(Similar to the Introduction in Quantitative Financial Economics by Cuthbertson & Nitzsche)
Synopsis of Various ‘Rates of Return’
In the world of finance there are many different types of assets. When analysing these, in an economic sense, we attempt to characterise them by reducing them to some of their more salient features, i.e. we attempt to model them. Thus, assets may vary in the amount of money required to obtain (invest in) them, the amount of periods these investments must be made over, the rate at which an investment grows from purchase of the asset until the point of redemption (maturity), the life-time of the investment, its liquidity (How quickly can one convert it to cash?), or the risk associated with the return on, and value of, the asset. Some assets appreciate in value (capital appreciation), some pay dividends; the nominal value of some dividends is known with (near) certainty (e.g. government bonds), other assets’ dividends are uncertain even in nominal terms (e.g. shares in firms). In real terms, all returns are uncertain, for we must adjust for the rate of inflation (i.e. one may be able to buy less with the proceeds of the investment), or the exchange rate.We are interested in finding the value of various types of financial assets, given these characteristics of risk, return, lifespan, liquidity…
Usually returns on investments are quoted in ‘simple annual’ terms, i.e. no attention is paid to compound interest (interest on interest). In what follows we will concern ourselves with the future value, FV, of an investment, that may vary in the frequency of interest/dividend payments, remaining life-span, risk characteristics, etc.
If, at some time t, we invest some amount Pt over a given time-period, and reap an amount Pt+1at time t+1, then our absolute nominal gain (over that time period) is the difference between Pt+1 and Pt.
(Absolute gain)t, t+1 = Pt+1 - Pt
We can also ascribe a natural rate of growth, r, to this investment, by setting the absolute growth relative to the initial value of the investment, i.e. by dividing it by the initial value.
Or, equivalently
Thus, if Pt is invested at a rate 1+ rt, we should reap Pt+1 = (1+ rt)Pt. Naturally, if we reinvested this at time t+1 at the prevailing rate of interest (or a project that would offer a growth rate of) rt+1, then we should have
If rt = rt+i = r (In this case, we say the term structure of interest is flat), then i periods from time t we should have
.
Thus, we are compounding our interest payments.By the reverse token, if we know the future value of some investment at time t + i, FVt+i, and we know the growth (interest) rate that existed in the period prior to this realisation, then we can work out the discounted present value, DPV, at time t of FVt+i.
Which is evident by dividing the foregoing equation by (1+r)i.
Assume we invest $x for n years at a certain rate of annual interest, r. If interest is paid out only once per annum (and is immediately re-invested at the same rate of interest), we compound annually.
Thus, the future value after n years is:
If interest is paid out m times per year and is immediately reinvested at the same rate of return as was the principal, the future value after n years with m payments per year is:
where r/m may be thought of as the periodic interest rate.
Aside:
It can be shown that continuous compounding (let m progress toward infinity) will leave us with the following expression:
FVnc = $x ern
[By taking logs] We can see that there exists a simple relationship between the quoted ‘simple annual’ rate, r, with m payments per year and the effective annual rate, rf. By definition, the following relationship must hold:
[1+rf]n = [1+r/m]nm
n*ln[1+rf] = n*m*ln[1+r/m]
1+rf = [1+r/m]m
And,
rfr
since rf takes compounding into account.
Discounted Present Value, Again
Let the annual interest on a ‘perfectly’ safe investment over n periods (years) be rs(n) [N.B.: at this stage rs is not raised to the power of n!!] Thus, rs(n) represents the annual interest that prevails if we borrow for n periods (years). We invest an amount $x and receive a payment in the future:
Thus, by inversion, if one were to receive with certainty in n years time FVn one should be willing to pay:
In the forgoing equation, that amount should be precisely $x.
The DPV of a stream of receipts, FVi (i=1 to n), which are certain, is simply the sum of the discounted present values of each of the payments (that are constituent to this stream).
We have assumed the term structure of interest rates is flat, i.e. they are all the same. Assume at the time of the investment, t=0, we can avail of a series of interest rates rs(i), which denotes the interest rate from t=0 to t=1 for i=1, and the rate from t=0 to t=2 for i = 2.
Thus,
Equivalently, for
[Aside: For risky investments we may adjust in order to take risk into account, i.e. we attribute a risk premium, rp(i) to period i.]
Investing in Projects
If a project has an initial investment cost KC, then an entrepreneur should only invest if the discounted value of the income stream he secures with this investment is at least as large as the capital cost required in order to invest in this scheme.
DPV KC
Or, equivalently if the net present value, NPV, must be non-negative
NPV = DPV – KC 0.
Given KC is constant and DPV falls as r rises (WHY??: discount factor!), we can determine a level of r such that NPV = 0. At this rate of interest, the return on the investment just allows us to pay interest and repayments. Let us denote that level of interest as y, the Internal Rate of Return (IRR).
Yield to Maturity
Zero Coupon Bonds
A zero-coupon, or pure discount, bond is an asset that has a fixed lifespan (time to maturity), at which point it provides a once-off payment, its par value, typically denoted by M. Thus, if M is paid out after one period, the zero has a one-period rate of return (equivalent to above), which is defined by M and the price paid for it, P.
Hence, P is the price paid at t=0 and M is the receipt at maturity, when t=1.
Viewed in terms of DPV,
or
Usually, zeroes (zero coupon bonds) have a short lifespan (3 months, e.g. T-Bills, to a year), and are highly liquid money market instruments. The return on a zero coupon bond can also be called spot rate, which is identical to its internal rate of return.
Coupon Paying Bonds
A level (non-callable) n-period, coupon paying bond pays fixed coupons, C, periodically (often bi-annually) until redemption, t=n, and pays a redemption price at expiration date, M. M is known with (assumed) certainty, and C is defined as a percentage of this par value, M. Hence, C is also known with (assumed) certainty. [N.B.: Even governments can default on debt, or can indirectly do so by allowing their exchange rate to depreciate)] Given the market price n periods from maturity, P(n), we can solve for y by using the following equation.
The yield to maturity is that constant rate of discount, which, at a point in time, equates the DPV of future payments with the current market price; therefore, it is also a form of internal rate of return. Implicitly, we assume the rate of reinvestment is equal to this discount rate. If P(n) falls, y increases; Why?
The running yield of a bond is defined as a single coupon payment relative to the market price [*100], which may not be too useful. However, for a perpetuity, a bond that is not redeemable and pays its coupons for ever, the rate of return is given by exactly that equation.
Note, that the YTM implicitly assumes the reinvestment rate is identical to the YTM itself. To see this, consider a two-period, coupon-paying bond. Its internal rate of return, y, is given by:
This clearly shows that, the future value of the initial outlay Pt at a growth rate of yis equal to a series of payments, the first of which must be reinvested at a rate equal to y.
Aside:
In the BKM book, you will note that the general expression is broken into two parts:
The left part of the sum is the summation of the present value of a T-period annuity. The right part of the summation is the present value of a lump sum payment in T periods time (be careful to distinguish the t, which represents a series of periods, from the T, which represents a specific time-period, i.e. the final period.
How can this be used to simplify calculations in general?
We can calculate the value of a t-period annuity. Consider the value of a perpetuity, which we have shown is approximately equal to . Furthermore, the present discounted value of a perpetuity that begins paying out T-periods from now should be .
Thus, the difference in present value between these investments should be equal to the present value of a T-period annuity:
.
Current Yield
The current yield is simply measured by the ratio of nominal annual coupon income to the price paid for the bond:
.
Thus, it neglects to take explicit account of the horizon to maturity or growth rates.
Realised Compound Yield
The realised compound yield provides us with an ex-post performance measure, i.e. it provides us with a rate of growth for our initial investment, taking explicit account of the actual reinvestment rates that prevailed over the life-span of the bond. It measures by what rate of compounded interest the initial investment would have to have grown in order to provide us with the same amount we obtained from holding the bond and reinvesting the coupons. Assume we held a two period bond, simple annual coupon, with a reinvestment rate of 10% over the second year, trading at par value:
C = 0.1M = 1000r = 0.1 n = 2
yRCY is the realised compound yield. In this case it is equal to the reinvestment rate and the coupon rate, which is also equal to the yield to maturity. It should be noted that if the coupon rates and reinvestment rates differ then the YTM will not equal the realised compound yield.
- If C = r => P = M
Why?
In this case the bond is a competitive investment, since C is fair compensation for the time-value of money.
Each coupon is r of M, and each coupon is reinvested at r. Hence, no further compensation should be necessary.
What if Cr? What is P to M. Explain this.
Appreciation/Depreciation of P vis-à-vis M will compensate (in economic terms) for differences in r and C.
Thus, the price will change over time in order to do this.
Non-Callable Zero Coupon Bond’s will appreciate in P over time until P = M at the point of maturity. This adjustment will increase over time, as it is exponential.
To see this, simply work out the present value of a Zero some periods from maturity (e.g. 10 periods) when interest rates are assumed to be known.
The One-Period Holding Period Return
The one-period holding period return, Ht+1, is defined as
,
where the first element in the sum captures the capital gain (loss) on the asset, while the latter captures the proportionate dividend yield. This implies,
.
Thus, ex post, for an n-period horizon, with initial investment A, the total value accrued (given all proceeds can be reinvested at a rate that is identical to the performance of the asset itself) should equal:
=>
For an n-period, coupon paying bond:
,
where is the market price at time t for a bond with a remaining lifespan of n-periods. This is referred to as the holding period yield, or HPY, which is useful for the evaluation of stocks and bonds.
YTM vs HPR
If the YTM stays unchanged then the HPR and the YTM are identical.
If yields fluctuate, so will the rate of return via unexpected changes in the interest rate.
If YTM ↑ => P↓
HPR < YTMinitial
YTM = f(Pt, M, C),
which are all known at time t (assuming no default).
HPR = g(Pt, Pt+1, C),
Which are not all observable at time t.
YTM can be used now, whereas HPR can at best be estimated, since future prices are a function of unexpected changes in the interest rate.
What about default?
Risk premia compensate for credit risk. [note: ECB statement on country debt.]
Stocks
Stocks are financial assets that have certain things in common with bonds, and certain distinguishing features. Essentially, stocks do not provide periodic payments (dividends) of a fixed, nominal size. In contrast, most (government) bonds do provide certain, fixed coupon payments, at least with close to certainty in well-developed economies. Their lifespan is not limited, unlike bonds, which have a specified period of time until redemption.
Hence, stocks provide the holder with the right to an uncertain income stream. Uncertainty implies variation in return, or risk. (As we shall see, statistical methods are often employed to evaluate/describe uncertain returns by employing expectations of return to measure expected proceeds and the variance of returns in order to capture the risk (variation or uncertainty) associated with these expectations.
Since stocks are inherently risky, their returns should not be discounted in the same fashion as are the certain returns on (near) risk-less assets, such as (government) bonds.
It is difficult to find a pertinent measure for the return on these assets. However, if there exists at time t an expectation of the one period HPY for t+1, qt = EtHt+1, then the fundamental value of that stock may be viewed as the DPV of the expected future dividends EtDt+j, deflated by discount factors (risk premia). The, the fundamental value at time t is
,
where qiis the one-period return between time period t+i-1 and t+i.
With perfect information and rational agents there should be no profitable systematic opportunities for profit making, i.e. Pt = Vt. If not, profits are possible by arbitrage.