY x 360/t
    • vs EAY: MM = [(1 + EAY)^(t/365) - 1] x (360/t)

Bond-Equivalent Yield (BEY)

Convention for restating semi-annual, quarterly, or monthly discount-bond or note yields into an annual yield. For example, if the yield of a semiannual-pay bond is 4%, its BEY is 8%. BEY allows securities whose payments are not annual to be compared with securities with annual yields.

BEY ignores the effect of compounding semiannual YTM. To take compounding effect into account (using semiannual and annual pay bonds as examples),

  • Convert a semiannual-pay bond's BEY to an annual-pay bond: [1+Semiannual BEY/2] ^ 2 - 1
  • Convert the equivalent annual yield of an annual-pay bond to a BEY: 2 x [ (1+YTM on the annual-pay bond)^0.5 - 1]

Category: C++ Quant > Tutorials

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