Role of interest rates in security valuation
Introduction
We have stated before that the market prices of securities and their fair value prices (FVP) can be poles apart (see Figure 14). This is where behavioral finance makes its appearance. We will not delve into this fascinating branch of Finance here. However, we would like to state the obvious: the strength of the herd instinct has a major impact on price discovery in the short term, and sometimes longer.
Interest rates play a central role in security valuation. All major assets (debt, shares and property) have a cash flow in the future. The future cash flows on these assets are all future values (FVs) and the valuation of these essentially amounts to the discounting (at an appropriate rate) the FVs to present value (PV = FVP).
We present examples of the calculation of the PVs of shares and bonds, but before this we present a simple illustration of the principle (see Figure 15). In this figure we assume we have an asset which has 12-months to maturity and a future value of LCC 110 000 (= the amount it will mature at). If this asset is valued at the current interest rate of 5% pa, its PV is LCC 104 761.90 (the FV is discounted at its current rate).
Figure 14: market price (MP) versus fair value price (FVP)
Figure 15: valuation of interest rate security (FV to PV): one period
Note the red dotted lines: if the discount rate is increased, the PV falls. Thus, generally in financial markets, when interest rates increase, the values of income-producing assets fall. The principle is the time value of money - the PV / FV concept.
Bonds
Bonds are long-term securities and typically35 pay interest in arrears six-monthly at a fixed rate called the coupon. This means that there are a number of cash flows in the future: the coupon payments and the principal amount that is payable at maturity. These are FV amounts that have to be discounted to PV. What rate does one use? It is the yield to maturity (ytm). It is a measure of the rate of return on a bond that has a number of coupons paid over a number of years and a face value payable at maturity. It may be seen as an average return over the life of a bond. Its reciprocal price may be described as the price that buyers are prepared to pay now (present value LCC) for a stream of regular payments and a lump sum at maturity.
Formally described, the ytm is the rate that equates the price of a bond with the present value of all the coupon payments and the present value of the principal amount (i.e. nominal / face value). Another way of stating this is: the price is merely the discounted value of the income streams (i.e. the coupon payments and redemption proceeds), discounted at the current market rate (ytm).
A basic example (where interest is payable once pa) may make this clear:
Settlement date: 30 / 9 / 2012 Maturity date: 30 / 9 / 2015
Coupon rate: 9% pa
Face value: LCC 1 000 000
Interest date: 30 / 9
ytm 8% pa.
Table 1 shows the cash flows that occur in this example; they are discounted using the now familiar PV-FV formula shown earlier, except that we now introduce compound interest [PV = FV / (1 + ytm / cp)y.cp] (y = years; cp = coupon payments per annum = 1 in this example):
Date |
Coupon payment |
Nominal / face value |
Compounding periods |
FV / (1 + ytm/cp)y.cp |
30/9/2013 |
LCC 90 000 |
- |
1 |
LCC 83 333.33 |
30/9/2014 |
LCC 90 000 |
- |
2 |
LCC 77 160.49 |
30/9/2015 |
LCC 90 000 |
- |
3 |
LCC 71 444.90 |
30/9/2015 |
- |
LCC 1 000 000 |
3 |
LCC 793 832.24 |
Total |
LCC 270 000 |
LCC 1 000 000 |
- |
LCC 1 025 770.96 |
Table 1: Bond valuation example
It will be evident that the value of the bond (PV) is LCC 1 025 770.96 (see also Figure 16 = price per LCC 1.0), and that the price of the bond is 1.02577096 or 102.577096%. The above may be written as the following formula for bonds (because there is only one coupon payment pa):
where:
cr = coupon rate pa ytm = yield to maturity.
Figure 16: valuation of interest rate security (Fv to Pv): multiple periods: fixed-rate bonds
Using the same numbers as above, i.e. coupon rate 9% pa and ytm 8% pa:
Price = (0.09 / 1.08) + (0.09 / 1.166400) + (0.09 / 1.259712) + (1 / 1.259712) = 0.08333333 + 0.07716049 + 0.0714449 + 0.79383224
= 1.02577096
= LCC 102.577096%.
It will be apparent that the coupon rate (0.09) for the periods and the face value that is paid at maturity (all FVs) are discounted at the ytm to PV.
Shares
In the share market the income on shares is not interest but dividends, and dividends grow, in many cases at a constant growth rate. The formula developed to account for this is the Gordon CGDDM (constant growth dividend discount model).
The PV of a share that has a past dividend of D0 and expected dividend growth rate (gr) is:
Because shares do not have a finite life, this translates to:
The interesting part of this formula is the rrr, which stands for required rate of return. This is a concept borrowed from the capital asset pricing model (CAPM).
According to the CAPM the rrr is equal to the rfr plus a multiple of the market risk premium as represented by the share's beta coefficient:
where
(3 = beta36
mr = market rate of return, i.e. the return observed over the period chosen
mr - rfr = the risk premium.
For example, if the rfr = 10%, the (3 = 1.7, the mr = 15%, then:
If, for example, the rrr required is 25%, the dividend now is LCC 10 (i.e. last dividend), and the gr is 5%, the fair market price (PV) is:
The investor will be prepared to pay no more than LCC 52.50 for the share. This is the PV (= FVP). 6.8.4 Derivatives
The interest rate (particularly the rfr) plays a major role in the valuation of derivative instruments, particularly forwards, futures and options.
With forwards and futures the relevance of PV-FV is the most obvious. The FVP of forwards and futures is determined as follows:
where
SP = spot price (i.e. the PV)
ir = interest rate for the forward period
t = term of the forward period.
Money market derivatives, such as repos and FRAs, are also forwards. With caps and floors the PV-FV concept is also applicable (even though they are option-like instruments).
The formula for valuing options is more elaborate (the Black-Scholes and Binomial valuation models are used), but a major input is the rfr (for the term of the option).