# How do investors know what the discount rate is?

Let's say that Jim buys a bond whose face value is 100\$ and has a 4.75% coupon, with coupon payments given annually, that matures in 5 years. Jim buys this bond today for 103.05\$.

Letting x represent the discount rate, I know that

``````4.75 * (1 - (1+x)^-5) / x + 100 * (1+x)^-5 = 103.05
``````

and (assuming Jim made a good trade) Wolfram Alpha tells me that the discount rate is at most .04064.

My question is: how did Jim know that the discount rate was at most this amount?

Edit: In response to Max's answer, I'm not looking for the calculations, instead, I'm more looking for the reasoning that Jim uses to say that "I should value this bond as if the discount rate was 4.064% ." Is he able to look this number up somewhere?

Just to make sure there's no confusion - discount rate here is not the discount is used to set the price of zero bond (e.g. price is 80, face value is 100, i.e. disount is 20% here), but the interest rate that turns the present value of the cash flow to 0.

You have the cash flows

``````t |   CF(t)
--+--------
0 | -103.50
1 |    4.75
2 |    4.75
3 |    4.75
4 |    4.75
5 |  104.75
``````

(a better representation of this cash flow here).

With the formula Σ [ t * CF(t) * (1+r)^(-t) ] = 0 you get r = 4.064%

Edit. Response to question's edit. If you want to start with discount rate and end up with the price, then you need to look up the credit default swap (CDS) spread for the bond issuer (market data). If CDS are not traded on this bond's issuer you need to look up his credit rating and corresponding credit spread. Discount rate would be the CDS spread + risk-free interest rate.

What you are calling "discount rate" is really the yield to maturity. The term discount rate actually has a specific meaning:

The discount rate is the interest rate charged to commercial banks and other depository institutions on loans they receive from their regional Federal Reserve Bank's lending facility--the discount window

The yield to maturity is a given. The Face value of the bond along with the coupon payments are fixed for a given bond. The YTM and current price of the bond vary inversely. The YTM will be different for government paper vs corporate bonds. It will vary from federal government to state, and by country to country. The YTM for each company gives you a hint as to how investors feel about that company.